# Sphericon, National Curve Bank, and Cardioid Art

Check out this re-run post from March, 2016! I still can’t get enough of Sphericons… Enjoy!

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!

# Squaring, Water Calculator, and Snap the Turtle

Welcome to this week’s Math Munch!

I’ve been really into squares lately. Maybe it’s because I recently ran across a new puzzle involving squares– something called Mrs. Perkin’s quilt.

69 by 69 Mrs. Perkin’s quilt.

The original version of the puzzle was published way back in 1907, and it went like this: “For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches.”

18 by 18 Mrs. Perkin’s quilt

Said in another way, if you have a 13 by 13 square, how can you divide it up into the smallest number of smaller squares? Don’t worry, you get to solve it yourself– I’m not including a picture of the solution to that version of the puzzle because there are so many beautiful pictures of solutions to the puzzle when you start with larger and smaller squares. Some are definitely more interesting than others. If you want to start simple, try the 4 by 4 version. I particularly like the look of the solution to the 18 by 18 version.

152 by 152 Mrs. Perkin’s quilt

Maybe you’re wondering where I got all these great pictures of Mrs. Perkin’s quits. And– wait a second– is that the solution to the 152 by 152 version? It sure is– and I got it from one of my favorite math websites, the Wolfram Demonstrations Project. The site is full of awesome visualizations of all kinds of things, from math problems to scans of the human brain. The Mrs. Perkin’s quilts demonstration solves the puzzle for up to a 1,098 by 1,098 square!

Next up, we here at Math Munch are big fans of unusual calculators. Marble calculators, domino calculators… what will we turn up next? Well, here for your strange calculator enjoyment is a water calculator! Check out this video to see how it works:

I might not want to rely on this calculator to do my homework, but it certainly is interesting!

Finally, meet Snap the Turtle! This cute little guy is here to teach you how to make beautiful math art stars using computer programming.

On the website Tynker, Snap can show you how to design a program to make intricate line drawings– and learn something about computer programming at the same time. Tynker’s goal is to teach kids to be programming “literate.” Combine computer programming with a little math and art (and a turtle)– what could be better?

I hope something grabbed your interest this week! Bon appetit!

# Combinatorial Games, Redistricting Game, and Graph Music

Welcome to this week’s Math Munch!

Have you ever played tic-tac-toe? If so, maybe you’ve noticed that unless you or your opponent makes a bad move, the game always ends in a tie! (Oops– spoiler alert!) Why is that? And what makes tic-tac-toe different from other games that have unpredictable outcomes, like Monopoly or the card game War?

Tic-tac-toe is similar to other kinds of game that mathematicians call combinatorial games— or games where there is no chance involved in the outcome and neither player has information that the other one doesn’t. This means that depending on who starts, where they go, and where each player decides to go next, the outcome is completely predictable and everyone playing could know what it is before it even happens. No surprises!

Now, this might also sound like NO FUN to you (why play the game at all if everyone knows what’s going to happen?) but I think it introduces a new kind of fun– figuring out what the outcomes could be! One of my favorite combinatorial games is the game NIM.

Here’s an example of a starting NIM board. If you go first, can you win? (Assuming your opponent never makes a mistake.)

NIM is a two-player game. You start with several piles or rows of objects (here they’re matches). On each turn, a player removes some objects from a pile– any number they want. BUT the player who’s forced to remove the last match loses!

There’s no chance in NIM– no dice determining how many matches you can remove, for example. Also both players know the rules and how many matches are in the piles at all times. That means that if you thought about it for a while, you could figure out who should win or lose any game of NIM. Maybe playing the game NIM isn’t super fun– but thinking about it like a puzzle is!

More versions of online NIM can be found here and here. And to read about combinatorial games we’ve written about in the past, check out this interview with mathematician Elwyn Berlekamp!

Next up, it’s presidential election time here again in the U.S.! Did you know that there’s a lot of mathematics behind what makes elections work? Four years ago, before the last presidential election, we shared a great series of YouTube videos about the math of elections.

A map in the redistricting game.

A big way that math gets involved in elections is through how politicians decide to draw districts, or regions of states that get to elect their own representative to the House of Representatives and elector to the Electoral College. The math behind drawing districts ranges from simple arithmetic to graph theory, or the field of math that deals with how parts of a shape or diagram are connected. To learn more about drawing election districts and the math behind it, check out the Re-Districting Game! In this game, you play the part of a map maker who works with the Congress, governor of your state, and courts to make a district map that meets everyone’s needs.

Finally, I recently ran across a series of graph music videos! What’s that? Videos in which a graph (made on Desmos) dances along to music, much like people would in a regular music video. Here’s one of my favorites:

The equations on the left-hand side of the screen create the images you see and the rhythm of the animation. Want to make your own graph music video? Share it with us!

Bon appetit!