solving problems scientific methodology

Solving Everyday Problems with the Scientific Method: Thinking Like a Scientist (Second Edition)

This book describes how one can use The Scientific Method to solve everyday problems including medical ailments, health issues, money management, traveling, shopping, cooking, household chores, etc. It illustrates how to exploit the information collected from our five senses, how to solve problems when no information is available for the present problem situation, how to increase our chances of success by redefining a problem, and how to extrapolate our capabilities by seeing a relationship among heretofore unrelated concepts. One should formulate a hypothesis as early as possible in order to have a sense of direction regarding which path to follow. Occasionally, by making wild conjectures, creative solutions can transpire. However, hypotheses need to be well-tested. Through this way, The Scientific Method can help readers solve problems in both familiar and unfamiliar situations. Containing real-life examples of how various problems are solved — for instance, how some observant patients cure their own illnesses when medical experts have failed — this book will train readers to observe what others may have missed and conceive what others may not have contemplated. With practice, they will be able to solve more problems than they could previously imagine. In this second edition, the authors have added some more theories which they hope can help in solving everyday problems. At the same time, they have updated the book by including quite a few examples which they think are interesting. Readership: General public interested in self-help books; undergraduates majoring in education and behavioral psychology; graduates and researchers with research interests in problem solving, creativity and scientific research methodology.

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What Is the Scientific Method?

The scientific method is a systematic way of conducting experiments or studies so that you can explore the things you observe in the world and answer questions about them. The scientific method, also known as the hypothetico-deductive method, is a series of steps that can help you accurately describe the things you observe or improve your understanding of them.

Ultimately, your goal when you use the scientific method is to:

• Find a cause-and-effect relationship by asking a question about something you observed
• Collect as much evidence as you can about what you observed, as this can help you explore the connection between your evidence and what you observed
• Determine if all your evidence can be combined to answer your question in a way that makes sense

Francis Bacon and René Descartes are usually credited with formalizing the process in the 16th and 17th centuries. The two philosophers argued that research shouldn’t be guided by preset metaphysical ideas of how reality works. They supported the use of inductive reasoning to come up with hypotheses and understand new things about reality.

Scientific Method Steps

The scientific method is a step-by-step problem-solving process. These steps include:

Observe the world around you. This will help you come up with a topic you are interested in and want to learn more about. In many cases, you already have a topic in mind because you have a related question for which you couldn't find an immediate answer.

Either way, you'll start the process by finding out what people before you already know about the topic, as well as any questions that people are still asking about. You may need to look up and read books and articles from academic journals or talk to other people so that you understand as much as you possibly can about your topic. This will help you with your next step.

Ask questions. Asking questions about what you observed and learned from reading and talking to others can help you figure out what the "problem" is. Scientists try to ask questions that are both interesting and specific and can be answered with the help of a fairly easy experiment or series of experiments. Your question should have one part (called a variable) that you can change in your experiment and another variable that you can measure. Your goal is to design an experiment that is a "fair test," which is when all the conditions in the experiment are kept the same except for the one you change (called the experimental or independent variable).

Form a hypothesis and make predictions based on it.  A hypothesis is an educated guess about the relationship between two or more variables in your question. A good hypothesis lets you predict what will happen when you test it in an experiment. Another important feature of a good hypothesis is that, if the hypothesis is wrong, you should be able to show that it's wrong. This is called falsifiability. If your experiment shows that your prediction is true, then your hypothesis is supported by your data.

Test your prediction by doing an experiment or making more observations.  The way you test your prediction depends on what you are studying. The best support comes from an experiment, but in some cases, it's too hard or impossible to change the variables in an experiment. Sometimes, you may need to do descriptive research where you gather more observations instead of doing an experiment. You will carefully gather notes and measurements during your experiments or studies, and you can share them with other people interested in the same question as you. Ideally, you will also repeat your experiment a couple more times because it's possible to get a result by chance, but it's less possible to get the same result more than once by chance.

Draw a conclusion. You will analyze what you already know about your topic from your literature research and the data gathered during your experiment. This will help you decide if the conclusion you draw from your data supports or contradicts your hypothesis. If your results contradict your hypothesis, you can use this observation to form a new hypothesis and make a new prediction. This is why scientific research is ongoing and scientific knowledge is changing all the time. It's very common for scientists to get results that don't support their hypotheses. In fact, you sometimes learn more about the world when your experiments don't support your hypotheses because it leads you to ask more questions. And this time around, you already know that one possible explanation is likely wrong.

Use your results to guide your next steps (iterate). For instance, if your hypothesis is supported, you may do more experiments to confirm it. Or you could come up with a hypothesis about why it works this way and design an experiment to test that. If your hypothesis is not supported, you can come up with another hypothesis and do experiments to test it. You'll rarely get the right hypothesis in one go. Most of the time, you'll have to go back to the hypothesis stage and try again. Every attempt offers you important information that helps you improve your next round of questions, hypotheses, and predictions.

Share your results. Scientific research isn't something you can do on your own; you must work with other people to do it.   You may be able to do an experiment or a series of experiments on your own, but you can't come up with all the ideas or do all the experiments by yourself .

Scientists and researchers usually share information by publishing it in a scientific journal or by presenting it to their colleagues during meetings and scientific conferences. These journals are read and the conferences are attended by other researchers who are interested in the same questions. If there's anything wrong with your hypothesis, prediction, experiment design, or conclusion, other researchers will likely find it and point it out to you.

It can be scary, but it's a critical part of doing scientific research. You must let your research be examined by other researchers who are as interested and knowledgeable about your question as you. This process helps other researchers by pointing out hypotheses that have been proved wrong and why they are wrong. It helps you by identifying flaws in your thinking or experiment design. And if you don't share what you've learned and let other people ask questions about it, it's not helpful to your or anyone else's understanding of what happens in the world.

Scientific Method Example

Here's an everyday example of how you can apply the scientific method to understand more about your world so you can solve your problems in a helpful way.

Let's say you put slices of bread in your toaster and press the button, but nothing happens. Your toaster isn't working, but you can't afford to buy a new one right now. You might be able to rescue it from the trash can if you can figure out what's wrong with it. So, let's figure out what's wrong with your toaster.

Observation. Your toaster isn't working to toast your bread.

Ask a question. In this case, you're asking, "Why isn't my toaster working?" You could even do a bit of preliminary research by looking in the owner's manual for your toaster. The manufacturer has likely tested your toaster model under many conditions, and they may have some ideas for where to start with your hypothesis.

Form a hypothesis and make predictions based on it. Your hypothesis should be a potential explanation or answer to the question that you can test to see if it's correct. One possible explanation that we could test is that the power outlet is broken. Our prediction is that if the outlet is broken, then plugging it into a different outlet should make the toaster work again.

Test your prediction by doing an experiment or making more observations. You plug the toaster into a different outlet and try to toast your bread.

If that works, then your hypothesis is supported by your experimental data. Results that support your hypothesis don't prove it right; they simply suggest that it's a likely explanation. This uncertainty arises because, in the real world, we can't rule out the possibility of mistakes, wrong assumptions, or weird coincidences affecting the results. If the toaster doesn’t work even after plugging it into a different outlet, then your hypothesis is not supported and it's likely the wrong explanation.

Use your results to guide your next steps (iteration). If your toaster worked, you may decide to do further tests to confirm it or revise it. For example, you could plug something else that you know is working into the first outlet to see if that stops working too. That would be further confirmation that your hypothesis is correct.

If your toaster failed to toast when plugged into the second outlet, you need a new hypothesis. For example, your next hypothesis might be that the toaster has a shorted wire. You could test this hypothesis directly if you have the right equipment and training, or you could take it to a repair shop where they could test that hypothesis for you.

Share your results. For this everyday example, you probably wouldn't want to write a paper, but you could share your problem-solving efforts with your housemates or anyone you hire to repair your outlet or help you test if the toaster has a short circuit.

What the Scientific Method Is Used For

The scientific method is useful whenever you need to reason logically about your questions and gather evidence to support your problem-solving efforts. So, you can use it in everyday life to answer many of your questions; however, when most people think of the scientific method, they likely think of using it to:

Describe how nature works . It can be hard to accurately describe how nature works because it's almost impossible to account for every variable that's involved in a natural process. Researchers may not even know about many of the variables that are involved. In some cases, all you can do is make assumptions. But you can use the scientific method to logically disprove wrong assumptions by identifying flaws in the reasoning.

Do scientific research in a laboratory to develop things such as new medicines.

Develop critical thinking skills.  Using the scientific method may help you develop critical thinking in your daily life because you learn to systematically ask questions and gather evidence to find answers. Without logical reasoning, you might be more likely to have a distorted perspective or bias. Bias is the inclination we all have to favor one perspective (usually our own) over another.

The scientific method doesn't perfectly solve the problem of bias, but it does make it harder for an entire field to be biased in the same direction. That's because it's unlikely that all the people working in a field have the same biases. It also helps make the biases of individuals more obvious because if you repeatedly misinterpret information in the same way in multiple experiments or over a period, the other people working on the same question will notice. If you don't correct your bias when others point it out to you, you'll lose your credibility. Other people might then stop believing what you have to say.

Why Is the Scientific Method Important?

When you use the scientific method, your goal is to do research in a fair, unbiased, and repeatable way. The scientific method helps meet these goals because:

It's a systematic approach to problem-solving. It can help you figure out where you're going wrong in your thinking and research if you're not getting helpful answers to your questions. Helpful answers solve problems and keep you moving forward. So, a systematic approach helps you improve your problem-solving abilities if you get stuck.

It can help you solve your problems.  The scientific method helps you isolate problems by focusing on what's important. In addition, it can help you make your solutions better every time you go through the process.

It helps you eliminate (or become aware of) your personal biases.  It can help you limit the influence of your own personal, preconceived notions . A big part of the process is considering what other people already know and think about your question. It also involves sharing what you've learned and letting other people ask about your methods and conclusions. At the end of the process, even if you still think your answer is best, you have considered what other people know and think about the question.

The scientific method is a systematic way of conducting experiments or studies so that you can explore the world around you and answer questions using reason and evidence. It's a step-by-step problem-solving process that involves: (1) observation, (2) asking questions, (3) forming hypotheses and making predictions, (4) testing your hypotheses through experiments or more observations, (5) using what you learned through experiment or observation to guide further investigation, and (6) sharing your results.

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What Are The Steps Of The Scientific Method?

Julia Simkus

Editor at Simply Psychology

BA (Hons) Psychology, Princeton University

Julia Simkus is a graduate of Princeton University with a Bachelor of Arts in Psychology. She is currently studying for a Master's Degree in Counseling for Mental Health and Wellness in September 2023. Julia's research has been published in peer reviewed journals.

Learn about our Editorial Process

Saul McLeod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

Science is not just knowledge. It is also a method for obtaining knowledge. Scientific understanding is organized into theories.

The scientific method is a step-by-step process used by researchers and scientists to determine if there is a relationship between two or more variables. Psychologists use this method to conduct psychological research, gather data, process information, and describe behaviors.

It involves careful observation, asking questions, formulating hypotheses, experimental testing, and refining hypotheses based on experimental findings.

How it is Used

The scientific method can be applied broadly in science across many different fields, such as chemistry, physics, geology, and psychology. In a typical application of this process, a researcher will develop a hypothesis, test this hypothesis, and then modify the hypothesis based on the outcomes of the experiment.

The process is then repeated with the modified hypothesis until the results align with the observed phenomena. Detailed steps of the scientific method are described below.

Keep in mind that the scientific method does not have to follow this fixed sequence of steps; rather, these steps represent a set of general principles or guidelines.

7 Steps of the Scientific Method

Psychology uses an empirical approach.

Empiricism (founded by John Locke) states that the only source of knowledge comes through our senses – e.g., sight, hearing, touch, etc.

Empirical evidence does not rely on argument or belief. Thus, empiricism is the view that all knowledge is based on or may come from direct observation and experience.

The empiricist approach of gaining knowledge through experience quickly became the scientific approach and greatly influenced the development of physics and chemistry in the 17th and 18th centuries.

Step 1: Make an Observation (Theory Construction)

Every researcher starts at the very beginning. Before diving in and exploring something, one must first determine what they will study – it seems simple enough!

By making observations, researchers can establish an area of interest. Once this topic of study has been chosen, a researcher should review existing literature to gain insight into what has already been tested and determine what questions remain unanswered.

This assessment will provide helpful information about what has already been comprehended about the specific topic and what questions remain, and if one can go and answer them.

Specifically, a literature review might implicate examining a substantial amount of documented material from academic journals to books dating back decades. The most appropriate information gathered by the researcher will be shown in the introduction section or abstract of the published study results.

The background material and knowledge will help the researcher with the first significant step in conducting a psychology study, which is formulating a research question.

This is the inductive phase of the scientific process. Observations yield information that is used to formulate theories as explanations. A theory is a well-developed set of ideas that propose an explanation for observed phenomena.

Inductive reasoning moves from specific premises to a general conclusion. It starts with observations of phenomena in the natural world and derives a general law.

Step 2: Ask a Question

Once a researcher has made observations and conducted background research, the next step is to ask a scientific question. A scientific question must be defined, testable, and measurable.

A useful approach to develop a scientific question is: “What is the effect of…?” or “How does X affect Y?”

To answer an experimental question, a researcher must identify two variables: the independent and dependent variables.

The independent variable is the variable manipulated (the cause), and the dependent variable is the variable being measured (the effect).

An example of a research question could be, “Is handwriting or typing more effective for retaining information?” Answering the research question and proposing a relationship between the two variables is discussed in the next step.

Step 3: Form a Hypothesis (Make Predictions)

A hypothesis is an educated guess about the relationship between two or more variables. A hypothesis is an attempt to answer your research question based on prior observation and background research. Theories tend to be too complex to be tested all at once; instead, researchers create hypotheses to test specific aspects of a theory.

For example, a researcher might ask about the connection between sleep and educational performance. Do students who get less sleep perform worse on tests at school?

It is crucial to think about different questions one might have about a particular topic to formulate a reasonable hypothesis. It would help if one also considered how one could investigate the causalities.

It is important that the hypothesis is both testable against reality and falsifiable. This means that it can be tested through an experiment and can be proven wrong.

The falsification principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory to be considered scientific, it must be able to be tested and conceivably proven false.

To test a hypothesis, we first assume that there is no difference between the populations from which the samples were taken. This is known as the null hypothesis and predicts that the independent variable will not influence the dependent variable.

Examples of “if…then…” Hypotheses:

• If one gets less than 6 hours of sleep, then one will do worse on tests than if one obtains more rest.
• If one drinks lots of water before going to bed, one will have to use the bathroom often at night.
• If one practices exercising and lighting weights, then one’s body will begin to build muscle.

The research hypothesis is often called the alternative hypothesis and predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and that they are significant in terms of supporting the theory being investigated.

Although one could state and write a scientific hypothesis in many ways, hypotheses are usually built like “if…then…” statements.

Step 4: Run an Experiment (Gather Data)

The next step in the scientific method is to test your hypothesis and collect data. A researcher will design an experiment to test the hypothesis and gather data that will either support or refute the hypothesis.

The exact research methods used to examine a hypothesis depend on what is being studied. A psychologist might utilize two primary forms of research, experimental research, and descriptive research.

The scientific method is objective in that researchers do not let preconceived ideas or biases influence the collection of data and is systematic in that experiments are conducted in a logical way.

Experimental Research

Experimental research is used to investigate cause-and-effect associations between two or more variables. This type of research systematically controls an independent variable and measures its effect on a specified dependent variable.

Experimental research involves manipulating an independent variable and measuring the effect(s) on the dependent variable. Repeating the experiment multiple times is important to confirm that your results are accurate and consistent.

One of the significant advantages of this method is that it permits researchers to determine if changes in one variable cause shifts in each other.

While experiments in psychology typically have many moving parts (and can be relatively complex), an easy investigation is rather fundamental. Still, it does allow researchers to specify cause-and-effect associations between variables.

Most simple experiments use a control group, which involves those who do not receive the treatment, and an experimental group, which involves those who do receive the treatment.

An example of experimental research would be when a pharmaceutical company wants to test a new drug. They give one group a placebo (control group) and the other the actual pill (experimental group).

Descriptive Research

Descriptive research is generally used when it is challenging or even impossible to control the variables in question. Examples of descriptive analysis include naturalistic observation, case studies , and correlation studies .

One example of descriptive research includes phone surveys that marketers often use. While they typically do not allow researchers to identify cause and effect, correlational studies are quite common in psychology research. They make it possible to spot associations between distinct variables and measure the solidity of those relationships.

Step 5: Analyze the Data and Draw Conclusions

Once a researcher has designed and done the investigation and collected sufficient data, it is time to inspect this gathered information and judge what has been found. Researchers can summarize the data, interpret the results, and draw conclusions based on this evidence using analyses and statistics.

Upon completion of the experiment, you can collect your measurements and analyze the data using statistics. Based on the outcomes, you will either reject or confirm your hypothesis.

Analyze the Data

So, how does a researcher determine what the results of their study mean? Statistical analysis can either support or refute a researcher’s hypothesis and can also be used to determine if the conclusions are statistically significant.

When outcomes are said to be “statistically significant,” it is improbable that these results are due to luck or chance. Based on these observations, investigators must then determine what the results mean.

An experiment will support a hypothesis in some circumstances, but sometimes it fails to be truthful in other cases.

What occurs if the developments of a psychology investigation do not endorse the researcher’s hypothesis? It does mean that the study was worthless. Simply because the findings fail to defend the researcher’s hypothesis does not mean that the examination is not helpful or instructive.

This kind of research plays a vital role in supporting scientists in developing unexplored questions and hypotheses to investigate in the future. After decisions have been made, the next step is to communicate the results with the rest of the scientific community.

This is an integral part of the process because it contributes to the general knowledge base and can assist other scientists in finding new research routes to explore.

If the hypothesis is not supported, a researcher should acknowledge the experiment’s results, formulate a new hypothesis, and develop a new experiment.

We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist that could refute a theory.

Draw Conclusions and Interpret the Data

When the empirical observations disagree with the hypothesis, a number of possibilities must be considered. It might be that the theory is incorrect, in which case it needs altering, so it fully explains the data.

Alternatively, it might be that the hypothesis was poorly derived from the original theory, in which case the scientists were expecting the wrong thing to happen.

It might also be that the research was poorly conducted, or used an inappropriate method, or there were factors in play that the researchers did not consider. This will begin the process of the scientific method again.

If the hypothesis is supported, the researcher can find more evidence to support their hypothesis or look for counter-evidence to strengthen their hypothesis further.

In either scenario, the researcher should share their results with the greater scientific community.

Step 6: Share Your Results

One of the final stages of the research cycle involves the publication of the research. Once the report is written, the researcher(s) may submit the work for publication in an appropriate journal.

Usually, this is done by writing up a study description and publishing the article in a professional or academic journal. The studies and conclusions of psychological work can be seen in peer-reviewed journals such as  Developmental Psychology , Psychological Bulletin, the  Journal of Social Psychology, and numerous others.

Scientists should report their findings by writing up a description of their study and any subsequent findings. This enables other researchers to build upon the present research or replicate the results.

As outlined by the American Psychological Association (APA), there is a typical structure of a journal article that follows a specified format. In these articles, researchers:

• Supply a brief narrative and background on previous research
• Give their hypothesis
• Specify who participated in the study and how they were chosen
• Provide operational definitions for each variable
• Explain the measures and methods used to collect data
• Describe how the data collected was interpreted
• Discuss what the outcomes mean

A detailed record of psychological studies and all scientific studies is vital to clearly explain the steps and procedures used throughout the study. So that other researchers can try this experiment too and replicate the results.

The editorial process utilized by academic and professional journals guarantees that each submitted article undergoes a thorough peer review to help assure that the study is scientifically sound. Once published, the investigation becomes another piece of the current puzzle of our knowledge “base” on that subject.

This last step is important because all results, whether they supported or did not support the hypothesis, can contribute to the scientific community. Publication of empirical observations leads to more ideas that are tested against the real world, and so on. In this sense, the scientific process is circular.

The editorial process utilized by academic and professional journals guarantees that each submitted article undergoes a thorough peer review to help assure that the study is scientifically sound.

Once published, the investigation becomes another piece of the current puzzle of our knowledge “base” on that subject.

By replicating studies, psychologists can reduce errors, validate theories, and gain a stronger understanding of a particular topic.

Step 7: Repeat the Scientific Method (Iteration)

Now, if one’s hypothesis turns out to be accurate, find more evidence or find counter-evidence. If one’s hypothesis is false, create a new hypothesis or try again.

One may wish to revise their first hypothesis to make a more niche experiment to design or a different specific question to test.

The amazingness of the scientific method is that it is a comprehensive and straightforward process that scientists, and everyone, can utilize over and over again.

So, draw conclusions and repeat because the scientific method is never-ending, and no result is ever considered perfect.

The scientific method is a process of:

• Making an observation.
• Forming a hypothesis.
• Making a prediction.
• Experimenting to test the hypothesis.

The procedure of repeating the scientific method is crucial to science and all fields of human knowledge.

Further Information

• Karl Popper – Falsification
• Thomas – Kuhn Paradigm Shift
• Positivism in Sociology: Definition, Theory & Examples
• Is Psychology a Science?
• Psychology as a Science (PDF)

List the 6 steps of the scientific methods in order

• Make an observation (theory construction)
• Ask a question. A scientific question must be defined, testable, and measurable.
• Form a hypothesis (make predictions)
• Run an experiment to test the hypothesis (gather data)
• Analyze the data and draw conclusions
• Share your results so that other researchers can make new hypotheses

What is the first step of the scientific method?

The first step of the scientific method is making an observation. This involves noticing and describing a phenomenon or group of phenomena that one finds interesting and wishes to explain.

Observations can occur in a natural setting or within the confines of a laboratory. The key point is that the observation provides the initial question or problem that the rest of the scientific method seeks to answer or solve.

What is the scientific method?

The scientific method is a step-by-step process that investigators can follow to determine if there is a causal connection between two or more variables.

Psychologists and other scientists regularly suggest motivations for human behavior. On a more casual level, people judge other people’s intentions, incentives, and actions daily.

While our standard assessments of human behavior are subjective and anecdotal, researchers use the scientific method to study psychology objectively and systematically.

All utilize a scientific method to study distinct aspects of people’s thinking and behavior. This process allows scientists to analyze and understand various psychological phenomena, but it also provides investigators and others a way to disseminate and debate the results of their studies.

The outcomes of these studies are often noted in popular media, which leads numerous to think about how or why researchers came to the findings they did.

Why Use the Six Steps of the Scientific Method

The goal of scientists is to understand better the world that surrounds us. Scientific research is the most critical tool for navigating and learning about our complex world.

Without it, we would be compelled to rely solely on intuition, other people’s power, and luck. We can eliminate our preconceived concepts and superstitions through methodical scientific research and gain an objective sense of ourselves and our world.

All psychological studies aim to explain, predict, and even control or impact mental behaviors or processes. So, psychologists use and repeat the scientific method (and its six steps) to perform and record essential psychological research.

So, psychologists focus on understanding behavior and the cognitive (mental) and physiological (body) processes underlying behavior.

In the real world, people use to understand the behavior of others, such as intuition and personal experience. The hallmark of scientific research is evidence to support a claim.

Scientific knowledge is empirical, meaning it is grounded in objective, tangible evidence that can be observed repeatedly, regardless of who is watching.

The scientific method is crucial because it minimizes the impact of bias or prejudice on the experimenter. Regardless of how hard one tries, even the best-intentioned scientists can’t escape discrimination. can’t

It stems from personal opinions and cultural beliefs, meaning any mortal filters data based on one’s experience. Sadly, this “filtering” process can cause a scientist to favor one outcome over another.

For an everyday person trying to solve a minor issue at home or work, succumbing to these biases is not such a big deal; in fact, most times, it is important.

But in the scientific community, where results must be inspected and reproduced, bias or discrimination must be avoided.

When to Use the Six Steps of the Scientific Method ?

One can use the scientific method anytime, anywhere! From the smallest conundrum to solving global problems, it is a process that can be applied to any science and any investigation.

Even if you are not considered a “scientist,” you will be surprised to know that people of all disciplines use it for all kinds of dilemmas.

Try to catch yourself next time you come by a question and see how you subconsciously or consciously use the scientific method.

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The 6 Scientific Method Steps and How to Use Them

General Education

When you’re faced with a scientific problem, solving it can seem like an impossible prospect. There are so many possible explanations for everything we see and experience—how can you possibly make sense of them all? Science has a simple answer: the scientific method.

The scientific method is a method of asking and answering questions about the world. These guiding principles give scientists a model to work through when trying to understand the world, but where did that model come from, and how does it work?

In this article, we’ll define the scientific method, discuss its long history, and cover each of the scientific method steps in detail.

What Is the Scientific Method?

At its most basic, the scientific method is a procedure for conducting scientific experiments. It’s a set model that scientists in a variety of fields can follow, going from initial observation to conclusion in a loose but concrete format.

The number of steps varies, but the process begins with an observation, progresses through an experiment, and concludes with analysis and sharing data. One of the most important pieces to the scientific method is skepticism —the goal is to find truth, not to confirm a particular thought. That requires reevaluation and repeated experimentation, as well as examining your thinking through rigorous study.

There are in fact multiple scientific methods, as the basic structure can be easily modified.  The one we typically learn about in school is the basic method, based in logic and problem solving, typically used in “hard” science fields like biology, chemistry, and physics. It may vary in other fields, such as psychology, but the basic premise of making observations, testing, and continuing to improve a theory from the results remain the same.

The History of the Scientific Method

The scientific method as we know it today is based on thousands of years of scientific study. Its development goes all the way back to ancient Mesopotamia, Greece, and India.

The Ancient World

In ancient Greece, Aristotle devised an inductive-deductive process , which weighs broad generalizations from data against conclusions reached by narrowing down possibilities from a general statement. However, he favored deductive reasoning, as it identifies causes, which he saw as more important.

Aristotle wrote a great deal about logic and many of his ideas about reasoning echo those found in the modern scientific method, such as ignoring circular evidence and limiting the number of middle terms between the beginning of an experiment and the end. Though his model isn’t the one that we use today, the reliance on logic and thorough testing are still key parts of science today.

The Middle Ages

The next big step toward the development of the modern scientific method came in the Middle Ages, particularly in the Islamic world. Ibn al-Haytham, a physicist from what we now know as Iraq, developed a method of testing, observing, and deducing for his research on vision. al-Haytham was critical of Aristotle’s lack of inductive reasoning, which played an important role in his own research.

Other scientists, including Abū Rayhān al-Bīrūnī, Ibn Sina, and Robert Grosseteste also developed models of scientific reasoning to test their own theories. Though they frequently disagreed with one another and Aristotle, those disagreements and refinements of their methods led to the scientific method we have today.

Following those major developments, particularly Grosseteste’s work, Roger Bacon developed his own cycle of observation (seeing that something occurs), hypothesis (making a guess about why that thing occurs), experimentation (testing that the thing occurs), and verification (an outside person ensuring that the result of the experiment is consistent).

After joining the Franciscan Order, Bacon was granted a special commission to write about science; typically, Friars were not allowed to write books or pamphlets. With this commission, Bacon outlined important tenets of the scientific method, including causes of error, methods of knowledge, and the differences between speculative and experimental science. He also used his own principles to investigate the causes of a rainbow, demonstrating the method’s effectiveness.

Scientific Revolution

Throughout the Renaissance, more great thinkers became involved in devising a thorough, rigorous method of scientific study. Francis Bacon brought inductive reasoning further into the method, whereas Descartes argued that the laws of the universe meant that deductive reasoning was sufficient. Galileo’s research was also inductive reasoning-heavy, as he believed that researchers could not account for every possible variable; therefore, repetition was necessary to eliminate faulty hypotheses and experiments.

All of this led to the birth of the Scientific Revolution , which took place during the sixteenth and seventeenth centuries. In 1660, a group of philosophers and physicians joined together to work on scientific advancement. After approval from England’s crown , the group became known as the Royal Society, which helped create a thriving scientific community and an early academic journal to help introduce rigorous study and peer review.

Previous generations of scientists had touched on the importance of induction and deduction, but Sir Isaac Newton proposed that both were equally important. This contribution helped establish the importance of multiple kinds of reasoning, leading to more rigorous study.

As science began to splinter into separate areas of study, it became necessary to define different methods for different fields. Karl Popper was a leader in this area—he established that science could be subject to error, sometimes intentionally. This was particularly tricky for “soft” sciences like psychology and social sciences, which require different methods. Popper’s theories furthered the divide between sciences like psychology and “hard” sciences like chemistry or physics.

Paul Feyerabend argued that Popper’s methods were too restrictive for certain fields, and followed a less restrictive method hinged on “anything goes,” as great scientists had made discoveries without the Scientific Method. Feyerabend suggested that throughout history scientists had adapted their methods as necessary, and that sometimes it would be necessary to break the rules. This approach suited social and behavioral scientists particularly well, leading to a more diverse range of models for scientists in multiple fields to use.

The Scientific Method Steps

Though different fields may have variations on the model, the basic scientific method is as follows:

#1: Make Observations 

Notice something, such as the air temperature during the winter, what happens when ice cream melts, or how your plants behave when you forget to water them.

#2: Ask a Question

Turn your observation into a question. Why is the temperature lower during the winter? Why does my ice cream melt? Why does my toast always fall butter-side down?

This step can also include doing some research. You may be able to find answers to these questions already, but you can still test them!

#3: Make a Hypothesis

A hypothesis is an educated guess of the answer to your question. Why does your toast always fall butter-side down? Maybe it’s because the butter makes that side of the bread heavier.

A good hypothesis leads to a prediction that you can test, phrased as an if/then statement. In this case, we can pick something like, “If toast is buttered, then it will hit the ground butter-first.”

#4: Experiment

Your experiment is designed to test whether your predication about what will happen is true. A good experiment will test one variable at a time —for example, we’re trying to test whether butter weighs down one side of toast, making it more likely to hit the ground first.

The unbuttered toast is our control variable. If we determine the chance that a slice of unbuttered toast, marked with a dot, will hit the ground on a particular side, we can compare those results to our buttered toast to see if there’s a correlation between the presence of butter and which way the toast falls.

If we decided not to toast the bread, that would be introducing a new question—whether or not toasting the bread has any impact on how it falls. Since that’s not part of our test, we’ll stick with determining whether the presence of butter has any impact on which side hits the ground first.

#5: Analyze Data

After our experiment, we discover that both buttered toast and unbuttered toast have a 50/50 chance of hitting the ground on the buttered or marked side when dropped from a consistent height, straight down. It looks like our hypothesis was incorrect—it’s not the butter that makes the toast hit the ground in a particular way, so it must be something else.

Since we didn’t get the desired result, it’s back to the drawing board. Our hypothesis wasn’t correct, so we’ll need to start fresh. Now that you think about it, your toast seems to hit the ground butter-first when it slides off your plate, not when you drop it from a consistent height. That can be the basis for your new experiment.

#6: Communicate Your Results

Good science needs verification. Your experiment should be replicable by other people, so you can put together a report about how you ran your experiment to see if other peoples’ findings are consistent with yours.

This may be useful for class or a science fair. Professional scientists may publish their findings in scientific journals, where other scientists can read and attempt their own versions of the same experiments. Being part of a scientific community helps your experiments be stronger because other people can see if there are flaws in your approach—such as if you tested with different kinds of bread, or sometimes used peanut butter instead of butter—that can lead you closer to a good answer.

A Scientific Method Example: Falling Toast

We’ve run through a quick recap of the scientific method steps, but let’s look a little deeper by trying again to figure out why toast so often falls butter side down.

#1: Make Observations

At the end of our last experiment, where we learned that butter doesn’t actually make toast more likely to hit the ground on that side, we remembered that the times when our toast hits the ground butter side first are usually when it’s falling off a plate.

The easiest question we can ask is, “Why is that?”

We can actually search this online and find a pretty detailed answer as to why this is true. But we’re budding scientists—we want to see it in action and verify it for ourselves! After all, good science should be replicable, and we have all the tools we need to test out what’s really going on.

Why do we think that buttered toast hits the ground butter-first? We know it’s not because it’s heavier, so we can strike that out. Maybe it’s because of the shape of our plate?

That’s something we can test. We’ll phrase our hypothesis as, “If my toast slides off my plate, then it will fall butter-side down.”

Just seeing that toast falls off a plate butter-side down isn’t enough for us. We want to know why, so we’re going to take things a step further—we’ll set up a slow-motion camera to capture what happens as the toast slides off the plate.

We’ll run the test ten times, each time tilting the same plate until the toast slides off. We’ll make note of each time the butter side lands first and see what’s happening on the video so we can see what’s going on.

When we review the footage, we’ll likely notice that the bread starts to flip when it slides off the edge, changing how it falls in a way that didn’t happen when we dropped it ourselves.

That answers our question, but it’s not the complete picture —how do other plates affect how often toast hits the ground butter-first? What if the toast is already butter-side down when it falls? These are things we can test in further experiments with new hypotheses!

Now that we have results, we can share them with others who can verify our results. As mentioned above, being part of the scientific community can lead to better results. If your results were wildly different from the established thinking about buttered toast, that might be cause for reevaluation. If they’re the same, they might lead others to make new discoveries about buttered toast. At the very least, you have a cool experiment you can share with your friends!

Key Scientific Method Tips

Though science can be complex, the benefit of the scientific method is that it gives you an easy-to-follow means of thinking about why and how things happen. To use it effectively, keep these things in mind!

Don’t Worry About Proving Your Hypothesis

One of the important things to remember about the scientific method is that it’s not necessarily meant to prove your hypothesis right. It’s great if you do manage to guess the reason for something right the first time, but the ultimate goal of an experiment is to find the true reason for your observation to occur, not to prove your hypothesis right.

Good science sometimes means that you’re wrong. That’s not a bad thing—a well-designed experiment with an unanticipated result can be just as revealing, if not more, than an experiment that confirms your hypothesis.

Be Prepared to Try Again

If the data from your experiment doesn’t match your hypothesis, that’s not a bad thing. You’ve eliminated one possible explanation, which brings you one step closer to discovering the truth.

The scientific method isn’t something you’re meant to do exactly once to prove a point. It’s meant to be repeated and adapted to bring you closer to a solution. Even if you can demonstrate truth in your hypothesis, a good scientist will run an experiment again to be sure that the results are replicable. You can even tweak a successful hypothesis to test another factor, such as if we redid our buttered toast experiment to find out whether different kinds of plates affect whether or not the toast falls butter-first. The more we test our hypothesis, the stronger it becomes!

What’s Next?

Want to learn more about the scientific method? These important high school science classes will no doubt cover it in a variety of different contexts.

Test your ability to follow the scientific method using these at-home science experiments for kids !

Need some proof that science is fun? Try making slime

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Melissa Brinks graduated from the University of Washington in 2014 with a Bachelor's in English with a creative writing emphasis. She has spent several years tutoring K-12 students in many subjects, including in SAT prep, to help them prepare for their college education.

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Multiple goals, multiple solutions, plenty of second-guessing and revising − here’s how science really works

Professor of Philosophy, University of Montana

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Soazig Le Bihan receives funding from the Maureen and Mike Mansfield Center at the University of Montana.

University of Montana provides funding as a member of The Conversation US.

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A man in a lab coat bends under a dim light, his strained eyes riveted onto a microscope. He’s powered only by caffeine and anticipation.

This solitary scientist will stay on task until he unveils the truth about the cause of the dangerous disease quickly spreading through his vulnerable city. Time is short, the stakes are high, and only he can save everyone. …

That kind of romanticized picture of science was standard for a long time. But it’s as far from actual scientific practice as a movie’s choreographed martial arts battle is from a real fistfight.

For most of the 20th century, philosophers of science like me maintained somewhat idealistic claims about what good science looks like. Over the past few decades, however, many of us have revised our views to better mirror actual scientific practice .

An update on what to expect from actual science is overdue. I often worry that when the public holds science to unrealistic standards, any scientific claim failing to live up to them arouses suspicion. While public trust is globally strong and has been for decades, it has been eroding. In November 2023, Americans’ trust in scientists was 14 points lower than it had been just prior to the COVID-19 pandemic, with its flurry of confusing and sometimes contradictory science-related messages.

When people’s expectations are not met about how science works, they may blame scientists. But modifying our expectations might be more useful. Here are three updates I think can help people better understand how science actually works. Hopefully, a better understanding of actual scientific practice will also shore up people’s trust in the process.

The many faces of scientific research

First, science is a complex endeavor involving multiple goals and associated activities.

Some scientists search for the causes underlying some observable effect, such as a decimated pine forest or the Earth’s global surface temperature increase .

Others may investigate the what rather than the why of things. For example, ecologists build models to estimate gray wolf abundance in Montana . Spotting predators is incredibly challenging. Counting all of them is impractical. Abundance models are neither complete nor 100% accurate – they offer estimates deemed good enough to set harvesting quotas. Perfect scientific models are just not in the cards .

Beyond the what and the why, scientists may focus on the how. For instance, the lives of people living with chronic illnesses can be improved by research on strategies for managing disease – to mitigate symptoms and improve function, even if the true causes of their disorders largely elude current medicine.

It’s understandable that some patients may grow frustrated or distrustful of medical providers unable to give clear answers about what causes their ailment. But it’s important to grasp that lots of scientific research focuses on how to effectively intervene in the world to reach some specific goals.

Simplistic views represent science as solely focused on providing causal explanations for the various phenomena we observe in this world. The truth is that scientists tackle all kinds of problems, which are best solved using different strategies and approaches and only sometimes involve full-fledged explanations.

Complex problems call for complex solutions

The second aspect of scientific practice worth underscoring is that, because scientists tackle complex problems, they don’t typically offer one unique, complete and perfect answer. Instead they consider multiple, partial and possibly conflicting solutions.

Scientific modeling strategies illustrate this point well. Scientific models typically are partial, simplified and sometimes deliberately unrealistic representations of a system of interest. Models can be physical, conceptual or mathematical. The critical point is that they represent target systems in ways that are useful in particular contexts of inquiry. Interestingly, considering multiple possible models is often the best strategy to tackle complex problems.

Scientists consider multiple models of biodiversity , atomic nuclei or climate change . Returning to wolf abundance estimates, multiple models can also fit the bill. Such models rely on various types of data, including acoustic surveys of wolf howls, genetic methods that use fecal samples from wolves, wolf sightings and photographic evidence, aerial surveys, snow track surveys and more.

Weighing the pros and cons of various possible solutions to the problem of interest is part and parcel of the scientific process. Interestingly, in some cases, using multiple conflicting models allows for better predictions than trying to unify all the models into one.

The public may be surprised and possibly suspicious when scientists push forward multiple models that rely on conflicting assumptions and make different predictions. People often think “real science” should provide definite, complete and foolproof answers to their questions. But given various limitations and the world’s complexity, keeping multiple perspectives in play is most often the best way for scientists to reach their goals and solve the problems at hand.

Science as a collective, contrarian endeavor

Finally, science is a collective endeavor, where healthy disagreement is a feature, not a bug.

The romanticized version of science pictures scientists working in isolation and establishing absolute truths. Instead, science is a social and contrarian process in which the community’s scrutiny ensures we have the best available knowledge. “Best available” does not mean “definitive,” but the best we have until we find out how to improve it. Science almost always allows for disagreements among experts.

Controversies are core to how science works at its best and are as old as Western science itself. In the 1600s, Descartes and Leibniz fought over how to best characterize the laws of dynamics and the nature of motion.

The long history of atomism provides a valuable perspective on how science is an intricate and winding process rather than a fast-delivery system of results set in stone. As Jean Baptiste Perrin conducted his 1908 experiments that seemingly settled all discussion regarding the existence of atoms and molecules, the questions of the atom’s properties were about to become the topic of decades of controversies with the birth of quantum physics.

The nature and structure of fundamental particles and associated fields have been the subject of scientific research for more than a century. Lively academic discussions abound concerning the difficult interpretation of quantum mechanics , the challenging unification of quantum physics and relativity , and the existence of the Higgs boson , among others.

Distrusting researchers for having healthy scientific disagreements is largely misguided.

A very human practice

To be clear, science is dysfunctional in some respects and contexts. Current institutions have incentives for counterproductive practices, including maximizing publication numbers . Like any human endeavor, science includes people with bad intent, including some trying to discredit legitimate scientific research . Finally, science is sometimes inappropriately influenced by various values in problematic ways.

These are all important considerations when evaluating the trustworthiness of particular scientific claims and recommendations. However, it is unfair, sometimes dangerous, to mistrust science for doing what it does at its best. Science is a multifaceted endeavor focused on solving complex problems that typically just don’t have simple solutions. Communities of experts scrutinize those solutions in hopes of providing the best available approach to tackling the problems of interest.

Science is also a fallible and collective process. Ignoring the realities of that process and holding science up to unrealistic standards may result in the public calling science out and losing trust in its reliability for the wrong reasons.

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Solving Everyday Problems with the Scientific Method

• By (author): 
• Don K Mak , 
• Angela T Mak , and 
• Anthony B Mak
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• Description
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This book describes how one can use The Scientific Method to solve everyday problems including medical ailments, health issues, money management, traveling, shopping, cooking, household chores, etc. It illustrates how to exploit the information collected from our five senses, how to solve problems when no information is available for the present problem situation, how to increase our chances of success by redefining a problem, and how to extrapolate our capabilities by seeing a relationship among heretofore unrelated concepts.

One should formulate a hypothesis as early as possible in order to have a sense of direction regarding which path to follow. Occasionally, by making wild conjectures, creative solutions can transpire. However, hypotheses need to be well-tested. Through this way, The Scientific Method can help readers solve problems in both familiar and unfamiliar situations. Containing real-life examples of how various problems are solved — for instance, how some observant patients cure their own illnesses when medical experts have failed — this book will train readers to observe what others may have missed and conceive what others may not have contemplated. With practice, they will be able to solve more problems than they could previously imagine.

In this second edition, the authors have added some more theories which they hope can help in solving everyday problems. At the same time, they have updated the book by including quite a few examples which they think are interesting.

Sample Chapter(s) Chapter 1: Prelude (63 KB)

• Preface to the Second Edition
• Preface to the First Edition
• The Scientific Method
• Observation
• Recognition
• Problem Situation and Problem Definition
• Induction and Deduction
• Alternative Solutions
• Mathematics
• Probable Value
• Bibliography

FRONT MATTER

• Pages: i–xvi

https://doi.org/10.1142/9789813145313_fmatter

• Claimers and Disclaimers

Chapter 1: Prelude

https://doi.org/10.1142/9789813145313_0001

The father put down the newspaper. It had been raining for the last two hours. The rain finally stopped, and the sky looked clear. After all this raining, the negative ions in the atmosphere would have increased, and the air would feel fresh. The father suggested the family of four should go for a stroll. There was a park just about fifteen minutes walk from their house.

Chapter 2: The Scientific Method

• Pages: 3–18

https://doi.org/10.1142/9789813145313_0002

In the history of philosophical ideation, scientific discoveries, and engineering inventions, it has almost never happened that a single person (or a single group of people) has come up with an idea or a similar idea that no one has ever dreamed of earlier, or at the same time. This person may not be aware of the previous findings, nor someone else in another part of the world has comparable ideas, and thus – his idea may be very original, as far as he is concerned. However, history tells us that it is highly unlikely that no one has already come up with some related concepts.

Chapter 3: Observation

• Pages: 19–56

https://doi.org/10.1142/9789813145313_0003

Observation is the first step of the Scientific Method. However, it can infiltrate the whole scientific process – from the initial perception of a phenomenon, to proposing a solution, and right down to experimentation, where observation of the results is significant.

Chapter 4: Hypothesis

• Pages: 57–95

https://doi.org/10.1142/9789813145313_0004

In scientific discipline, a hypothesis is a set of propositions set forth to explain the occurrence of certain phenomena. In daily language, a hypothesis can be interpreted as an assumption or guess. In this book, we employ both these definitions. Within the context of the first definition, we search for an explanation of why the problem occurs to begin with. Within the context of the second definition, we look for a plausible solution to the problem.

Chapter 5: Experiment

• Pages: 96–121

https://doi.org/10.1142/9789813145313_0005

In scientific discipline, an experiment is a test under controlled conditions to investigate the validity of a hypothesis. In everyday language, experiment can be interpreted as a testing of an idea. In this book, we employ both these definitions. Within the context of the first definition, we attempt to confirm whether an explanation of an observation is correct. Within the context of the second definition, we check whether a proposed idea for a solution is valid.

Chapter 6: Recognition

• Pages: 122–144

https://doi.org/10.1142/9789813145313_0006

Before we can solve any problem, we need to recognize that a problem exists in the first place. That may seem obvious, but while some problems stick out like thorns in a bush, others are hidden like plants in a forest. As such, not only do we need to tune up our observational skills to see that a problem does exist; we should also sharpen our thinking to anticipate that a problem may arise. Thus, recognition can be considered to be a combination of observing and hypothesizing.

Chapter 7: Problem Situation and Problem Definition

• Pages: 145–152

https://doi.org/10.1142/9789813145313_0007

For just about any situation, we can look at it from different perspectives. Take the example of a piece of rock, it will look different from the eyes of a landscaper, an architect, a geologist and an artist.

Chapter 8: Induction and Deduction

• Pages: 153–164

https://doi.org/10.1142/9789813145313_0008

Once a problem has been defined, we need to find a solution. To determine which route we can take, we will have to take a look at the knowledge that we already have in hand, and we may want to search for more information when necessary. It is therefore, much more convenient if we already have an arsenal of tools that have been stored neatly and categorized in our mind. That simply means, that we should have been observing our surroundings, and preferably have come up with some general principles that can guide us in the present problem.

Chapter 9: Alternative Solutions

• Pages: 165–193

https://doi.org/10.1142/9789813145313_0009

While there are various ways to view a problem situation, and thus define a problem differently, there are also different ways to solve a problem once it is defined. Some of the solutions may be better than others. If we have the option of not requiring to make a snap judgement, we should wait till we have come up with several plausible solutions, and then decide which one would be the best. How do we know which solution is the best? We will discuss that in the chapter on Probable Value. Generally speaking, we should train ourselves to come up with a few suggestions, and weigh the pros and cons of each resolution. This would be equivalent to coming up with different hypotheses, and judging which one would provide an optimal result.

Chapter 10: Relation

• Pages: 194–225

https://doi.org/10.1142/9789813145313_0010

Relation is the connection and association among different objects, events, and ideas. Problem solving, quite often, is connected with the ability to see the various relations among diversified concepts. Understanding the affiliation of a mixture of notions can be considered as hypothesizing the existence of certain correlation.

Chapter 11: Mathematics

• Pages: 226–306

https://doi.org/10.1142/9789813145313_0011

Mathematics, even some simple arithmetic, is so important in solving some of the everyday problems, that we think a whole chapter should be written on it.

Chapter 12: Probable Value

• Pages: 307–318

https://doi.org/10.1142/9789813145313_0012

For a certain problem, we may come up with several plausible solutions. Which path should we take? Each path would only have certain chance or probability of success in resolving the problem. If each path or solution has a different reward, we can define the probable value of each path to be the multiplication of the reward by the probability. We should, most likely, choose the path that has the highest probable value. (The term “probable value” is coined by us. The idea is appropriated from the term “expected value” in Statistics. In this sense, expected value can be considered as the sum of all probable values.).

Chapter 13: Epilogue

• Pages: 319–322

https://doi.org/10.1142/9789813145313_0013

We run into problems every day. Even when we do not encounter any problems, it does not mean that they do not exist. Sometimes, we wish we could be able to recognize them earlier. The scientific method of observation, hypothesis, and experiment can help us recognize, define, and solve our problems.

BACK MATTER

• Pages: 323–332

https://doi.org/10.1142/9789813145313_bmatter

Praise for the First Edition:

“The book was fun: a clever and entertaining introduction to basic logical thinking and maths.”

“This ingenious and entertaining volume should be useful to anyone in the general public interested in self-help books; undergraduate students majoring in education or behavioral psychology; and graduates and researchers interested in problem-solving, creativity, and scientific research methodology.”

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How to Use the Scientific Method in Everyday Life

How to Set Up a Controlled Science Experiment

The scientific method is a procedure consisting of a series of steps with the goal of problem-solving and information-gathering. The scientific method begins with the recognition of a problem and a clear elaboration or description of the problem itself. A process of experimentation and data collection then follows. The final steps consist of the formulation and testing of a hypothesis or potential solution and conclusion. For people unaccustomed to using the scientific method, the process may seem abstract and unapproachable. With a little consideration and observation, any problem encountered in daily life is a potential possibility to use the scientific method.

Locate or identify a problem to solve. Your personal environment is a good place to start, either in the workplace, the home, or your town or city.

Describe the problem in detail. Make quantifiable observations, such as number of times of occurrence, duration, specific physical measurements, and so on.

Form a hypothesis about what the possible cause of the problem might be, or what a potential solution could be. Check if the previously collected data suggests a pattern or possible cause.

Test your hypothesis either through further observation of the problem or by creating an experiment that highlights the aspect of the problem you wish to test. For example, if you suspect a faulty wire is the cause of a light not working, you must find a way to isolate and test whether or not the wire is actually the cause.

Repeat the steps of observation, hypothesis formation and testing until you reach a conclusion that is reinforced by supporting data or directly solves the problem at hand.

• The scientific method is best suited to solving problems without direct or simple answers. For example, a light bulb that burns out may simply need to be replaced. A light bulb that works intermittently is a much more suitable candidate for use of the scientific method, because of all of the potential causes of it not working.

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• Britannica Online Encyclopedia: Scientific Method; June 2011

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Alex Jakubik began his writing career in 2000 with book-cover summaries for Barnes & Noble. He has also authored concert programs and travel blogs, and worked both nationally and internationally in the arts. Jakubik holds a Bachelor of Music degree from Indiana University and a Master of Music from Yale University.

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Microbe Notes

Scientific Method: Definition, Steps, Examples, Uses

Sir Francis Bacon, an English philosopher, developed modern scientific research and scientific methods. He is also known as “the Father of modern science.”

He was influenced by Galileo Galilei and Nicholas Copernicus’ writings throughout his study.

The scientific method is a powerful analytical or problem-solving method of learning more about the natural world.  

The scientific method is a combined method, which consists of theoretical knowledge and practical experimentation by using scientific instruments, analysis and comparisons of results, and then peer reviews.

• The scientific method is a procedure that the scientists use to conduct research.
• Scientific investigators play a crucial role in following a series of steps such as asking questions, setting hypothesis to answer questions, performing multiple experiments to confirm the reliability of data/ results, data collection and interpretation, and developing conclusions based on the hypothesis.

Table of Contents

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Steps of Scientific Method

There are seven steps of the scientific method such as:

• Make an observation
• Ask a question
• Background research/ Research the topic
• Formulate a hypothesis
• Conduct an experiment to test the hypothesis
• Data record and analysis
• Draw a conclusion

1. Make an observation

• Before asking a question, you need a proper observation to get information about some topic, which may help to identify the question. 
• Proper observation in the area of investigation or about something you are interested in is required, whether you recognize it or not. 

2. Ask a question

• The scientific method follows a step by asking a question. Based on what you observe, Asking questions starts with Wh- such as What, When, Who, Which, Why, How, Does or Where? 
• A question helps to identify a core problem and form a hypothesis . The question should be relatable and specific as much as possible. 
• Why is this thing happening?
• What is the reason behind this?
• How does this happen?
• Does it need to happen?

3. Background research/ Research the topic

• Background research on the experiment/ topic is necessary to analyze and answer the questions. 
• Many scientists are employing various techniques and equipment, such as libraries and Internet research (research papers, articles, journals, etc.), that push how to investigate, design, and understand the experiment. 
• In addition, you can learn from other experiences, research, or experiments, which helps you not repeat the same mistakes and be aware of doing things further. 
• It helps to predict what will happen in the future. It also helps to understand the theory and background history of the experiment.

4. Formulate a hypothesis

• A Hypothesis is an idea or a guess to explain a specific occurrence, natural event, or particular experience based on prior observation.
• It is another step in the scientific method. A hypothesis allows you to make a prediction. Scientists predict what will be the outcome. 
• It outlines the objectives of the experiment, the variables used, and the expected outcome of the experiment. The hypothesis must be either falsifiable or testable. It also answers the previous question. 
• A hypothesis needs to be testable by gathering evidence. A hypothesis needs to be testable to perform an experiment, whether the evidence supports the hypothesis or not. 

5. Conduct an experiment to test a hypothesis

• After formulating a hypothesis, you must design and conduct an experiment. Experiments are the process of investigations to prove or disprove the hypothesis.
• Two variables play a crucial role in conducting experiments to test the hypothesis. 
• They are Independent variables (Can be manipulated or controlled by the person, or you can change while experimenting) and dependent variables (one you measure, which may be affected by the independent variable).
• They both are the cause and effect. The dependent variable is dependent on the independent variable. 
• All the variables must be under control to ensure that they have no impact on the result.
• You can also set another type of hypothesis, such as a “null hypothesis” or “no difference” hypothesis. 

There is no difference in the intense rain and crop destruction.

6. Data Record and Analysis

• During the experiment, data needs to be recorded and collected. Data is a set of values. It should be represented quantitatively (measured in numbers) or qualitatively (an explanation of outcomes).
• After the data collection, you can interpret the data by drawing a chart or constructing a table or graph to show the result. 
• After the data representation, you can analyze or interpret the data to understand the meaning of the data. 
• You can compare the results with other experiments visually or in graphics form. 

7. Draw a Conclusion

• Your Conclusion always showcases whether the experiments support the prediction and hypothesis or contradict.
• Scientists will analyze the experiment’s results and develop a new hypothesis based on the data they collect if they discover that their experiment did not support their hypothesis or that their prediction is not supported.
• While we conclude the experiment, all the collected results will be analyzed, which helps to interpret the hypothesis.
• Did your experiments support or reject your hypothesis? 
• Does your hypothesis prove or disprove your study? 
• Did your results show a strong correlation? 
• Was there any way to change the thing to make a better experiment?
• Are there things that need to be studied further? 
• If your hypothesis is supported, then that is fine. You can carry on. 
• But If not, do not try to manipulate the result or try to change the result. 
• Keep the result to its original form, or you can further repeat the experiment to get better results.

Application of Scientific Method

• It is essential in many sectors, such as social sciences, empirical sciences, statistics, biology, chemistry, and physics. It can be used in the laboratory.
• Scientific methods lead to discoveries, innovations, and improvements in various disciplines.
• The scientific method can be used to solve problems, explain the phenomena of the study, and find and test solutions.
• Scientific methods guarantee that the findings are based on evidence, making the study reliable and replicable and allowing research to occur objectively and systematically.
• The Editors of Encyclopaedia Britannica. (2024, March 14). Scientific method | Definition, Steps, & Application. Retrieved from https://www.britannica.com/science/scientific-method
• Biology Dictionary. (2020, November 6). Scientific method. Retrieved from https://biologydictionary.net/scientific-method/
• Bailey, R. (2019, August 21). Scientific method. Retrieved from https://www.thoughtco.com/scientific-method-p2-373335
• Buddies, S., & Buddies, S. (2023, August 17). Writing a Science Fair Project research plan. Retrieved from https://www.sciencebuddies.org/science-fair-projects/science-fair/writing-a-science-fair-project-research-plan
• Buddies, S., & Buddies, S. (2024, January 25). Steps of the scientific method. Retrieved from https://www.sciencebuddies.org/science-fair-projects/science-fair/steps-of-the-scientific-method
• Helmenstine, A. (2023, January 1). Steps of the scientific method. Retrieved from https://sciencenotes.org/steps-scientific-method/
• Cartwright, M., & Greer, R. (2023). Scientific method. World History Encyclopedia . Retrieved from https://www.worldhistory.org/Scientific_Method/
• https://www.extension.purdue.edu/extmedia/ID/ID-507-w.pdf
• GeeksforGeeks. (2024, April 18). Applications of scientific methods. Retrieved from https://www.geeksforgeeks.org/applications-of-scientific-methods/

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Scientific Method Steps in Psychology Research

Steps, Uses, and Key Terms

Verywell / Theresa Chiechi

How do researchers investigate psychological phenomena? They utilize a process known as the scientific method to study different aspects of how people think and behave.

When conducting research, the scientific method steps to follow are:

• Observe what you want to investigate
• Ask a research question and make predictions
• Test the hypothesis and collect data
• Examine the results and draw conclusions
• Report and share the results 

This process not only allows scientists to investigate and understand different psychological phenomena but also provides researchers and others a way to share and discuss the results of their studies.

Generally, there are five main steps in the scientific method, although some may break down this process into six or seven steps. An additional step in the process can also include developing new research questions based on your findings.

What Is the Scientific Method?

What is the scientific method and how is it used in psychology?

The scientific method consists of five steps. It is essentially a step-by-step process that researchers can follow to determine if there is some type of relationship between two or more variables.

By knowing the steps of the scientific method, you can better understand the process researchers go through to arrive at conclusions about human behavior.

Scientific Method Steps

While research studies can vary, these are the basic steps that psychologists and scientists use when investigating human behavior.

The following are the scientific method steps:

Step 1. Make an Observation

Before a researcher can begin, they must choose a topic to study. Once an area of interest has been chosen, the researchers must then conduct a thorough review of the existing literature on the subject. This review will provide valuable information about what has already been learned about the topic and what questions remain to be answered.

A literature review might involve looking at a considerable amount of written material from both books and academic journals dating back decades.

The relevant information collected by the researcher will be presented in the introduction section of the final published study results. This background material will also help the researcher with the first major step in conducting a psychology study: formulating a hypothesis.

Step 2. Ask a Question

Once a researcher has observed something and gained some background information on the topic, the next step is to ask a question. The researcher will form a hypothesis, which is an educated guess about the relationship between two or more variables

For example, a researcher might ask a question about the relationship between sleep and academic performance: Do students who get more sleep perform better on tests at school?

In order to formulate a good hypothesis, it is important to think about different questions you might have about a particular topic.

You should also consider how you could investigate the causes. Falsifiability is an important part of any valid hypothesis. In other words, if a hypothesis was false, there needs to be a way for scientists to demonstrate that it is false.

Step 3. Test Your Hypothesis and Collect Data

Once you have a solid hypothesis, the next step of the scientific method is to put this hunch to the test by collecting data. The exact methods used to investigate a hypothesis depend on exactly what is being studied. There are two basic forms of research that a psychologist might utilize: descriptive research or experimental research.

Descriptive research is typically used when it would be difficult or even impossible to manipulate the variables in question. Examples of descriptive research include case studies, naturalistic observation , and correlation studies. Phone surveys that are often used by marketers are one example of descriptive research.

Correlational studies are quite common in psychology research. While they do not allow researchers to determine cause-and-effect, they do make it possible to spot relationships between different variables and to measure the strength of those relationships. 

Experimental research is used to explore cause-and-effect relationships between two or more variables. This type of research involves systematically manipulating an independent variable and then measuring the effect that it has on a defined dependent variable .

One of the major advantages of this method is that it allows researchers to actually determine if changes in one variable actually cause changes in another.

While psychology experiments are often quite complex, a simple experiment is fairly basic but does allow researchers to determine cause-and-effect relationships between variables. Most simple experiments use a control group (those who do not receive the treatment) and an experimental group (those who do receive the treatment).

Step 4. Examine the Results and Draw Conclusions

Once a researcher has designed the study and collected the data, it is time to examine this information and draw conclusions about what has been found.  Using statistics , researchers can summarize the data, analyze the results, and draw conclusions based on this evidence.

So how does a researcher decide what the results of a study mean? Not only can statistical analysis support (or refute) the researcher’s hypothesis; it can also be used to determine if the findings are statistically significant.

When results are said to be statistically significant, it means that it is unlikely that these results are due to chance.

Based on these observations, researchers must then determine what the results mean. In some cases, an experiment will support a hypothesis, but in other cases, it will fail to support the hypothesis.

So what happens if the results of a psychology experiment do not support the researcher's hypothesis? Does this mean that the study was worthless?

Just because the findings fail to support the hypothesis does not mean that the research is not useful or informative. In fact, such research plays an important role in helping scientists develop new questions and hypotheses to explore in the future.

After conclusions have been drawn, the next step is to share the results with the rest of the scientific community. This is an important part of the process because it contributes to the overall knowledge base and can help other scientists find new research avenues to explore.

Step 5. Report the Results

The final step in a psychology study is to report the findings. This is often done by writing up a description of the study and publishing the article in an academic or professional journal. The results of psychological studies can be seen in peer-reviewed journals such as  Psychological Bulletin , the  Journal of Social Psychology ,  Developmental Psychology , and many others.

The structure of a journal article follows a specified format that has been outlined by the  American Psychological Association (APA) . In these articles, researchers:

• Provide a brief history and background on previous research
• Present their hypothesis
• Identify who participated in the study and how they were selected
• Provide operational definitions for each variable
• Describe the measures and procedures that were used to collect data
• Explain how the information collected was analyzed
• Discuss what the results mean

Why is such a detailed record of a psychological study so important? By clearly explaining the steps and procedures used throughout the study, other researchers can then replicate the results. The editorial process employed by academic and professional journals ensures that each article that is submitted undergoes a thorough peer review, which helps ensure that the study is scientifically sound.

Once published, the study becomes another piece of the existing puzzle of our knowledge base on that topic.

Before you begin exploring the scientific method steps, here's a review of some key terms and definitions that you should be familiar with:

• Falsifiable : The variables can be measured so that if a hypothesis is false, it can be proven false
• Hypothesis : An educated guess about the possible relationship between two or more variables
• Variable : A factor or element that can change in observable and measurable ways
• Operational definition : A full description of exactly how variables are defined, how they will be manipulated, and how they will be measured

Uses for the Scientific Method

The  goals of psychological studies  are to describe, explain, predict and perhaps influence mental processes or behaviors. In order to do this, psychologists utilize the scientific method to conduct psychological research. The scientific method is a set of principles and procedures that are used by researchers to develop questions, collect data, and reach conclusions.

Goals of Scientific Research in Psychology

Researchers seek not only to describe behaviors and explain why these behaviors occur; they also strive to create research that can be used to predict and even change human behavior.

Psychologists and other social scientists regularly propose explanations for human behavior. On a more informal level, people make judgments about the intentions, motivations , and actions of others on a daily basis.

While the everyday judgments we make about human behavior are subjective and anecdotal, researchers use the scientific method to study psychology in an objective and systematic way. The results of these studies are often reported in popular media, which leads many to wonder just how or why researchers arrived at the conclusions they did.

Examples of the Scientific Method

Now that you're familiar with the scientific method steps, it's useful to see how each step could work with a real-life example.

Say, for instance, that researchers set out to discover what the relationship is between psychotherapy and anxiety .

• Step 1. Make an observation : The researchers choose to focus their study on adults ages 25 to 40 with generalized anxiety disorder.
• Step 2. Ask a question : The question they want to answer in their study is: Do weekly psychotherapy sessions reduce symptoms in adults ages 25 to 40 with generalized anxiety disorder?
• Step 3. Test your hypothesis : Researchers collect data on participants' anxiety symptoms . They work with therapists to create a consistent program that all participants undergo. Group 1 may attend therapy once per week, whereas group 2 does not attend therapy.
• Step 4. Examine the results : Participants record their symptoms and any changes over a period of three months. After this period, people in group 1 report significant improvements in their anxiety symptoms, whereas those in group 2 report no significant changes.
• Step 5. Report the results : Researchers write a report that includes their hypothesis, information on participants, variables, procedure, and conclusions drawn from the study. In this case, they say that "Weekly therapy sessions are shown to reduce anxiety symptoms in adults ages 25 to 40."

Of course, there are many details that go into planning and executing a study such as this. But this general outline gives you an idea of how an idea is formulated and tested, and how researchers arrive at results using the scientific method.

Erol A. How to conduct scientific research ? Noro Psikiyatr Ars . 2017;54(2):97-98. doi:10.5152/npa.2017.0120102

University of Minnesota. Psychologists use the scientific method to guide their research .

Shaughnessy, JJ, Zechmeister, EB, & Zechmeister, JS. Research Methods In Psychology . New York: McGraw Hill Education; 2015.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Multiple goals, multiple solutions, plenty of second-guessing and revising—here's how science really works

by Soazig Le Bihan, The Conversation

A man in a lab coat bends under a dim light, his strained eyes riveted onto a microscope. He's powered only by caffeine and anticipation.

This solitary scientist will stay on task until he unveils the truth about the cause of the dangerous disease quickly spreading through his vulnerable city. Time is short, the stakes are high, and only he can save everyone.…

That kind of romanticized picture of science was standard for a long time. But it's as far from actual scientific practice as a movie's choreographed martial arts battle is from a real fistfight.

For most of the 20th century, philosophers of science like me maintained somewhat idealistic claims about what good science looks like. Over the past few decades, however, many of us have revised our views to better mirror actual scientific practice .

An update on what to expect from actual science is overdue. I often worry that when the public holds science to unrealistic standards, any scientific claim failing to live up to them arouses suspicion. While public trust is globally strong and has been for decades, it has been eroding. In November 2023, Americans' trust in scientists was 14 points lower than it had been just prior to the COVID-19 pandemic, with its flurry of confusing and sometimes contradictory science-related messages.

When people's expectations are not met about how science works, they may blame scientists. But modifying our expectations might be more useful. Here are three updates I think can help people better understand how science actually works. Hopefully, a better understanding of actual scientific practice will also shore up people's trust in the process.

The many faces of scientific research

First, science is a complex endeavor involving multiple goals and associated activities.

Some scientists search for the causes underlying some observable effect, such as a decimated pine forest or the Earth's global surface temperature increase .

Others may investigate the what rather than the why of things. For example, ecologists build models to estimate gray wolf abundance in Montana . Spotting predators is incredibly challenging. Counting all of them is impractical. Abundance models are neither complete nor 100% accurate—they offer estimates deemed good enough to set harvesting quotas. Perfect scientific models are just not in the cards .

Beyond the what and the why, scientists may focus on the how. For instance, the lives of people living with chronic illnesses can be improved by research on strategies for managing disease —to mitigate symptoms and improve function, even if the true causes of their disorders largely elude current medicine.

It's understandable that some patients may grow frustrated or distrustful of medical providers unable to give clear answers about what causes their ailment. But it's important to grasp that lots of scientific research focuses on how to effectively intervene in the world to reach some specific goals.

Simplistic views represent science as solely focused on providing causal explanations for the various phenomena we observe in this world. The truth is that scientists tackle all kinds of problems, which are best solved using different strategies and approaches and only sometimes involve full-fledged explanations.

Complex problems call for complex solutions

The second aspect of scientific practice worth underscoring is that, because scientists tackle complex problems, they don't typically offer one unique, complete and perfect answer. Instead they consider multiple, partial and possibly conflicting solutions.

Scientific modeling strategies illustrate this point well. Scientific models typically are partial, simplified and sometimes deliberately unrealistic representations of a system of interest. Models can be physical, conceptual or mathematical. The critical point is that they represent target systems in ways that are useful in particular contexts of inquiry. Interestingly, considering multiple possible models is often the best strategy to tackle complex problems.

Scientists consider multiple models of biodiversity , atomic nuclei or climate change . Returning to wolf abundance estimates, multiple models can also fit the bill. Such models rely on various types of data, including acoustic surveys of wolf howls, genetic methods that use fecal samples from wolves, wolf sightings and photographic evidence, aerial surveys, snow track surveys and more.

Weighing the pros and cons of various possible solutions to the problem of interest is part and parcel of the scientific process. Interestingly, in some cases, using multiple conflicting models allows for better predictions than trying to unify all the models into one.

The public may be surprised and possibly suspicious when scientists push forward multiple models that rely on conflicting assumptions and make different predictions. People often think "real science" should provide definite, complete and foolproof answers to their questions. But given various limitations and the world's complexity, keeping multiple perspectives in play is most often the best way for scientists to reach their goals and solve the problems at hand.

Science as a collective, contrarian endeavor

Finally, science is a collective endeavor, where healthy disagreement is a feature, not a bug.

The romanticized version of science pictures scientists working in isolation and establishing absolute truths. Instead, science is a social and contrarian process in which the community's scrutiny ensures we have the best available knowledge. "Best available" does not mean "definitive," but the best we have until we find out how to improve it. Science almost always allows for disagreements among experts.

Controversies are core to how science works at its best and are as old as Western science itself. In the 1600s, Descartes and Leibniz fought over how to best characterize the laws of dynamics and the nature of motion.

The long history of atomism provides a valuable perspective on how science is an intricate and winding process rather than a fast-delivery system of results set in stone. As Jean Baptiste Perrin conducted his 1908 experiments that seemingly settled all discussion regarding the existence of atoms and molecules, the questions of the atom's properties were about to become the topic of decades of controversies with the birth of quantum physics.

The nature and structure of fundamental particles and associated fields have been the subject of scientific research for more than a century. Lively academic discussions abound concerning the difficult interpretation of quantum mechanics , the challenging unification of quantum physics and relativity , and the existence of the Higgs boson , among others.

Distrusting researchers for having healthy scientific disagreements is largely misguided.

A very human practice

To be clear, science is dysfunctional in some respects and contexts. Current institutions have incentives for counterproductive practices, including maximizing publication numbers . Like any human endeavor, science includes people with bad intent, including some trying to discredit legitimate scientific research . Finally, science is sometimes inappropriately influenced by various values in problematic ways.

These are all important considerations when evaluating the trustworthiness of particular scientific claims and recommendations. However, it is unfair, sometimes dangerous, to mistrust science for doing what it does at its best. Science is a multifaceted endeavor focused on solving complex problems that typically just don't have simple solutions. Communities of experts scrutinize those solutions in hopes of providing the best available approach to tackling the problems of interest.

Science is also a fallible and collective process. Ignoring the realities of that process and holding science up to unrealistic standards may result in the public calling science out and losing trust in its reliability for the wrong reasons.

Provided by The Conversation

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From large labs to small teams, mentorship thrives

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Each year, new MIT graduate students are tasked with the momentous decision of choosing a research group that will serve as their home for the next several years. Among many questions they face: join an established research effort, or work with a new faculty member in a growing group?

Professors Cynthia Breazeal, leading a group of over 30 students, and Ming Guo, with a lab of fewer than 10, demonstrate that excellent mentorship can thrive in a research group of any size.

Cynthia Breazeal: Flexible leadership

Cynthia Breazeal is a professor of media arts and sciences at MIT, where she founded and directs the Personal Robots group at the MIT Media Lab. She is also the MIT dean for digital learning, leading MIT Open Learning’s business and research and engagement units. Breazeal is a pioneer of social robotics and human-robot interaction, and her research group investigates social robots applied to education, pediatrics, health and wellness, and aging.

Breazeal’s focus on taking multidisciplinary approaches to her research has resulted in an inclusive and supportive lab environment. Moreover, she does not shy away from taking students with unconventional backgrounds.

One nominator joined Breazeal's lab as a design researcher without a computer science background. However, Breazeal recognized the value of their work within the context of her lab’s research directions. “I was a bit of an oddball in the group”, the nominator modestly recounts, “but had joined to help make the work in the group more human-centered.”

Throughout the student's academic journey, Breazeal offered unwavering support, whether by connecting them with experts to solve specific problems or guiding them through the academic job search process.

Over the Covid-19 pandemic, Breazeal prioritized gathering student feedback through a survey about how she could best support her research group. In response to this input, Breazeal established the Senior Research Team (SRT) within her group.

The SRT includes PhD holders such as postdocs and research scientists who provide personalized mentorship to one or two graduate students per semester. The SRT members serve as dedicated advocates and points of contact, with weekly check-ins to address questions within the lab. Additionally, SRT members meet by themselves weekly to discuss student concerns and bring up urgent issues with Breazeal directly. Lastly, students can sign up for meetings with Breazeal and participate in paper review sessions with her and co-authors.

In the nominator’s opinion, this new system was implemented because Breazeal cares about her students and her lab culture. With over 30 members in her group, Breazeal cannot provide hands-on support for everyone daily, but she still deeply cares about each person's experience in the lab. The nominator shared that Breazeal “understands as she progresses in her career, she needs to make sure that she is changing and creating new systems for her research group to continue to operate smoothly.”

Ming Guo: Emphasizing learning over achievement

Ming Guo is an associate professor in the Department of Mechanical Engineering. Guo’s group works at the interface of mechanics, physics, and cell biology, seeking to understand how physical properties and biological function affect each other in cellular systems.

A key aspect of Guo’s mentorship style is his ability to foster an environment where students feel comfortable expressing their difficulties. He actively shows empathy for his students’ lives outside of the lab, often reaching out to provide support during challenging times. When one nominator found themselves faced with significant personal difficulties, Guo made a point to check in regularly, ensuring the student had a support network of friends and labmates.

Guo champions his students both academically and personally. For instance, when a collaborating lab placed unrealistic expectations on a student’s experimental output, Guo openly praised the student’s efforts and achievements in a joint meeting, alleviating pressure and highlighting the student’s hard work.

In addition, Guo encourages vulnerable conversations about issues affecting students, such as political developments and racial inequities. During the graduate student unionization process, he fostered open discussion, showing genuine interest in understanding the challenges faced by graduate students and using these insights to better support them.

In Guo’s research group, learning and development are prioritized over achievements and goals. When students encounter challenges in their research, Guo helps them maintain perspective by validating their struggles and recognizing the skills they acquire through difficult experiments. By celebrating their progress and emphasizing the importance of the learning process, he ensures that students understand the value of their experiences beyond outcomes. This approach not only boosts their confidence, but also fosters a deeper appreciation for the scientific process and their own development as researchers.

Guo says that he feels most energized and happy when he talks to students. He looks forward to the new ideas that they present. One nominator commented on how much Guo enjoys giving feedback at group meetings: “Sometimes he isn’t convinced in the beginning, but he has cultivated our lab atmosphere to be conducive to extended discussion.”

The nominator continues, “When things do work and become really interesting, he is extremely excited with us and pushes us to share our own ideas with the wider research community.” 

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August 7, 2024

Mathematicians Reinvent the Wheel in Higher Dimensions to Solve Decades-Old Geometry Problem

A new mathematical technique shows how to build small objects in any dimension that roll like a wheel, expanding our understanding of higher dimensional space

By Max Springer

Malte Mueller/Getty Images

Mathematicians are “reinventing the wheel” by giving it a new shape . Their newly imagined wheel looks like a many-dimensional guitar pick, and it could theoretically roll in ways beyond our three-dimensional understanding. This breakthrough solves a decades-old geometry problem by showing how to build objects in dimensions that we cannot envision .

“It’s a stunning theory,” says Gil Kalai, a professor at the Einstein Institute of Mathematics in Israel, who was not involved with the study. The results prove that these unfathomable objects can be constructed in any dimension at a fraction of the size of more traditional rolling shapes, such as circles or spheres.

Wheels roll because they are objects with “constant width”—they appear to be the same width from every angle. This geometric property allows wheels to maintain a constant distance between two parallel planes, such as the ground and a car, as they move. Essentially, a shape has constant width if it can roll smoothly without wobbling. For example, place a tennis ball between your parallel hands and rotate it around—you’ll see that your hands never get closer together or farther away because the ball has constant-width geometry. An oblong shape such as an egg would fail that test.

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Amanda Montañez

Circles and spheres are simple, intuitive examples of constant-width shapes, and humans have been using them to facilitate movement for millennia. These are special kinds of constant-width shapes called “ balls ”—shapes where all boundary points are the same distance from the center. This is a circle in two dimensions and a sphere in three, and the concept extends into higher dimensions that we can’t readily visualize.

Because the boundary points are all positioned at a fixed distance away from one central point, these balls are hefty: they have the maximum possible volume for a constant-width shape in any dimension. But being so voluminous isn’t always ideal. In the 1980s mathematician Oded Schramm posed the question: How can we find constant-width shapes with the minimum volume in any dimension? That “is a very basic question,” Kalai says, one mathematicians have been interested in solving ever since. “But nobody had any method of how to probe it.”

The problem remained until this June, when an international team of mathematicians proposed a new way to construct constant-width shapes . The researchers’ approach, involving the intersection of an infinite number of n -dimensional balls, was posted on the preprint server arXiv.org in a concise three-page proof . “The recipe itself is very simple,” says study co-author Andriy Bondarenko, a professor of mathematics at the Norwegian University of Science and Technology. Although using and analyzing this recipe is relatively straightforward, it took the researchers years to “[understand] why we should consider this recipe in the first place.”

Wheels in Flatland

This latest work pioneers the investigation of constant-width shapes in any dimension, but designing wheels in two or three dimensions is itself not a new problem. For these easily understood lower dimensions, mathematicians have discovered many constant-width shapes with smaller volumes. In two dimensions, the Reuleaux triangle has the smallest area of constant-width shapes. You can draw this shape yourself using a sort of three-way Venn diagram. First, draw an equilateral triangle, then add three circles of equal radius around each corner. At the center of these circles, you’ll find a rounded shape that rolls like a circle with only a fraction of the size.

In three dimensions, you can use a similar method: start with a regular tetrahedron, a shape formed by four equilateral triangles, and add a sphere to each of its vertices. The resulting shape at the center of these overlapping spheres is known as the Reuleaux tetrahedron. It’s not exactly constant-width—it comes close, but the edges stick out too much. Some minor sanding down, however, gives rise to a constant-width shape. This can be done two different ways to form two slightly different shapes called Meissner bodies.

But the simple formulas behind these shapes offer no insights into how to build in four or more dimensions, which is beyond human perception. “It’s incredibly hard to generalize the Reuleaux construction,” Bondarenko says. “If it were easy, someone would have done it before.”

Wheels in Higher Dimensions

The latest work provides a general algorithm for constructing constant-width objects in any dimension by extending the Reuleaux intersection method. The team of mathematicians used a related Venn-diagram-like approach to yield the desired new shape—a geometrically anomalous nugget at the center of higher-dimensional space.

To represent this in two dimensions, again draw an equilateral triangle, followed by a circle centered around one of the triangle’s vertices, with a radius as long as each of the triangle’s legs. Then imagine moving that circle so that its center point follows the outline of the triangle, going up each leg and past each vertex before returning to where it started. As the circle moves, there are some places that it consistently occupies. This intersection of the infinitely many positions of the moving circle forms a familiar shape: the Reuleaux wheel, a generalization of the Reuleaux triangle. “By doing this, you essentially have the same construction as with the Reuleaux triangle, except here we take the intersection of infinitely many balls rather than just three or four,” explains Andrii Arman, a mathematician at the University of Manitoba and a co-author of the study.

This simple, computable technique can reveal an object of constant width in any dimension, so long as the boundary we drag our circle around is selected properly for each dimension. According to the new research, selecting this boundary in any dimension boils down to a basic recipe. In two dimensions, we trace the circle around a smaller quarter circle instead of an equilateral triangle. In three, we narrow this further from a fourth to an eighth of a sphere, and this pattern extends into higher dimensions by increasing powers of 2. Making such a boundary in n dimensions and moving a corresponding n -dimensional ball along it traces a higher-dimensional Venn diagram, the center of which the authors demonstrate must always have a width of exactly 2, yielding their new shape in any dimension.

This infinite, rather than finite, approach to building shapes not only ensures constant width but also makes computing their volume in higher-dimensional space straightforward. In comparison, prior constructions involve estimating an integral over many variables, while the latest work involves only two variables regardless of the shape’s dimension. “It’s really difficult to estimate volume in high dimensions,” Kalai says, yet “this whole [proof] is fairly simple and so elegant.”

The volume of the new object is 0.9 n times smaller than an n -dimensional ball, which means that the volume decreases exponentially with each additional dimension. While the shapes decrease in size at an increasing rate when moving into higher-dimensional space, they are not the smallest possible objects that maintain constant width. “It’s conjectured that the Meissner bodies possess the smallest possible volume” in three dimensions, Bondarenko says, adding, “Our result is only 0.14 percent bigger than that.”

Kalai suggests that creating these shapes in higher dimensions with an infinite series “may be the dawn of [a new] era in the study of sets with constant width.” With the original problem now verified, “we are in uncharted territory,” he concludes, but armed with these new methods, “there is some hope to tackle many new problems.”

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• Published: 09 August 2024

N-level complex helical structure modeling method

• Lijuan Zhao 1 ,
• Tianyi Zhang 1 ,
• Tianxiang Wang 1 ,
• Bo Xie 2 &
• Feng Gao 2  

Scientific Reports volume  14 , Article number:  18549 ( 2024 ) Cite this article

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• Engineering
• Mechanical engineering

This paper primarily explores the modeling method of n-level complex helical structures with coal mining machine cables as the research object. The paper first elaborately introduces the modeling method of n-level helix curves based on parametric equations and coordinate transformations, and compensates for the n-level helix curves with corrected pitch, which can obtain more accurate n-level helix curves and improve the accuracy of n-level helix curves modeling. Subsequently, based on this high-precision n-level helix curves modeling method, the paper elaborates on the method of solving pitch and twisting radius of multi-layer helical structure. Calculation scripts were written based on the above methods, which can be used to batch calculate the twisting radius and pitch of each layer structure in multi-layer structures when satisfying the conditions of in-layer tangency, inter-layer tangency, and extrusion deformation, and retain the actual results through logical judgment. Then, based on the above two methods, the paper developed a modeling method for braided structures based on piecewise functions containing fifth-order polynomials, which can effectively avoid the problem of insufficiently dense arrangement of braided lines and easy interference in traditional methods. Finally, a set of modeling tools was developed using C# and Python in Grasshopper to implement the modeling algorithm. Taking the MCPT-1.9/3.3 3120 + 170 + 4 * 10 coal mining machine cable as an example. The cable was modeled using both the method proposed in this paper and the traditional method. Comparative data shows that the method proposed in this paper can reduce errors by 3.31E6 times in the second-level and above helical structures. In addition this paper compares the standard line length, measured line length, and the line length established by the proposed model, showing that the relative errors are both less than 0.1941%. This paper provides a new, systematic, high-precision, and full-process cable modeling method, in which all parameters except the process parameters are accurately solved by equations. It lays a theoretical foundation for the high-precision simulation and intelligent sensing cables, which is of great significance for improving the safety, stability, and efficient development of the coal mining industry. The research results of the paper can not only be applied to the modeling of coal mining machine cables but also can be extended to the modeling of other complex multi-layer helical structures.

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Introduction.

In modern engineering, cables play an indispensable role in power transmission, communication, and control systems. Among these, mining machine cables are responsible for providing power to the coal mining machine and transmitting control signals. Improving the comprehensive performance of mining machine cables is of great significance for ensuring safe coal mining, stabilizing coal supply, and improving mining efficiency. It is also a crucial guarantee for achieving intelligent fully mechanized coal mining under complex conditions.

Dynamic simulation of coal mining machine cable characteristics is crucial for performance evaluation and design optimization to enhance durability and reliability. Accurate cable model construction is key in this regard.

Cables are complex n-level helical structures, and some progress has been made in the field of modeling. Wang et al. analyzed the structure of round strand wire rope and proposed a general mathematical model to describe the paths of conductors with single and second-level helix forms 1 . Jiang et al. established a finite element model of seven-strand wire rope to analyze the termination effect 2 , and subsequently proposed an accurate and general finite element model for wire rope to analyze the tensile, shear, bending, torsion, contact, and frictional effects during loading 3 . Wang Guilan et al. established the spatial curve equations of wires in double-twist wire rope based on differential geometry and analyzed the influence of typical twist combinations on the curvature of wires in the rope 4 , 5 , 6 . Elata et al. proposed a new model to simulate the mechanical response of independent wire ropes 7 . Erdonmez et al. proposed a more realistic three-dimensional modeling method for independent wire ropes with a secondary helix structure 8 , 9 , and then they discussed the geometry of the tertiary helix based on the modeling of primary and secondary helix structures and extended it to the construction of n-level helix equations 10 . Wu Juan et al. derived the spatial vector expression of secondary helix lines through the Frenet-Serret frame, constructed a finite element analysis model of wire rope, and studied the stress distribution of wire rope 11 . Hu Yujiao established a submarine cable model by establishing a centerline using key points and studied its mechanical behavior 12 . Zhongbin modeled prefabricated spiral joints using parameter equations and studied the changes in gripping force using Abaqus 13 . Zhang et al. constructed a three-dimensional model of coal mining machine cables using the trajpar function in ProE and analyzed the dynamic characteristics of coal mining machines using homogenization theory and the volume averaging principle 14 .

Although some scholars have made certain achievements in the field of modeling n-level complex helical structures, traditional modeling methods have shortcomings in systematically and comprehensively constructing high-precision complex multi-layer helical structures and braided structures. This study aims to introduce a systematic, full-process, high-precision method to model complex multi-layer helical structure cables represented by coal mining machine cables, in order to overcome the limitations of existing methods and improve the accuracy and practicality of modeling complex multi-layer helical structures.

Modeling method of n-level helix curve based on parameter equations

Cables are typical complex n-level helical structures, and their accurate modeling is a complex and challenging task. N-level helical structures are formed by n-level helix curves as centerlines, and one of the key aspects is to establish accurate n-level helix curves.

Method for constructing first-level helix curve

In the Cartesian coordinate system, the first-level helix curve is swept by points moving linearly along the z-axis while undergoing circular motion in the x-y plane (the normal plane of the z-axis), as depicted in Fig. 1 .

Schematic diagram of the first-level helix curve and its Frenet-Serret frame.

All points on this first-level helix curve can be represented by the parametric equation shown in Eq. ( 1 ).

In Eq. ( 1 ), \(\theta_{1}\) represents the phase angle of the first-level helix curve, \(t_{1}\) denotes the distance traveled by the point along the z-direction of the base coordinate, \(R_{1}\) is the twisting radius of the first-level helix curve, \(L_{1}\) is the helical pitch of the first-level helix curve, and when \(flg_{1}\) equals 0, it indicates left-handed helicity, while when \(flg_{1}\) equals 1, it indicates right-handed helicity.

Revised method for constructing second-level helix curves

The second-level helix curve is swept by points undergoing circular motion in the normal plane of the first-level helix curve. Utilizing the theory of differential geometry, a set of Frenet-Serret frames is established for the first-level helix curve. The Frenet-Serret frame is an important mathematical tool for describing the geometric properties of spatial curves. It is based on three mutually perpendicular unit vectors: tangent, normal, and binormal vectors. Taking the tangent vector of the curve as the z-axis, the principal normal vector of the curve as the y-axis, and the binormal vector as the x-axis, a frame is constructed, as illustrated in Fig. 1 .

Using the arc length of the first-level helix curve as the parameter, we establish the parametric equation. Here, \(t_{2}\) represents the distance traveled by the point in the direction of progression along the curve. \(R_{2}\) denotes the twisting radius of the second-level helix curve, and \(L_{2}\) represents the helical pitch of the second-level helix curve. \(t_{2}\) can be obtained by Eq. ( 2 ):

The direction vectors of the Frenet-Serret frame for the first-level helix curve are respectively given by Eqs. ( 3 ) and ( 4 ):

The coordinates of the second-level helix curve on the Frenet-Serret frame of the first-level helix curve are given by Eq. ( 5 ):

Using coordinate transformation, the coordinates of the first-level helix curve's Frenet-Serret frame are transformed to the base coordinate system. The coordinates of the second-level helix curve in the base coordinate system are given by Eq. ( 6 ):

In Eq. ( 6 ):

The arc length of the constructed second-level helix curve is given by Eq. ( 8 ):

Using this conventional approach, the parameters are assigned according to Eq. ( 9 ) second-level helix curves are constructed separately for \(flg_{2} = 0\) and \(flg_{2} = 1\) , and their lengths are calculated. The second-level helix curves are illustrated in Fig. 2 . The arc length of the left-handed second-level helix curve is 65.813 mm, while the arc length of the right-handed second-level helix curve is 54.611 mm. Additionally, we can calculate the standard arc lengths of the first-level helix curve and the second-level helix curve as 37.242 mm and 59.809 mm, respectively, using Eq. ( 10 ).

presents a comparison of the second-level helix curves before and after introducing the corrected pitch.

Research has found that the lengths of the left-handed and right-handed second-level helix curves are not equal and do not match the theoretical arc length. The reason for this phenomenon is that the Frenet-Serret frame undergoes rotations in three directions as it progresses along the curve. These rotations are superimposed on the next-level helix, resulting in differences in arc length between the left-handed and right-handed next-level helix curves, and neither of them matches the theoretically calculated arc length. On the other hand, the first-level helix curve can be considered to exist on a set of frames along the central axis, which is a straight line. The axes of this set of frames are mutually parallel, without bending or twisting. Therefore, the arc lengths of the left-handed and right-handed first-level helix curves are both equal to the standard arc length.

Taking the coal mining machine cable as an example, according to its manufacturing process, all helical structures are twisted into the cable under the condition of first-level twisting. The method proposed in this paper, which adjusts the pitch in non-first-level helix curves to compensate for the lengths of double and higher helix curves, can effectively solve the distortion problem of the twisted structure model caused by using traditional methods to construct second-level helix curves and higher helix curves.

For the second-level helix curve, we replace \(L_{2}\) in Eq. ( 8 ) with the unknown variable \(L_{2}{\prime}\) .We construct an integral equation using \(LEN_{2}\) and \(LSUM_{2}\) , and solving this integral equation yields the corrected pitch \(L_{2}^{\prime}\) .The constructed integral equation is shown in Eq. ( 11 ):

After solving for the corrected pitch, we can substitute it back into Eq. ( 6 ) to obtain the corrected second-level helix curve with compensated length, as shown in Fig. 2 . Assigning values to each parameter according to Eq. ( 9 ), the numerical solutions for the corrected pitch of the left-handed and right-handed second-level helix curves are found to be 10.106 mm and 8.746 mm, respectively.

Generalization of corrected second-level helix curves to corrected n-level helix curves

Similarly, the coordinates of the n-level helix curve on the Frenet-Serret frame twisted at the (n-1)-level are given by Eq. ( 12 ):

Using coordinate transformation, the coordinates of the n-level helix curve are transformed from the (n-1)-level Frenet-Serret frame to the base coordinate system. The equation for the n-level helix curve in the base coordinate system is given by Eq. ( 13 ):

As shown in Fig. 3 , according to the cable manufacturing process, we can recursively calculate the standard length \(LEN_{n}\) of each level helix curve using Eq. ( 14 ).

The recursive calculation method for the standard length of the helix curve.

Based on the equation of the n-level helix curve in the base coordinate system and the standard length of the n-level helix curve, we can construct an integral equation to solve for \(L_{n}^{\prime}\) . The integral equation is given by Eq. ( 15 ):

Finally, bringing \(L_{n}^{\prime}\) back to \({}_{ }^{B} P_{n}\) yields the compensated length of the n-level helix curve. Taking the third-level helix curve as an example, assigning values to each parameter according to Eq. ( 16 ) and establishing the third-level helix curve and the corrected third-level helix curve, as shown in Fig. 4 . The comparative data are presented in Table 1 . The results indicate that after introducing the corrected pitch, the length error of the second-level helix curve decreases from 5.084106 to −0.98E−4 mm, and the length error of the third-level helix curve decreases from −3.5902 to 1.89E−4 mm.

Comparison diagram of the third-level helix curve.

The establishment of the helix curve is a recursive process. A second-level helix curve is obtained recursively from the first-level helix curve using the aforementioned algorithm, and a third-level helix curve is obtained recursively from the second-level helix curve using the same algorithm. Similarly, an n-level helix curve is obtained recursively from the (n-1)-level helix curve using the aforementioned algorithm. Therefore, we can easily extend a third-level helix curve to an n-level helix curve recursively in the same manner. Due to space limitations, we only present detailed data for the first-level to third-level helix curves. To supplement this, we compared the line length accuracy of the fourth-level to tenth-level helix curves. The results show that the line length error of the tenth-level helix curve established using the method in this paper is still within the order of 1e−4 compared to the theoretical line length. This indicates that the helix curves of the third level and above established using the method in this paper can still ensure good accuracy.

In summary, by introducing the corrected pitch obtained through the integral equation, it is possible to effectively compensate for the length errors of multi-level helix curves caused by the rotation of the Frenet-Serret frame, providing an accurate and reliable basis for cable modeling. However, the aforementioned methods for modeling multi-level helix curves are all based on known values of the twisting radius and pitch. In practical engineering, it is often impossible to directly obtain the actual values of the twisting radius and pitch during the cable manufacturing process. Therefore, it is necessary to calculate the twisting radius \(R_{n}\) and pitch \(L_{n}\) of each level helix curve through geometric relationships and cable process parameters, and calculate the standard length \(LEN_{n}\) of each level helix curve based on the twisting radius and pitch.

Method for determining pitch and twisting radius of multi-layer helical structures

In multi-layer helical structures modeling, determining the pitch and twisting radius of each helix based on the actual object is indispensable. Taking a certain type of mining machine cable as the engineering object, its core line, power unit, and control unit are all relatively dense multi-layer helical structures. The core line is composed of multiple layers of fine wires, while the power unit and control unit are composed of multiple layers of core lines, as shown in Fig. 5 .

Cable structure diagram.

Taking the core line as an example, the twisting radius of each layer of helical structure is related to the diameter of the wires and the number of wires in each layer. Traditional modeling methods generally calculate the twisting radius of each layer of helical structure through a circle tangent to each other 15 , 16 , 17 , while the pitch during modeling is often variable. Although this method is convenient, the cross-section of each wire on the overall plane of the core line is elliptical due to the different normal planes of each wire, as shown in Fig. 6 . When tangential within the layer, if the twisting radius is calculated roughly as a circle according to the cross-sectional pattern, it will result in a smaller calculated twisting radius and interfere with the model, thereby reducing the accuracy of the model.

Twisted wire section pattern.

As shown in Fig. 7 , there are mainly three distributions of twisted wires in multi-layer twisted structures: (1) meeting the condition of “in-layer tangency”, that is, the twisted wires within the layer are tangential to each other; (2) meeting the condition of “inter-layer tangency”, that is, each twisted wire in the layer is tangential to the outer circle of the previous layer; (3) meeting the condition of “compression deformation”, that is, the mutual compression of twisted wires within this layer leads to the deformation of the insulation part.

Schematic diagram of the distribution of twisted wires.

Calculation of twisting radius and pitch for first-level helix curve multi-layer twisted structure

First-level helix curve multi-layer twisted structure meeting the condition of “in-layer tangency”.

Assuming that under the condition of “in-layer tangency”, the mth layer is composed of n m strands of twisted wires. The long axis of the ellipse is \(r_{p\_m}\) , the short axis is \(r_{x\_m}\) (wire radius), the pitch is \(L_{m}\) , and the twisting radius is \(R_{m}\) . As shown in Fig. 6 , when there is in-layer tangency, solving the twisting radius of the mth layer twist structure essentially involves finding the radius of the circle formed by the centers of \(n_{m}\) ellipses when they are tangent to each other. The shape of the ellipse determines the twisting radius \(R_{m}\) , and this shape is also related to the pitch and twisting radius of the twisted wire, so it is necessary to establish a system of equations for solving.

As shown in Fig. 8 a and b, we can flatten the cylinder surface where the wire centerline lies to determine the inclination angle \(\varphi_{m} = {\text{arctan}}\left( {L_{m} /2\pi R_{m} } \right)\) of the wire using geometric relationships. We establish a coordinate system \(f\) at the center of one ellipse, with the coordinates of a tangent point on the ellipse denoted as \(\left( {{}_{ }^{f} x_{m} ,{}_{ }^{f} y_{m} } \right)\) , as shown in Fig. 8 c. By using the relationship between \(r_{p\_m}\) and \(r_{x\_m}\) , the ellipse equation, the coordinates of the ellipse tangent point, and the slope of the tangent line to the ellipse, we can construct a system of equations based on geometric relationships and relationships between variables to solve for \(L_{m}\) and \(R_{m}\) . Additionally, the cable industry generally uses the outer diameter \(R_{o\_m}\) and the pitch-diameter ratio \(j_{m}\) to describe the twist structure. To better match engineering practice, we introduce the pitch- diameter ratio, a parameter widely used in the industry. With the introduction of the pitch-diameter ratio, \(L_{m}\) and \(R_{m}\) can be solved by Eq. ( 17 ).

Schematic diagram of the geometric relationship of stranded wires on the m layer.

Using the above system of equations, we only need to input the pitch-diameter ratio, wire radius, and the number of wires in the mth layer to calculate the twisting radius and pitch of the mth layer in the multi-layer twisted structure when it satisfies the condition of “in-layer tangency” within the layer.

First-level helix curve multi-layer twisted structure meeting the condition of “inter-layer tangency”

When satisfying the condition of “inter-layer tangency”, i.e., when the mth layer is tangential to the (m-1)th layer, the twisting radius of the mth layer is equal to the radius of the outer envelope circle of the cross-section of the (m-1)th layer plus the wire radius of the mth layer. As shown in Fig. 9 , which represents a particular case where \(R_{m} \ne R_{m - 1} + r_{x\_m} + r_{x\_m - 1}\) , we can derive a general equation for calculating the twisting radius of the mth layer based on the twisting radius and pitch of the (m-1)th layer, as shown in Eq. ( 18 ).

Schematic diagram of the outer envelope circle of layer m-1.

The method for solving the pitch in this case is shown in Eq. ( 19 ).

First-level helix curve multi-layer twisted structure meeting the condition of “compression deformation”

The method for solving the pitch and the twisting radius under the condition of “compression deformation”, that is, when the insulation undergoes deformation due to compression, is shown in Eq. ( 20 ).

Using the three methods mentioned above, we can calculate the twisting radius and pitch of first-level helix multi-layer twisted structures under conditions of “in-layer tangency”, “inter-layer tangency”, and “compression deformation”. When the sum of the twisting radius calculated using the “in-layer tangency” condition and the wire radius is greater than the difference between the outer radius and the wire radius, it indicates that “compression deformation” conditions are met. In cases where “compression deformation” conditions are not met, if the twisting radius calculated using the “in-layer tangency” condition is less than the twisting radius calculated using the “inter-layer tangency” condition, it indicates that the wires in the mth layer are not sufficient to completely cover the (m-1)th layer when each wire is in tangency. In this case, the data calculated using the “in-layer tangency” condition should be discarded, and the data calculated using the “inter-layer tangency” condition should be used instead. Conversely, if the twisting radius calculated using the “in-layer tangency” condition is greater than the twisting radius calculated using the “inter-layer tangency” condition, it indicates that the wires in the mth layer sufficiently cover the (m-1)th layer when each wire is in tangency. In this case, the data calculated using the “in-layer tangency ”condition should be used.

For the convenience of batch calculation of multi-layer twisted structures, we have developed a script to compute the required pitch and twisting radius for modeling purposes. This script takes as input the number of winding layers, arrays representing the number of wires per layer, arrays representing the wire radius per layer, arrays representing the pitch-diameter ratio per layer, and arrays representing the outer diameter per layer. The script can automatically calculate the twisting radius, pitch, and distribution condition of windings for each layer in the first-level helix multi-layer twisted structure. The program flowchart of the script is illustrated in Fig. 10 .

Flowchart of the calculation script of multi-layer twisted structure.

Since the pitch-diameter ratio data is obtained before each level of winding is stranded into cable, utilizing the aforementioned script allows us to obtain the pitch and twisting radius of each layer's structure before cable-laying. Furthermore, by utilizing the twisting radius and pitch before stranding into cable, we can recursively solve for the standard wire length of each part of the twisted structure using formula ( 14 ).

Calculation of twisting radius and pitch for n-level helix curve multi-layer twisted structure

N-level helix curve multi-layer twisted structure is a winding structures which center line is (n-1)-level helix curve, where the center line of each wire in this structure is an n-level helix curve. Fig. 11 shows the first-level helix curve multi-layer twisted structure and the second-level helix curve multi-layer twisted structure. Sections " Revised method for constructing second - level helix curves " and " Generalization of corrected second - level helix curves to corrected n - level helix curves " mention that for second-level and higher-level helix curve, integral equations are needed to solve the corrected pitch. However, this corrected pitch is intended to compensate for errors caused by the twisting and bending of the Frenet-Serret frame. Therefore, after introducing the corrected pitch, the inclination of the wires in the n-level helix curve twisted structure relative to the overall center line (the pitch angle of the wires) will not change compared to before stranding. Therefore, the cross-sectional shape obtained by the normal plane of the overall center line before and after stranded into cable will not change. In summary, the calculation of the twisting radius of the multi-layer twisted structure after stranded to cable can still use the calculation script in Fig. 10 , and the corrected pitch can be directly solved using formula 15 .

Fiist-level and second-level helix curve multi-layer twisted structure.

Construction of the braid structure mode

The braided layer of the cable is a cross-woven structure made of metal wires or synthetic materials, covering the outer surface of its insulation layer. This layer is mainly used to provide electromagnetic shielding and mechanical protection, preventing electromagnetic interference from affecting the cable signal, and increasing the durability of the cable.

Traditional modeling of braided structures generally uses rose curves to describe the centerline of the braided structure. For general braided structures, the sine function in the rose curve can be replaced with a piecewise function of sine function and circular arc 18 , or the circle in the sine function of the rose curve can be replaced with an ellipse 19 to transform the traditional rose curve into a modified rose curve, ensuring that the peaks of the rose curve have a certain width to accommodate multiple strands of braided wires arranged side by side. However, in engineering practice, we found that both methods have defects. The approach based on the sine function and circular arc needs to separately discuss single-strand braided structures and multi-strand braided structures, and interference may occur in the rising and falling segments of the braided wire when the braided wire is thick. As for the approach based on the modified rose curve, the braided structure it establishes will have a certain gap between the upper and lower layers at the intersection segment, and it is difficult to calculate a suitable major axis for the ellipse. To solve these problems, this paper proposes a piecewise function composed of a fifth-degree polynomial and circular arc to replace the rose curve in traditional methods. The fifth-degree polynomial allows direct control of the starting and ending points of the path, as well as their tangent directions and curvature, making precise adjustments to the path more convenient with such endpoint control.

The essence of the braided wire modeling method in this paper is to superimpose a piecewise function that controls oscillations on the base helix curve. Before establishing the braided wire, it is necessary to first solve for the pitch and twisting radius of the basic helix. The pitch of the base helix curve is denoted as \(L_{b}\) and the twisting radius as \(R_{b}\) , which can be determined by Eq. ( 21 ). In Eq. ( 21 ), \(R_{o\_b}\) represents the outer diameter of the braided structure, and \(j_{b}\) represents the pitch-diameter ratio of the braided structure. The base helix curve and the braided wire are shown in Fig. 12 .

Braided structure base helix curve and braided wire.

As shown in Fig. 13 , the piecewise function controlling oscillations consists of 3 arc segments and 2 quintic polynomial segments. The braided structure includes both left-hand and right-hand braided wires, with the number of clusters in each direction denoted as p and the number of wires per cluster as q. The period of the piecewise function controlling oscillations is \(2\pi /p\) . The cross-sectional shape of the braided wire arc segment on the centerline normal plane of the braided structure is an ellipse, with the major and minor axes corresponding to the radius of the braided wire \(r_{x\_b}\) and the major axis being \(r_{p\_b}\) .

Schematic diagram of piecewise function.

Additionally, for the n-level helix curve braided structure (a braided structure with the centerline being an n-1 level helix curve), it is also necessary to introduce integral equations to solve for the corrected pitch \(L_{b} {^{\prime}}\) . The corrected pitch of the basic helix curve of the n-level helix curve braided structure after stranding can be determined using Eq. ( 15 ).

Similar to Eq. ( 17 ), the central angle \(phase\_b\) occupied by each braided wire on the normal plane of the centerline in the braided structure can be determined using the equation in Eq. ( 22 ):

As shown in Fig. 14 , after projecting all arc lengths onto the middle layer of the braid, the arc length \(r_{t\_b}\) occupied by the width of a single left-handed braided wire on the right-handed braided wire can be determined using Eq. ( 23 ), where \(\varphi_{b}\) represents the inclination angle of the braided wire.

Schematic diagram of braided wire occupying arc length.

The angle \(phase\_x\) at which \(r_{t\_b}\) is converted into the rotation angle of the braided wire around the center line can be determined using Eq. ( 24 ):

Referring to the method described in Eq. ( 12 ), the coordinates of the \(q_{x}\) braided wire in the \(p_{x}\) group on the Frenet-Serret frame of the center line can be represented as \({}_{ }^{{F_{b - 1} }} P_{{b\_p_{x} q_{x} }}\) , as shown in Eq. ( 25 ). When \(flg_{b} = 0\) , it represents a left-handed braided wire, while \(flg_{b} = 1\) represents a right-handed braided wire, where \(p_{x} = 0\sim p - 1\) , \(q_{x} = 0\sim q - 1\) .

In Eq. ( 25 ), \(r\_b\left( \alpha \right)\) is a segmented periodic function that controls the up-and-down fluctuations of the braided wire, and it can be calculated using Eq. ( 26 ).

In Eq. ( 25 ), \(a,b,c,d,e,f\) are the coefficients of the fifth-degree polynomial, which can be solved using the system of equations given in Eq. ( 27 ).

In Eq. ( 25 ), \(\theta_{b} \left( {p_{x} ,q_{x} } \right)\) is a parameter that determines the position of the weaving line along the overall circumference of the structure and can be calculated using Eq. ( 28 ).

In Eq. ( 25 ), \(\theta_{z} \left( {q_{x} } \right)\) is a parameter controlling the phase of the weaving line's oscillation and can be calculated using Eq. ( 29 ).

Finally, the coordinates in the Frenet-Serret frame need to be transformed into the base coordinate system using a coordinate transformation method. By applying the method described in Eq. ( 13 ), the coordinates of the weaving line in the base coordinate system can be obtained, as shown in Eq. ( 30 ).

Based on the above method, modeling of both single-strand and multi-strand weaving structures can be easily handled. As shown in Fig. 15 , all parameters except for the process parameters are obtained through equations. In comparison to traditional methods, the modeling approach proposed in this paper does not differentiate between single-strand and multi-strand weaving structures. It is more unified and, at a sufficient density, avoids interference phenomena.

Various braided structures.

Comparison of modeling results

We utilized the algorithm proposed in this paper to develop a complete modeling toolset using C# script combined with Python script in Grasshopper. This toolset efficiently solves the equations in the algorithm using libraries such as numpy and scipy in Python script, and then builds the model efficiently using C# script, balancing functionality and modeling efficiency. Figure 16 illustrates the process flow of modeling a certain type of cable using this modeling toolset.

Flowchart of modeling a certain type of cable.

Specifically, this toolset primarily includes the "Twisting radius and pitch calculation tool", "Helix curve generation tool", "Braided structure generation tool" and "Insulation layer generation tool" as illustrated in Fig. 17 . These tools are described as follows:

"Twisting radius and pitch calculation tool": This module leverages Python to solve the equations presented in Section " Method for determining pitch and twisting radius of multi-layer helical structures ", enabling the calculation of twisting radius and pitch based on known parameters such as the "diameter-to-pitch ratio."

Modules within the modeling toolset.

“Helix curve generation tool” : Utilizing C# scripts, this module implements the corrected helix curve generation algorithm detailed in Section " Modeling method of n-level helix curve based on parameter equations ". The script calculates points on the corrected helix curve and generates the helix curve through these discrete points, thereby creating an n+1 level corrected helix curve from an n-level corrected helix curve.

"Braided structure generation tool" : This module employs both Python and C# scripts to realize the braiding line generation algorithm described in Section " Construction of the braid structure mode ". Python is used to solve the parameters of the quintic polynomial and the central angle occupied by the braiding line (phase_b). C# is then used to generate discrete points on the modified braiding line, which are used to construct the braiding line.

"Insulation layer generation tool" : This module utilizes built-in functions within Grasshopper to generate multiple insulation layers with non-standard cross-sections. It can batch-generate multiple hollow insulation layers that press against each other.

Using both the modeling method proposed in this paper and the traditional modeling method, we modeled the MCPT-1.9/3.3 3120 + 170 + 4 * 10 type of shearer cable with the folded yarns as the smallest unit, with a modeling length of 480mm (one pitch lenth). The process parameters of the cable are shown in Table 2 , and the modeling parameters obtained using the method proposed in this paper are shown in Table 3 . Thirteen different centerlines were selected (as shown in Fig. 18 ) to compare the errors of the two modeling methods, and the data are shown in Table 4 . The error comparison between the traditional method (“Traditional error”) and the method proposed in this paper (“Paper error”) is shown in Fig. 19 .

MCPT-1.9/3.3 3 * 120 + 1 * 70 + 4 * 10 coal mining machine cable rendering.

The traditional method and this paper’s method error comparison.

From Table 4 and Fig. 19 , it can be observed that the error between the length of the first-level helix curve and the standard wire length in the models established by both methods is within 1e−4 mm, indicating that there is little difference in accuracy between the two methods for the first-level helix curve, and both methods achieve high accuracy. For the second-level and higher-level helix curves, the length accuracy of the modeling method using corrected pitch and twisting radius introduced in this paper can still be maintained below 1e−4 mm. In contrast, for the traditional modeling method without introducing corrected pitch and twisting radius, the maximum error can reach −149.606188 mm, a difference in maximum error of 3.31E6 times. This demonstrates that the research method proposed in this paper can effectively reduce modeling errors and maintain a high level of accuracy in the model.

Alongside this, we cut a 1000 mm section of MCPT-1.9/3.3 3120 + 170 + 4 * 10 coal mining machine cable and dissected it. Each dissected part was measured five times, and the average value was taken to obtain the measured length. We organized the standard line length values (“Standard length”), measured line length values (“Measured length”), and the line length values obtained using the method in this paper (“Paper length”). Absolute and relative errors between the measured line length and the line length obtained using this paper's method (“Measure absolute error”, “Measure relative error”, “Paper absolute error” and “Paper relative error”) were calculated and summarized, as shown in Table 5 . The comparison of the standard line length, measured line length, and the line length obtained using the method in this paper is shown in Fig. 20 .

Comparison of Standard length, measured length and paper length.

The data in Table 5 indicate that the "Measure relative error" is all below 0.195%, demonstrating that the measured lengths are relatively accurate. The "Paper relative error" is all below 0.22 parts per million, indicating that the cable model lengths established using the method in this paper have a high level of accuracy. As depicted in Fig. 20 , comparing the values of "Standard length," "Measured length," and "Paper length," the relative errors between "Paper length" and both "Standard length" and "Measured length" are all less than 0.1941%. This indicates that the model established by the method in this paper exhibits good fidelity and high accuracy. This demonstrates that the method presented in this paper can effectively reduce the errors of traditional methods and significantly improve the authenticity of the model.

The paper first elaborates on the parametric modeling method of n-level helix curves and proposes compensating n-level helix curves with corrected pitches. The errors between the lengths of the second-level and third-level helical curves before and after introducing the corrected pitch are compared, and the comparative results show that the proposed research method in this paper can obtain accurate n-level helix curves, solving the problem of inaccurate center lines in n-level twisted structures. Next, equations were established for multi-layer twisted structures under conditions of “in-layer tangency”, “inter-layer tangency”, and “compression deformation”, respectively. The twisting radius and pitch under these three conditions were obtained by the pitch-diameter ratio, and an automatic calculation script was written to batch calculate the twisting radius and pitch of each layer structure in multi-layer twisted structures. Subsequently, we proposed a modeling method for braided structures based on a segmented function composed of fifth-degree polynomials and circular arcs, replacing the sine function in traditional braiding modeling methods. This method effectively avoids issues of insufficient contact between braided lines and interference between braided lines encountered in traditional methods. Finally, we developed a modeling tool using C# and Python in Grasshopper to implement the modeling algorithm. Taking the MCPT-1.9/3.3 3120 + 170 + 4 * 10 coal mining machine cable as an example, we modeled it using both the research method proposed in this paper and the traditional method. Comparative results show that the method proposed in this paper can reduce errors by up to 3.31E6 times in second-level and higher-level helical structures. Additionally, this paper compares the standard line length, measured line length, and the line length established by the model developed using the method proposed in this study. The results indicate that the relative errors between the line length established by the proposed method and the standard line length, as well as the measured line length, are both less than 0.1941%.This indicates that this paper provides a new, systematic, and high-precision modeling method for multi-level complex twisted structures, laying the foundation for high-precision modeling and simulation of cables. This method eliminates the manual adjustment of modeling parameters in traditional methods. Except for process parameters, all other parameters are automatically derived from equations, simplifying the modeling process while improving accuracy and efficiency. Moreover, the research results of this paper can be applied not only to the modeling of coal mining machine cables but also to the modeling of other complex multi-layer twisted structures.

The results of this study have been applied to the optimization of cable structures. Through mechanical performance simulation and electromagnetic performance simulation, we have successfully enhanced the overall performance of the cables. The optimized cables have demonstrated significantly improved electromagnetic characteristics and are expected to have a longer service life. Furthermore, the results of this study will be further extended to the field of fiber optic sensing cables, providing a theoretical foundation for constructing mathematical models and developing information calculation algorithms. This lays a solid foundation for the realization of intelligent sensing cables and contributes key technologies for the comprehensive intelligentization of underground operations. Therefore, this study is of profound theoretical and practical significance for enhancing the safety and stability of the coal industry and promoting its transformation towards high-quality development.

Data availability

All data generated or analysed during this study are included in this published article and its supplementary information files.

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Exploring two new iterative methods for solving absolute value equations

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Numerous challenges arising in diverse fields such as scientific computing, operations research, management science, and engineering can be addressed by solving absolute value equations (AVEs). This paper introduces two innovative methods called the fixed point method and the modified generalized Gauss-Seidel method for solving AVEs. We delve into the convergence analysis of these methods under appropriate assumptions. Furthermore, we conduct three sets of numerical experiments to assess the practical feasibility of our proposed methods. The results obtained are encouraging and pave the way for further study in this domain.

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Ali, R., Zhang, Z. Exploring two new iterative methods for solving absolute value equations. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02211-3

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