# Fold and Cut, My Favorite Spaces, and Hook

Welcome to this week’s Math Munch!

Before you watch this video, think about this question: Do you think you could fold a piece paper so that you could cut a square out of it using exactly one straight cut? How about a triangle? Hexagon? Christmas tree shape??

Give it a try. Then watch this video:

Pattern for a very angular swan, by Erik Demaine

Surprised? As you may have seen in the video, using the “fold and cut” process you can make any shape with straight sides! Isn’t that crazy? I learned about this a few years ago, and now cutting weird shapes out of paper using just one cut is one of my favorite things to do.

The person who proved this amazing result is one of my favorite mathematicians, Erik Demaine. (You might remember him from our post a few years ago about origami mazes.) I think it’s really interesting that this idea that’s now a mathematical theorem appeared throughout history as a magic trick and a method for cutting out five-pointed stars to make American flags. Check out this website about the fold and cut problem to learn more about the history of the theorem, Demaine’s method for cutting out any straight-edged shape, and other related problems.

Evelyn wearing a Borromean ring cowl. Sweet!

I found out about this video from another favorite mathematician of mine, Evelyn Lamb. Evelyn writes a blog about math for Scientific American called Roots of Unity that’s really fun to read. Check it out if you get the chance!

She has a series of posts called “A Few of My Favorite Spaces” (cue Sound of Music song, “My Favorite Things”). Favorite spaces, you may ask? I’m not familiar with spaces plural. There’s more than just regular old 3D space? Yes, in fact there are! And if you read Evelyn’s blog you’ll learn about how mathematicians like to invent new spaces with bizarre properties– and sometime find out that what they thought was a completely new space actually resembles something very familiar.

Such as… The “house with two rooms.” As I understand it, this a box (“house”) with two floors and two tunnels in it– one punched from the top of the box and another from the bottom. The top tunnel lets you get from the roof of the house to the ground floor; the bottom tunnel lets you get from below the house to the second floor.

If you want to see someone making this crazy house in Minecraft and hear a much better explanation of what the house is like, here’s a video!

Ok, so what’s the point? Well, it turns out you can squish (just squish– no ripping or gluing) this house all the way down to a single point. This means that in topology (the type of math that involves a lot of squishing), the crazy tunnel house space is the same as the really boring space of just one point. I might want to live in a house with all these tunnels– but I definitely don’t want to live in a point. But in topology-world, they’re the same space. Huh.

To learn more about the house with two rooms (aka, point) and other crazy spaces, check out Evelyn’s blog!

Finally, speaking of squishing things down to a point, I want to show you a fun new game I found that involves a lot of squishing– Hook! Here’s a trailer video for the game:

You can find this game online at Kongregate. Enjoy!

Bon appetit!

# HYPERNOM, Euclid the Game, and Math Quilts

Welcome to this week’s Math Munch! And, welcome to a new school year! Back to school means back to Math Munch– and we’re super excited to share some great new things that we found over the summer.  The first of which is…

… THIS.

(GIF hoisted from the amazing Aperiodical)

That’s an image from this crazy new game called HYPERNOM, invented by some of our favorite people– Vi Hart, Henry Segerman, and Andrea Hawksley!

Noming through tasty tasty tetrahedra.

In this game, you wiggle around in a projection of 4-dimensional space, eating (or, better put, NOMING– NOM NOM NOM) 4-dimensional objects. Such as the dodecahedra (polyhedron with faces made from regular pentagons) that come together to form the 4-dimensional shape (called a polytope) you’re moving around in.

This game is MINDBLOWING. Really. You can play it on your computer– but I got to play it wearing a helmet that plunged me into the fourth dimension and left me feeling very dizzy.

The math behind HYPERNOM is kind of complicated but VERY interesting. If you’d like to learn more about the game and the 4-dimensional math it involves, check out this post from Aperiodical. Or, watch the talk that Vi, Henry, and Andrea gave about HYPERNOM at this year’s Bridges Mathematical Art conference!

Next up, the Math Munch team went back to school a few weeks ago, too– literally! And this member of the Math Munch team is taking a math class! My homework assignment last week was to play a new game called Euclid: The Game.

On my way to constructing an equilateral triangle. What should I do next?

The game is pretty much exactly what it sounds like. You get to use just a straight-edge and compass (but a virtual straight-edge and compass, powered by Geogebra, because it’s a computer game!) to make Euclid’s constructions. For instance, the first challenge is to make an equilateral triangle– and all you can do is draw circles and lines! How would you do that?

I love this game for learning geometry because it lets you see how Euclid and his mathematicians peers thought about geometry– but you don’t have to use a real compass! The game saves your constructions so you can use them later– so if you ever want to make an equilateral triangle again, you don’t have to start from scratch. The game also gives you points if you make your construction with the least number of steps or without using any new tools. Give it a try!

Finally, I recently ran across the beautiful mathematical quilts of Elaine Ellison. Elaine is a former high school math teacher from Indiana who now creates and gives talks about making mathematical quilts. Her quilts explore some of the most interesting types of mathematics– from tessellations (like the Escher-inspired fish tessellation quilt to the left), to conic sections, to strange geometric spaces.

 “Poincare Plane” “Parabolas”

Elaine has a website and a YouTube channel devoted to her gorgeous quilts. Check them out! Here’s a taster:

# 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!

# Sequence Day, Penguins, and a little folding

Welcome to this week’s Math Munch! And… happy Sequence Day!

If you didn’t know that today was Sequence Day, don’t feel bad– I didn’t know until I ran across this article written by Aziz Inan, an electrical engineering professor at the University of Portland. Why is today Sequence Day? Well, because all of the digits in the sequence 0, 1, 2, 3, 4, 5 appear in today’s date– 3-4-2015!

This particular Sequence Day isn’t super special. There will be another one with the exact same sequence on April 3 (4-3-2015). But, according to Aziz, April 3 will be the last Sequence Day of this year– and the last until 2031! Aziz made this chart of all the Sequence Days that will happen this century. There are 48 all together– and if you look carefully at the chart, you may notice some interesting patterns.

See how the Sequence Days mostly occur at the beginnings of decades, in the first half of the month, and never later in the year than June? Why do you think that might be? Also, the last Sequence Day of the 21st century is in 2065. That means we’ll have to wait almost 40 years for the next Sequence Day after that– until 2103! But, in the scheme of things, this actually isn’t so bad– there were no Sequence Days at all in the last century. (Why might that be?)

We all know about days like Pi Day (coming up soon on 3-14-15 — and it’s a special one because we’ve got those two additional digits this year!), but, as Aziz likes to show, lots of days can be mathematical holidays– if you just look carefully enough. Maybe you’ll find a mathematical holiday of your own! If you do, let us know. We love any excuse to have a party!

Next up, you think penguins are cute, right? Well, take a look at this:

You may have heard the narrator say, “Something more organized is going on.” Well, several mathematicians wondered what that more organized thing was… and it turns out to be very mathematical!

How many penguins do you see?

Francois Blanchette, a mathematician at the University of California, Merced, had the idea to use math to study how penguins keep warm while watching penguin movies like this one. He studies the math of something called fluid dynamics, which, basically, is how things like water and air flow. Francois and several other mathematicians at the University of Erlangen-Nuremberg in Germany noticed that when one penguin in a huddle moves just a little bit, it triggers a chain reaction in which all of the other penguins move in an organized way to keep warm. Their tiny movements cause the huddle to organize into the best shape for all penguins to keep warm during the cold of winter.

Huddle up, little guy!

Scientists and mathematicians are only now realizing all of the amazing ways that math comes into play in the lives of animals, especially in large groups. It seems that penguins are only the beginning! To learn more about the organization of large groups of animals, I suggest you check out this awesome PBS documentary about animal swarms.

Finally, we haven’t heard from Vi Hart in a while. If you’ve been feeling the need for some math art fun in your life, check out this video I dug up from the archives. Origami meets Pythagorean Theorem– what could be better?

Stay warm, and bon appetit!

# Braids, Hacktastic, and Rock Climbing

Welcome to this week’s Math Munch!

Math hair braiding art by So Yoon Lym, shown at the 2014 Joint Mathematics Meetings.

First up, a little about one of my favorite things to do (and part of what got me into math in the first place!): hair braiding. If you’ve ever done a complicated braid in someone’s hair before, you might have had an inkling that something mathematical was going on. Well, you’re right! Mathematicians Gloria Ford Gilmer and Ron Eglash have spent much of their careers studying and teaching about the math that goes into hair braiding.

See the tessellation?

In their research, Gloria and Ron investigate how math can improve hair braiding, how hair braiding can improve math, and how the overlap between the two can teach us about how different cultures use and understand math. As Gloria shows in her article on math and braids, tessellations are very important to braided designs.

And so are fractals! Ron studies how fractals are used in African and African American designs, including in the layouts of towns, tile patterns, and cornrow braids. (Watch his TED Talk to learn more!) On his beautiful website dedicated to the math of cornrows, Ron shows how braiders use tools essential to making fractals to design their braids.

Just like when making a fractal, braid designers repeat the same shape while shifting, rotating, reflecting, and shrinking it. You can design your own mathematical cornrow braid using Ron’s braid programming app! If you’ve ever used Scratch, this app will look very familiar. I made the spiral braid on the right using the app. Next challenge: try to make your braid on a real head of hair…

Next up, a little about something I wish I could do: make awesome 3D-printed art! Here’s a blog that might help me (and you) get started. Mathematician Laura Taalman (who calls herself @mathgrrl on Twitter) writes a blog called Hacktastic all about making math designs, using a 3D-printer and many other tools. She has designs for all kinds of awesome things, from Menger sponges to trigonometric bracelets. One of the best things about Laura’s site is that she tells you the story behind how she came up with her designs, along with all the instructions and code you’ll ever need to make her designs yourself.

Skip Garibaldi, climbing

Finally, a little about something I’m trying to learn to do better: rock climbing! Mathematician Skip Garibaldi loves both math and rock climbing– so he decided to combine his interests for the better of each. In this video, Skip discusses some of the mathematical ideas important to rock climbing– including some essential to a type of climbing that I find most intimidating, lead climbing. Check it out!

Bon appetit!

# Harriss Spiral, Math Snacks, and SET

Happy New Year, and welcome to this week’s Math Munch!

The Harriss Spiral

Exciting news, folks! The Golden Ratio curve– that beautiful spiral that everyone adores– has evolved. And not into a freak of nature, either! Into something– dare I say it?– even more beautiful…

Meet the Harriss spiral. It was discovered/invented by mathematician and artist Edmund Harriss (featured before here and here) when he began playing around with golden rectangles. A golden rectangle is a very special rectangle, whose sides are in a particular proportion. You can read more about them here— but what’s most important to this new discovery is what you can do with them. If you make a square inside a golden rectangle you get another golden rectangle– and continuing to make squares and new golden rectangles inside of ever-shrinking golden rectangles, and drawing arcs through the squares, is one way to make the beautiful golden ratio spiral.

Edmund Harriss decided to get creative. What would happen, he wondered, if he cut the golden rectangle into two similar rectangles (same shape, just one is a scaled-down version of the other) and a square? And then what if he did the same thing to the new rectangles, again and again to make a fractal? Edmund’s new golden rectangle fractal makes this pattern, and when you draw a spiral through it, you get a lovely branching shape.

But don’t take my word for it. Math journalist Alex Bellos broke the news just this week in his article in The Guardian. His article explains much, much more than I can here– check it out to learn many more wonderful things about the Harriss spiral (and other spirals that Harriss has created…)!

(Bonus: Here’s a GeoGebra demonstration created by John Golden that builds the Harriss Spiral. It’s awesome!)

Next up is a site that sounds quite a lot like Math Munch. But it’s all games and cartoons, all the time. (Maybe that means you’ll like it better…) Check out Math Snacks, a site developed by a group of math educators at New Mexico State University. They worked hard to create games and animations that are both fun and full of interesting math.

One of my favorite games on Math Snacks is called Game Over Gopher.In this game, you have to save your carrot from an army of gophers by placing little machines that feed the gophers. Where’s the math, you may wonder? Placing the gopher-feeders and the other equipment that can help you save the carrot requires you to think carefully about geometry and coordinates.

Finally, speaking of games, here’s one of my favorites. I love to play SET, and I recently found a way to play online– either against a friend or against the computer. Click on this link to start your own game!

To play SET, you deal 12 cards. Then, you try to find a group of 3 cards that all share and all don’t share the same characteristics. For example, in the picture to the right– do you see the cards with the empty red ovals? They’re all the same shape, shading, and color (oval, empty, red), but they’re all different numbers (1, 2, and 3). Can you find any other sets in the picture? (Hint: One involves purple.)

Want to hone your SET skills without competing? Here’s a daily SET puzzle to challenge you.

Enjoy the games (and maybe invent a spiral of your own) and bon appetit!

# Grothendieck, Circle Packing, and String Art

Welcome to this week’s Math Munch!

This week brought some sad news to the mathematical world. Alexander Grothendieck, known by many as the greatest mathematician of the past century, passed away on November 13th. You may not have heard of him, but many mathematicians say that the work he did in math was as influential as the work Albert Einstein did in physics.

One of the things that make Grothendieck so interesting is, of course, the math he did. Grothendieck was always very creative. When he was in high school, he preferred to do math problems he made up on his own over the problems assigned by his teacher. “These were the book’s problems, and not my problems,” he said.

When he was young, inspired by some gaps he found in definitions in his geometry book about measuring lengths and areas, Grothendieck re-created some of the most important mathematical ideas of the beginning of the twentieth century. Maybe this sounds silly to you– why re-invent something that’s already been done? But, to Grothendieck, the most important part was that he’d done the whole thing by himself. He’d figured out something in his own way. He later wrote that this experience showed him what being a mathematician was like:

Without having been told, I nevertheless knew ‘in
my gut’ that I was a mathematician: someone who
‘does’ math, in the fullest sense of the word…

During his years as a mathematician, Grothendieck worked on connecting different parts of math (a project requiring a lot of creativity)– algebra, geometry, topology, and calculus, among others.

Alexander Grothendieck as a kid

The other thing that makes Grothendieck so interesting is his life story. As a kid, Grothendieck and his parents fled from Germany to France to escape the Nazis. As an adult, Grothendieck spoke out strongly for peace. He used his fame to take a stand against the wars of the second half of the twentieth century. This eventually led him to step away from the world of mathematicians– which many regretted. But he left behind work that changed all of mathematics for the better.

If you’d like to learn more about Grothendieck’s fascinating life and work, check out this great (but long) article from the American Mathematical Society. This article provides a shorter history, including a great statement Grothendieck made about his feelings on creativity in mathematics. Grothendieck was a very private person, so many of his mathematical writings aren’t available online– but the Grothendieck Circle has done their best to collect everything written about him.

A pretty circle packing

Next up, a little something for you to play with! We were studying circle packing problems in one of my classes this week. Did you know that you can fit exactly six circles snugly around another circle of the same size? But, if you try to fit circles snugly around a circle twice as large, it doesn’t work? I wonder why that is…

I did it!

Anyway, my class inspired me to look for a circle packing game– and I found one! In this game, simply called Circle Packing, you have to fit all of the smaller circles into the larger circle– without any of them touching! It’s pretty tricky, and really fun.

Finally, the Math Munch team got something wonderful in the mail (email, I guess) this week! Math art made by Julia Dweck’s 5th grade math class! Julia’s class has been working hard to make parabolic curve string art– curves made by drawing (or stringing, in this case) many, many straight lines. They plotted each curve precisely before stringing it, to make sure it was both mathematically and artistically perfect. The pieces they made are so creative and beautiful. We’re proud to feature them on our site!

You can see the whole collection of string art pieces made by Julia’s class on our Readers’ Gallery String Art page. And, want to know more about how the 5th graders made their String Art? Have any questions for Julia and her students about their love of math and the connections they see between math and art? Write your questions here and we’ll send them to Julia’s students!

Have any math art of your own? Send it to mathmunchteam@gmail.com, and we’ll post it in the Readers’ Gallery!

Bon appetit!