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.

House with 2 roomsSuch 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…


(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. Mmmm!

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.

Playing hypernomThis 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?

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!

Ellison tessellation quiltFinally, 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.

Ellison poincare quilt

“Poincare Plane”

Ellison parabola quilt


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

We hope you’re enjoying your return to school! We definitely are. Bon appetit!

Web Applets, Space Fillers, and Sisters

Welcome to this week’s Math Munch!

Recently I’ve been running across tons of neat, slick math applets. I feel like they all go together. What do they have in common? Maybe you’ll be able to tell me.

First up, you can tinker with some planetary gears. Then try out these chorded polygons. And then how about some threaded lines?

plantearygears chords shapes

Ready for some more? Because with these sorts of visualizations, Dan Anderson has been on fire lately. Dan is a high school math teacher in New York state. He and his students had fifteen minutes of fame last year when they investigated whether or not Double Stuf Oreos really have double the stuf.

Here is Dan’s page on OpenProcessing. (Processing is the computer language in which Dan programs his applets.) And check out the images and gifs on Dan’s Tumblr. Here’s a sampling!

tumblr_nm56rdMlvl1uppablo1_r3_400 tumblr_noqxoi8EsC1uppablo1_400 tumblr_nolvf9dSt61uppablo1_400

Dan also coordinates Daily Desmos, which we’ve feature previously. Check out the latest periodic and “obfuscation” challenges!

That’s a chunk of math to chew on already, but we’re just getting started! Next up, check out the space-filling artwork of John Shier.

doublecircles eyes
 fish  hearts

John’s artwork places onto the canvas shapes of smaller and smaller sizes. Notice that the circles below fill in gaps, but they don’t touch each other, they way circles do in an Apollonian gasket.

circle_prog_1B_AnimeYou can learn more about John’s space-filling shapes on this page and find further details in this paper.

Thanks for making us this sweet banner, John!

Thanks for making us this sweet banner, John!

Last up this week, head to this site to watch an awesome trailer of a film about Julia Robinson. The short clip focuses on Julia’s work on Hilbert’s tenth problem. It includes interviews with a number of people who knew Julia, including her sister Constance Reid. Constance wrote extensively about mathematics and mathematicians. I’ve read her biography of Hilbert and can highly recommend it. You can read more about Julia and Constance here and here.

Julia Robinson

Julia Robinson

Julia's sister, Constance Reid

Julia’s sister, Constance Reid

Julia and Constance as young girls.

Julia and Constance as young girls.

You might enjoy visiting the site of the Julia Robinson Mathematics Festival. Check to see if a festival will be hosted in your area sometime soon, or find out how you can run one yourself!

With May wrapped up and June getting started, I hope you have a lot of math to look forward to this summer. Bon appetit!

Continents, Math Explorers’ Club, and “I use math for…”

Welcome to this week’s Math Munch!


Steven Strogatz.

All of our munches this week come from the recent tweets of mathematician, author, and friend of the blog Steven Strogatz. Steve works at Cornell University as an applied mathematician, tackling questions like “If people shared taxis with strangers, how much money could be saved?” and “What caused London’s Millennium Bridge to wobble on its opening day?”

On top of his research, Steve is great at sharing math with others. (This week I learned one great piece of math from him, and then another, and suddenly there was a very clear theme to my post!) Steve has written for the New York Times and was recently awarded the Lewis Thomas Prize as someone “whose voice and vision can tell us about science’s aesthetic and philosophical dimensions, providing not merely new information but cause for reflection, even revelation.”

NMFLogo_Horiz_RGB_300DPI2This Saturday, Steve will be presenting at the first-ever National Math Festival. The free and fun main event is at the Smithsonian in Washington, DC, and there are related math events all around the country this weekend. Check and see if there’s one near you!

Here are a few pieces of math that Steve liked recently. I liked them as well, and I hope you will, too.

First up, check out this lovely image:

tesselation1-blog480It appeared on Numberplay and was created by Hamid Naderi Yeganeh, a student at University of Qom in Iran. Look at the way the smaller and smaller tiles fit together to make the design. It’s sort of like a rep-tile, or this scaly spiral. And do those shapes look familiar? Hamid was inspired by the shapes of the continents of Africa and South America (if you catch my continental drift). Maybe you can create your own Pangaea-inspired tiling.

If you think that’s cool, you should definitely check out Numberplay, where there’s a new math puzzle to enjoy each week!

Next, up check out the Math Explorers’ Club, a collection of great math activities for people of all ages. The Club is a project of Cornell University’s math department, where Steve teaches.

The first item every sold on the auction site eBay. Click through for the story!

The first item every sold on the auction site eBay. Click through for the story!

One of the bits of math that jumped out to me was this page about auctions. There’s so much strategy and scheming that’s involved in auctions! I remember being blown away when I first learned about Vickrey auctions, where the winner pays not what they bid but what the second-highest bidder did!

If auctions aren’t your thing, there’s lots more great math to browse at the Math Explorer’s Club—everything from chaos and fractals to error correcting codes. Even Ehrenfeucht-Fraïssé games, which are brand-new to me!

And finally this week: have you ever wondered “What will I ever use math for?” Well, SIAM—the Society for Industrial and Applied Mathematics—has just the video for you. They asked people attending one of their meetings to finish the sentence, “I use math for…”. Here are 32 of their answers in just 60 seconds.

Thanks for sharing all this great math, Steve! And bon appetit, everyone!

The Colorspace Atlas, allRGB, and Hyperbolic Puzzles

Welcome to this week’s Math Munch!

Update: A few weeks ago we met Dearing Wang, mathematical artist and creator of Dearing Draws. Now you can read a Math Munch Q&A with Dearing Wang.

OK, first up in this week’s post, do you remember when we talked about the six dimensions of color and the RGB color system? Well either way, consider this:


Artist Tauba Auerbach (one of my absolute favorite contemporary artists) made a book that contains every possible color!!! Tauba calls it “The RGB Colorspace Atlas.” The book is a perfect 8″ by 8″ by 8″ cube, matching the classic RGB color cube.

RGB_Cube_Show_lowgamma_cutout_aThe primary colors of light (red, blue, and green) increase as you move in each of the three directions. This leaves white and black at opposite corners of the cube, and all the wonderful colors spread around throughout the cube, with the primary and secondary colors on the other corners. You can read more here, if you like.

The book shows cross-sections moving through a single axis, so Tauba really had 3 choices for how the pages should flip through the cube. In fact, she made all three books!  Jonathan Turner made simulations of all three axes however, so we can see each one if we like. Can you tell which one is open in the pictures above?

That’s the Red Axis. Compare that to the Green Axis and Blue Axis.

For computer graphics, RGB color codes are ordered triples of numbers like (120, 15, 28). Each number says how much of each color should be included in the mix.  There are 256 possible values for each one, with values from 0 to 255. [Examples: (0,0,0) is black. (255,255,255) is white.  (255,0,0) is red. (127,0,0) is a red that’s half as bright.] Since there are only so many number combinations, computers have exactly 16,777,216 possible colors. That’s where allRGB comes in.


Starry Night


Hilbert Coloring


Escher LIzards

As they say, “The objective of allRGB is simple: To create images with one pixel for every RGB color (16777216); not one color missing, and not one color twice.” AllRGB is a bounded concept, since there are only finitely many ways to rearrange those 16777216 pixels. But of course there are a HUUUUGGGEEEE number of ways to rearrange them, so there’s lots to see. (In fact if you wrote a 1 with 100 million zeroes after it, that number would still be smaller than the number of allRGB pictures!! And that’s only part of the story)  Click the pictures above for zoomable versions as well as descriptions of their creation.

hyperbolic maze 1 hyperbolic maze

We’ve posted a little before about hyperbolic geometry. Very very briefly, the hyperbolic plane is a 2D surface where some of our usual intuition gets a little warped. For example, two lines can be parallel to the same line but not parallel to each other, which seems a little awkward. Click the images above to really experience what it’s like to walk through a hyperbolic world. David Madore created these hyperbolic “mazes,” which give you a birds eye view as you walk through a strange new land.

Finally, you might enjoy this old Numberplay puzzle with a hyperbolic feel, based on the movements of whales.

Gary Antonick asks "What is the fewest-bun path between the two white buns? (The two white buns are the first and last — or 40th — buns in the top row."

Gary Antonick asks “What is the fewest-bun path between the two white buns? (The two white buns are the first and last — or 40th — buns in the top row.”

What do buns have to do with whales and hyperbolic geometry? You’ll just have to click and find out.

Have a great week and bon appetit!

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.

Wooden sphericonIt 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:How to make a sphericon

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

Sphericon pattern

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?

ncbmastertitleI 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?

epicycloidaThere 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 cardioid

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!

Pi Digit, Pi Patterns, and Pi Day Anthem


Painting by Renée Othot for Simon Plouffe’s birthday.

Welcome to this week’s Math Munch!

It’s here—the Pi Day of the Century happens on Saturday: 3-14-15!

How will you celebrate? You might check to see if there are any festivities happening in your area. There might be an event at a library, museum, school, or university near you.

(Here are some pi day events in NYC, Baltimore, San Francisco, Philadelphia, Houston, and Charlotte.)


John Conway at the pi recitation contest in Princeton.

John Conway at the pi recitation contest in Princeton.

There’s a huge celebration here in Princeton—in part because Pi Day is also Albert Einstein’s birthday, and Albert lived in Princeton for the last 22 years of his life. One event involves kids reciting digits of pi and and is hosted by John Conway and his son, a two-time winner of the contest. I’m looking forward to attending! But as has been noted, memorizing digits of pi isn’t the most mathematical of activities. As Evelyn Lamb relays,

I do feel compelled to point out that besides base 10 being an arbitrary way of representing pi, one of the reasons I’m not fond of digit reciting contests is that, to steal an analogy I read somewhere, memorizing digits of pi is to math as memorizing the order of letters in Robert Frost’s poems is to literature. It’s not an intellectually meaningful activity.

I haven’t memorized very many digits of pi, but I have memorized a digit of pi that no one else has. Ever. In the history of the world. Probably no one has ever even thought about this digit of pi.

And you can have your own secret digit, too—all thanks to Simon Plouffe‘s amazing formula.


Simon’s formula shows that pi can be calculated chunk by chunk in base 16 (or hexadecimal). A single digit of pi can be plucked out of the number without calculating the ones that come before it.

Wikipedia observes:

The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the nth digit of π without calculating all of the preceding n − 1 digits.

Check out some of Simon's math art!

Check out some of Simon’s math art!

Simon is a mathematician who was born in Quebec. In addition to his work on the digits of irrational numbers, he also helped Neil Sloane with his Encyclopedia of Integer Sequences, which soon online and became the OEIS (previously). Simon is currently a Trustee of the OEIS Foundation.

There is a wonderful article by Simon and his colleagues David Bailey, Jonathan Borwein, and Peter Borwein called The Quest for Pi. They describe the history of the computation of digits of pi, as well as a description of the discovery of their digit-plucking formula.

According to the Guinness Book of World Records, the most digits that someone has memorized and recited is 67,890. Unofficial records go up to 100,000 digit. So just to be safe, I’ve used an algorithm by Fabrice Bellard based on Simon’s formula to calculate the 314159th digit of pi. (Details here and here.) No one in the world has this digit of pi memorized except for me.

Ready to hear my secret digit of pi? Lean in and I’ll whisper it to you.

The 314159th digit of pi is…7. But let’s keep that just between you and me!

And just to be sure, I used this website to verify the 314159th digit. You can use the site to try to find any digit sequence in the first 200 million digits of pi.

Aziz and Peter's patterns.

Aziz & Peter’s patterns.

Next up: we met Aziz Inan in last week’s post. This week, in honor of Pi Day, check out some of the numerical coincidences Aziz has discovered in the early digits in pi. Aziz and his colleague Peter Osterberg wrote an article about their findings. By themselves, these observations are nifty little patterns. Maybe you’ll find some more of your own. (This kind of thing reminds me of the Strong Law of Small Numbers.) As Aziz and Peter note at the end of the article, perhaps the study of such little patterns will one day help to show that pi is a normal number.

And last up this week, to get your jam on as Saturday approaches, here’s the brand new Pi Day Anthem by the recently featured John Sims and the inimitable Vi Hart.

Bon appetit!