Applied Mathematics is a way to use math to solve problems in the real world, like in economics, engineering, and medicine. 

The purpose of my project is to see what would happen if a basic sine function is added to the graph of a circle or an ellipse. I believe that a pattern will be created according to how the circle "bumps" around the curve. 

What happens when you add a sine wave to the equation of a circle? You use graphing software to overlay a modified circle on top of the original. The sine function creates bumps that bulge outside the curve.

You count the bumps for each period value up to 25. The number of bumps is always odd. As the period increases, the count either stays the same or jumps by two or four. You test circles with radii from two through six and repeat the process with ellipses.

The bump count follows a formula based on the radius and the period of the sine function. When you increase the amplitude instead of the period, the bump count stays the same but the shape begins to break apart.

Project Title: Morphing Circles

Objectives/Goals

The purpose of my project is to see what would happen if a basic sine function
is added to the graph of a
circle or an ellipse. I believe that a pattern will be created according to how
the circle "bumps" around the
curve.

Methods/Materials

Using the computer software "Nucalc," I graphed a regular circle. I overlaid it
with a circle with the same
radius but with a sine function added to the equation of that circle. I noticed
that adding a sine function to
a circle creates bumps, so I defined a "bump" as a curve that is outside but
still intersecting the original
circle. I made t-charts to record the amount of bumps I saw as I increased the
period of the sine function. I
recorded the number of bumps up to the 25th period. I made t-charts using
circles with radii of two
through six. With ellipses, I followed the same procedures, and I used ellipses
of different sizes for my
t-charts.

Results

As I made my t-charts, I noticed that as the period increased, the number of
bumps either stayed the same,
increased by two, or increased by four. The number of bumps was always an odd
number. When I
overlaid the "morphed" circle with the sine curve graphed separately, I saw that
the number of times the
sine curve crossed the x-axis while inside the original circle was equal to the
number of bumps the
morphed circle has. In addition, when I increased the amplitude of the sine
function that was added to the
circle, the number of bumps stayed the same, but the morphed circle began to
break apart.

Conclusions/Discussion

My data supports my hypothesis because numerical patterns did exist in my
t-charts. I found out that the
number of bumps on a circle uses the greatest integer function and 2rb/Pi, where
r is the radius of the
circle, and b is the period of the added sine function. With ellipses, the
number of bumps uses the greatest
integer function and 2cb/Pi, where c is the radius along the x-axis for the
ellipse. Based on all the data and
research I have gathered, I was able to see a relationship between the number of
bumps a sine curve
creates on a circle when added together and the period of the sine function.

Summary Statement

My project is about what would happen if a sine function is added to a circular
graph.

Help Received

My mother helped cut out some of my work for my board, my project advisor, Diana
Herrington,
proofread my work and provided me with some necessary materials, and my father
provided the
transportation for me to go buy the needed materials for my project.

Sine Waves Added to Circles

Morphing Circles

<h1><strong>1000 Science Fair Project Ideas by Subject</strong></h1><p>Looking for a science fair project idea? Each of our science fair project idea has complete instructions with video. Get ideas for your science projects, find extra resources for your class' scientific projects, or try a fun experiment at home! It doesn't matter what grade you're in, there's something for everyone!</p><p>You can find your perfect science fair project from the different science subjects below or from our <a href="https://www.all-science-fair-projects.com/category0.html#bottom">science fair project idea lists!</a></p>

<h2>Science Fair Project Idea Lists</h2><p>Looking for other ways to find good ideas for science fair projects? Check out these lists of science projects!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>Easy Science Fair Projects</h3><p><img src="/images2/flex/176_popular 2 (2).png" data-size="100" data-float="left" style="width:100px; float:left">Explore the exciting world of science with an easy science fair project that you can do at home using simple materials. Any one of these easy project ideas will teach you important scientific concepts and help you have fun while learning. An easy science fair project is a great way to introduce the wonders of science to kids and adults alike!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><ul><li><p>Changing the Color of Flowers</p></li><li><p>Baking Soda Volcano</p></li><li><p>Swimming Raisins</p></li><li><p>Egg in Vinegar</p></li><li><p>Color of Skittles Experiment</p></li><li><p>Jumping Rice Krispies</p></li><li><p>Mixing Colors of Light</p></li><li><p>Camouflage and Hide</p></li><li><p>Light and Dark on Seed Germination</p></li><li><p>Make a Parachute</p></li></ul><p>More science project details at <a href="https://www.all-science-fair-projects.com/science_fair_project_list/12">Easy Science Fair Projects <strong>→</strong></a></p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>The Top 10 Best Science Fair Projects</h3><p><img src="/images2/flex/177_popular 2 (3).png" data-size="100" data-float="left" style="width:100px; float:left">To find the best science fair project for you, we've picked the best science projects from 10 different science categories, all with very interesting topics. If you're looking for more science projects at another difficulty level or topic, you can explore science fair project ideas by grade level at the top of this page!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><ul><li><p>Jet Lag in Hamsters (circadian rhythm, Biology science fair project)</p></li><li><p>Egg with Vinegar (acid-base reaction, Chemistry science fair project)</p></li><li><p>Galileo's Inclined Plane Experiment (acceleration, Physics science fair project)</p></li><li><p>Plant's Favorite Color for Photosynthesis (Photosynthesis, Botany science fair project)</p></li><li><p>Color and Taste (color perception, Psychology science fair project)</p></li><li><p>Does Dog Saliva Kill Bacteria? 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With just a few simple things, you can be fascinated by science and have tons of fun with these popular science projects!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><ul><li><p>Explosive Elephant Toothpaste</p></li><li><p>Egg in Vinegar</p></li><li><p>Electrifying Lemon Battery</p></li><li><p>Making Oobleck: Liquid or Solid?</p></li><li><p>Egg Drop Project</p></li><li><p>Colors of Skittles Experiment</p></li><li><p>Cloud in a Bottle</p></li><li><p>Baking Soda Volcano</p></li><li><p>Egg in a Bottle</p></li><li><p>Plastic Milk</p></li></ul><p>Project details and more cool science projects at <a href="https://www.all-science-fair-projects.com/science_fair_project_list/1/Popular_Science_Experiments_At_Home">Popular Science Fair Projects <strong>→</strong></a></p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>One-Day Science Fair Project Ideas</h3><p><img src="/images2/flex/172_popular 2 (1).png" data-size="100" data-float="left" style="width:100px; float:left">Quick experiments can be a great option for a science fair project. In this list, we've highlighted one-day science fair project ideas for elementary school (easy), middle school (medium), or high school (hard). These experiments are fast and easy to do, but the science topic packs a punch!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h4>Acid-base reactions: Mixing acids with bases quickly makes carbon dioxide gas!</h4><ul><li><p><a href="https://www.all-science-fair-projects.com/project830.html"><strong>Rocket Film Canisters (easy) </strong></a>Make a rocket fly sky high with just baking soda and vinegar!</p></li><li><p><a href="https://www.all-science-fair-projects.com/project829.html"><strong>Make Your Own PH Indicator (medium) </strong></a>Learn how to make your own acid-base indicator using red cabbage and use it to test different fluids!</p></li><li><p><a href="https://www.all-science-fair-projects.com/project1327.html"><strong>Testing Antacids (hard) </strong></a>Find out which antacid can neutralize the most acid and get relief from heartburn and gastritis!</p></li></ul><h4>Chromatography reactions: Separating out colors as liquid moves through the paper!</h4><ul><li><p><a href="https://www.all-science-fair-projects.com/project826.html"><strong>Colors of Skittles Experiment (easy) </strong></a>Did you know that some Skittles have a secret rainbow hiding inside them?</p></li><li><p><a href="https://www.all-science-fair-projects.com/project678.html"><strong>Uncovering Fall Colors (medium) </strong></a>Have you ever wondered why leaves change color in the fall? 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Science museums, natural history museums, space museums, and discovery museums are all great places to do that! You can discover all sorts of interesting things about science and find cool project ideas for science fairs.</p><p>There may be free admission days or free passes to a science museum near you! Check your credit card for offers, your local library for free museum passes, and nearby science museums for free entrance days.</p><p>So what are you waiting for? <a href="https://en.wikipedia.org/wiki/List_of_science_museums">Find a science museum near you</a> and prepare to be awed by all that you can learn there!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h4>Look around and observe</h4><p><img src="/images2/flex/117_5.png" data-size="100" data-float="left" style="width:100px; float:left">Do you know how scientists start their cool projects? They often look around and notice things that are interesting or make them curious. 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Check them out at <a href="https://www.all-science-fair-projects.com/science_fair_project_list/12">Easy Science Fair Projects <strong>→</strong></a></p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3 style="text-align: start"><strong>How do I start a science fair project?</strong></h3><p><img src="/images2/flex/199_science coach.png" data-size="100" data-float="left" style="width:100px; float:left">No matter if you're in high school, elementary or middle school, science fair projects are a fun and unique way to discover science by asking a question and doing a science project. We'll help you find a science fair project idea and guide you through the process of doing and presenting your science fair project with <a href="https://www.all-science-fair-projects.com/science_fair_projects_coach/science-project-coach.html">Science Fair Coach's Science Fair Project Guide</a>.</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>What should I do after I have a science fair project idea?</h3><p><img src="/images2/flex/143_intro science fair.png" data-size="100" data-float="left" style="width:100px; float:left">If you have a science fair project idea, Science Fair Coach can help you turn it into a full science fair project! Starting at the <a href="https://www.all-science-fair-projects.com/pgpage/4">Introduction to Science Fairs</a>, check out the section on <strong>How to do a science fair project</strong> for a step-by-step guide!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><p><img src="/images2/flex/203_compared to scientific method.png" data-size="100" data-float="left" style="width:100px; float:left">We'll guide you through every step of the <a href="https://www.all-science-fair-projects.com/pgpage/5">scientific method</a>, from designing and conducting your experiment to collecting and analyzing your findings.</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><p><img src="/images2/flex/202_engineering design process.png" data-size="100" data-float="left" style="width:100px; float:left">If you're working on an engineering project, you can head over to <a href="https://www.all-science-fair-projects.com/pgpage/9">The Guide to the Engineering Design Process</a>.</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><p> <img src="/images2/flex/150_science fair poster.png" data-size="100" data-float="left" style="width:100px; float:left">Finally, we'll help you showcase your results on a <a href="https://www.all-science-fair-projects.com/pgpage/7">science fair board</a> for others to learn from and enjoy.</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3 style="text-align: start"><strong>How do I make a science fair board?</strong></h3><p style="text-align: start"><img src="/images2/flex/150_science fair poster.png" data-size="100" data-float="left" style="width:100px; float:left">Your science fair board is where you show off your science fair project for everyone to see and learn from. With <a href="https://www.all-science-fair-projects.com/science_fair_projects_coach/science-project-display-board-6a.html">Science Fair Coach's step-by-step guide</a>, you can make sure your science fair board stands out and impresses at the science fair!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>Where can I find a science fair competition?</h3><p><img src="/images2/flex/179_Intro Science Fair Projects 32.png" data-size="100" data-float="left" style="width:100px; float:left">Science fair competitions are an excellent opportunity to explore science. One of the of the premier science advocacy organizations, the Society for Science, is associated with more than 400 science fair competitions in almost every US state, as well as over 70 other countries, regions, and territories. You can find out if there's a <a href="https://findafair.societyforscience.org/">Society of Science affiliated science fair</a> near you!</p><p>The <a href="http://physics.usc.edu/~gould/ScienceFairs/">The www Virtual Library: Science Fairs website</a> also has a collection of science fairs from all over the world, as well as national, state, regional, local, and virtual competitions!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat">

Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Sine Waves Added to Circles

Hypothesis

Method & Materials

Results

Why do this project?

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