# Sphericon, National Curve Bank, and Cardioid String Art

Welcome to this week’s Math Munch!

Behold the Sphericon!

What is that? Well, it rolls like a sphere, but is made of two cones attached with a twist– hence, the spheri-con! The one in the video is made out of pie (not sure why…), but you can make sphericons out of all kinds of materials.

It was developed by a few people at different times– like many brilliant new objects. But it entered the world of math when mathematician Ian Stewart wrote about it in his column in Scientific American. The wooden sphericon was made by Steve Mathias, an engineer from Sacramento, California, who read Ian’s article and thought sphericons would be fun to make. To learn more about how Steve made those beautiful wooden sphericons, check out his site!

Even if you’re not a woodworker, like Steve, you can still make your own sphericon. You can start with two cones and make one this way, by attaching the cones at their bases, slicing the whole thing in half, rotating one of the halves 90 degrees, and attaching again:

Or you can print out this image, cut it out, fold it up, and glue (click on the image for a larger printable size):

If you do make your own sphericon (which I recommend, because they’re really cool), watch the path it makes as it rolls. See how it wiggles? What shape do you think the path is?

I found out about the sphericon while browsing through an awesome website– the National Curve Bank. It’s just what it sounds like– an online bank full of curves! You can even make a deposit– though, unlike a real bank, you can take out as many curves as you like. The goal of the National Curve Bank is to provide great pictures and animations of curves that you’d never find in a normal math book. Think of how hard it would be to understand how a sphericon works if you couldn’t watch a video of it rolling?

There are lots of great animations of curves and other shapes in the National Curve Bank– like the sphericon! Another of my favorites is the “cycloid family.” A cycloid is the curve traced by a point on a circle as the circle rolls– like if you attached a pen to the wheel of your bike and rode it next to a wall, so that the pen drew on the wall. It’s a pretty cool curve– but there are lots of other related curves that are even cooler. The epicycloid (image on the right) is the curve made by the pen on your bike wheel if you rode the bike around a circle. Nice!

You should explore the National Curve Bank yourself, and find your own favorite curve! Let us know in the comments if you find one you like.

String art cardioid

Finally, to round out this week’s post on circle-y curves (pun intended), check out another of my favorite curves– the cardioid. A cardioid looks like a heart (hence the name). There are lots of ways to make a cardioid (some of which we posted about for Valentine’s Day a few years ago). But my favorite way is to make it out of string!

String art is really fun. If you’ve never done any string art, check out the images made by Julia Dweck’s class that we posted last year. Or, try making your own string art cardioid! This site shows you how to draw circles, ovals, cardioids, and spirals using just straight lines– you could follow the same instructions, replacing the straight lines you’d draw with pieces of string attached to tacks! If you’re not sure how the string part would work, check out this site for basic string art instructions.

Bon appetit!

# Demonstrations, a Number Tree, and Brainfilling Curves

Welcome to this week’s Math Munch!

Maybe you’re headed back to school this week. (We are!) Or maybe you’ve been back for a few weeks now. Or maybe you’ve been out of school for years. No matter which one it is, we hope that this new school year will bring many new mathematical delights your way!

A website that’s worth returning to again and again is the Wolfram Demonstrations Project (WDP). Since it was founded in 2007, users of the software package Mathematica have been uploading “demonstrations” to this website—amazing illuminations of some of the gems of mathematics and the sciences.

Each demonstration is an interactive applet. Some are very simple, like one that will factor any number up to 10000 for you. Others are complex, like this one that “plots orbits of the Hopalong map.”

Some demonstrations are great for visualizing facts about math, like these:

 Any Quadrilateral Can Tile A Proof of Euler’s Formula Cube Net or Not?

There’s also a whole category of demonstrations that can be used as MArTH—mathematical art—tools, including these:

 Rotate and Fold Back Polygons Arranged in a Circle Turtle Fractals

With over 8000 demonstrations to explore and new ones being added all the time, you can see why the Wolfram Demonstrations Project is worth returning to again and again!

Jeffrey Ventrella’s Number Tree

Next up, check out this number tree. It was created by Jeffrey Ventrella, an innovator, artist, and computer programmer who lives in San Francisco. His number tree arranges the numbers from 1 to 100 according to their largest proper factors. For instance, the factors of 18 are 18, 9, 6, 3, 2, and 1. Once we toss out 18 itself as being “improper”—a.k.a. “uninteresting”—the largest factor of 18 is 9. This in turn has as its largest factor 3, and 3 goes down to 1. Chains of factors like this one make up Jeffrey’s tree. It has a wonderful accumulative feeling to it—it’s great to watch how patterns and complexity build up over time.

(On this theme, WDP also has a demonstrations about trees and about prime factorization graphs.)

Cloctal: “a fractal design that visualizes the passage of time”

There’s lots more math to explore on Jeffrey’s website. His piece Cloctal—a fractal clock—is one of my favorites. What I’d like to feature here, though, is the diverse and intricate work Jeffrey has done with plane-filling and space-filling curves.  You can find many examples at fractalcurves.com, Jeffrey’s website that’s chock full of great links.

Jeffrey recently completed a book called Brainfilling Curves. It’s “a visual math expedition, lead by a lifelong fractal explorer.” According to the description, the book picks up where Mandelbrot left off and develops an intuitive scheme for understanding an “infinite universe of fractal beauty.”

An example of a “brainfilling curve” from Jeffrey’s “root8” family

The title comes from the idea that nature uses space-filling curves quite often, to pack intestines into your gut or lots and lots of tissue into the brain you’re using to read this right now! Hopefully you’re finding all of this math quite brainfilling as well.

(And just one more example of why WDP is worth revisiting: here’s a demonstration that depicts the space-filling Hilbert and Moore curves. So much good stuff!)

Finally, here’s a video that Jeffrey made about brainfilling curves. You can find more on his YouTube channel.

Bon appetit!