Have you ever played with shapes and patterns? That's what geometry is all about! It's the study of lines, angles, shapes, and space. From drawing cool designs to building structures, geometry is an important part of our world.

To determine whether it is possible to construct shapes with infinite perimeters but finite areas.

Can a shape have an edge that goes on forever but still fit inside a small space? A fractal called the Koch snowflake does exactly that. You start with a simple triangle and add smaller triangles to each side. Each new layer makes the edge longer but adds only a tiny bit of area.

You use geometry software to build the snowflake through many stages. Then you use power series (a way to add up shrinking numbers) to calculate the final area. The snowflake's area ends up only 1.6 times the original triangle.

You also try the same idea with a square shape. The square version produces a figure with an infinite edge but an area only 2.0 times the original square.

Title: Endless Snowflake: Constructing Shapes with Infinite Perimeters and
Finite Areas

Objectives/Goals

To determine whether it is possible to construct shapes with infinite perimeters
but finite areas.

Methods/Materials

Using the Geometer's Sketchpad program and a Koch curve, I began with an
equilateral triangle and grew
a snowflake through various generations which, if taken to infinity, would have
an infinite perimeter but a
finite area. Using power series, I calculated what that area would be. I then
decided to develop a new
fractal curve based on a square shape rather than the traditional triangular
Koch curve.

Results

I calculated that this fractal pattern for the triangle would produce a figure
of infinite perimeter, but whose
area is only 1.6 times the area of the original triangle. Meanwhile, for the
square, the area for an infinite
perimeter shape would be 2.0 times the area of the original square.

Conclusions/Discussion

Ordinarily, when shapes are magnified, area grows faster than perimeter.
However, using the idea of
convergent series, it is possible to add ever-smaller increments of area such
that while the perimeter grows
to infinity, the sum of the areas remains finite. It may be possible to
generalize this approach to three
dimensions, producing a shape of infinite surface area and finite volume.

Summary Statement

My project uses the ideas of fractals and power series to construct a Koch
snowflake, and to explore new
families of curves with infinite perimeters and finite areas.


Fractals and Finite Area from Infinite Perimeters

Constructing Shapes with Infinite Perimeters and Finite Areas

<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. 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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. 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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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Fractals and Finite Area from Infinite Perimeters

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