Tag Archives: fractals

Tsoro Yematatu, Fano’s Plane, and GIFs

Welcome to this week’s Math Munch!

Board and pieces for tsoro yematatu.

Here’s a little game with a big name: tsoro yematatu. If you enjoyed Paul’s recent post about tic-tac-toe, I think you’ll like tsoro yematatu a lot.

I ran across this game on a website called Behind the Glass. The site is run by the Cincinnati Art Museum. (What is it with me and art museums lately?) The museum uses Behind the Glass to curate many pieces of African art and culture, including four mathematical games that are played in Africa.

The simplest of these is tsoro yematatu. It has its origin in Zimbabwe. Like tic-tac-toe, the goal is to get three of your pieces in a row, but the board is “pinched” and you can move your pieces. Here’s an applet where you can play a modified version of the game against a computer opponent. While the game still feels similar to tic-tac-toe, there are brand-new elements of strategy.

Tsoro yematatu reminds me of one that I played as a kid called Nine Men’s Morris. I learned about it and many other games—including go—from a delightful book called The Book of Classic Board Games. Kat Mangione—a teacher, mom, and game-lover who lives in Tennessee—has compiled a wonderful collection of in-a-row games. And wouldn’t you know, she includes Nine Men’s Morris, tsoro yematatu, tic-tac-toe, and dara—another of the African games from Behind the Glass.

The Fano plane.

The Fano plane.

The board for tsoro yematatu also reminds me of the Fano plane. This mathematical object is very symmetric—even more than meets the eye. Notice that each point is on three lines and that each line passes through three points. The Fano plane is one of many projective planes—mathematical objects that are “pinched” in the sense that they have vanishing points. They are close cousins of perspective drawings, which you can check out in these videos.

Can you invent a game that can be played on the Fano plane?

Closely related to the Fano plane is an object called the Klein quartic. They have the same symmetries—168 of them. Felix Klein discovered not only the Klein quartic and the famous Klein bottle, but also the gorgeous Kleinian groups and the Beltrami-Klein model. He’s one of my biggest mathematical heroes.

The Klein quartic.

The Klein quartic.

This article about the Klein quartic by mathematician John Baez contains some wonderful images. The math gets plenty tough as the article goes on, but in a thoughtfully-written article there is something for everyone. One good way to learn about new mathematics is to read as far as you can into a piece of writing and then to do a little research on the part where you get stuck.

If you’ve enjoyed the animation of the Klein quartic, then I bet my last find this week will be up your alley, too. It’s a Tumblr by David Whyte and Brian Fitzpatrick called Bees & Bombs. David and Brian create some fantastic GIFs that can expand your mathematical imagination.

This one is called Pass ‘Em On. I find it entrancing—there’s so much to see. You can follow individual dots, or hexagons, or triangles. What do you see?


This one is called Blue Tiles. It makes me wonder what kind of game could be played on a shape-shifting checkerboard. It also reminds me of parquet deformations.


A few of my other favorites are Spacedots and Dancing Squares. Some of David and Brian’s animations are interactive, like Pointers. They have even made some GIFs that are inspired by Tilman Zitzmann’s work over at Geometry Daily (previously).

I hope you enjoy checking out all of these new variations on some familiar mathematical objects. Bon appetit!

Reflection Sheet – Tsoro Yematatu, Fano’s Plane, and GIFs

Partial Cubes, Open Cubes, and Spidrons

Welcome to this week’s Math Munch!

Recently the videos that Paul and I made about the Yoshimoto Cube got shared around a bit on the web. That got me to thinking again about splitting cubes apart, because the Yoshimoto Cube is made up of two pieces that are each half of a cube.

A part of Wall Drawing #601 by Sol LeWitt

A part of Wall Drawing #601
by Sol LeWitt

A friend of mine once shared with me some drawings of cubes by the artist Sol LeWitt. The cubes were drawn as solid objects, but parts of them were cut away and removed. It was fun trying to figure out what fraction of a cube remained.

On the web, I found a beautiful image that Sol made called Wall Drawing #601. In the clipping of it to the left, I see 7/8 of a cube and 3/4 of a cube. Do you? You can view the whole of this piece by Sol on the website of the Greater Des Moines Public Art Foundation.

The Cube Vinco by Vaclav Obsivac.

The Cube Vinco by Vaclav Obsivac.

There are other kinds of objects that break a cube into pieces in this way, like this tricky puzzle by Vaclav Obsivac and this “shaved” Rubik’s cube modification. Maybe you’ll design a cube dissection of your own!

As I further researched Sol LeWitt’s art, I found that he had investigated partial cubes in other ways, too. My favorite of Sol’s tinkerings is the sculpture installation called “Variations of Incomplete Cubes“. You can check out this piece of artwork on the SFMOMA site, as well as in the video below.

In the video, a diagram appears that Sol made of all of the incomplete open cubes. He carefully listed out and arranged these pictures to make sure that he had found them all—a very mathematical task. It reminds me of the list of rectangle subdivisions I wrote about in this post.


Sol’s diagram got me to thinking and making: what other shapes might have interesting “incomplete open” variations? I started working on tetrahedra. I think I might try to find and make them all. How about you?

Two open tetrahedra I made. Can you find some more?

Two open tetrahedra I made. Can you find some more?

Finally, as I browsed Google Images for “half cube”, one image in particular jumped out at me.


What are those?!?!

Dániel's original spidron from 1979

Dániel’s original spidron from 1979

These lovely rose-shaped objects are called spidrons—or more precisely, they appear to be half-cubes built out of fold-up spidrons. What are spidrons? I had never heard of them, but there’s one pictured to the right and they have their own Wikipedia article.

The first person who modeled a spidron was Dániel Erdély, a Hungarian designer and artist. Dániel started to work with spidrons as a part of a homework assignment from Ernő Rubik—that’s right, the man who invented the Rubik’s cube.

A cube with spidron faces.

A cube with spidron faces.

Two halves of an icosahedron.

Two halves of an icosahedron.

A hornflake.

A hornflake.

Here are two how-to videos that can help you to make a 3D spidron—the first step to making lovely shapes like those pictured above. The first video shows how to get set up with a template, and the second is brought to you by Dániel himself! Watching these folded spidrons spiral and spring is amazing. There’s more to see and read about spidrons in this Science News article and on Dániel’s website.

And how about a sphidron? Or a hornflake—perhaps a cousin to the flowsnake? So many cool shapes!

To my delight, I found that Dániel has created a video called Yoshimoto Spidronised—bringing my cube splitting adventure back around full circle. You’ll find it below. Bon appetit!

Reflection Sheet – Partial Cubes, Open Cubes, and Spidrons

Andrew Hoyer, Cameron Browne, & Sphere Inversion

Welcome to this week’s Math Munch! Fractals, origami, math art, games, and a mind-bending video are all ahead, so let’s get into it.

Andrew Hoyer

Andrew Hoyer

First up, let’s take a look at the work of Andrew Hoyer.  According to his website, he’s a “software engineer in his mid-twenties living it up in sometimes sunny San Francisco.” I came across his work when I found his beautiful and completely engaging introduction to simple fractals.  (Go on! Click. Then read, experiment and play!)

Cantor Set

A Cantor set

At the bottom of that page, Andrew links to a wonderful, long list of fractals, arranged by Hausdorff dimension, which is a way of measuring fractals as being something like 2.5 dimensions.  A line is 1 dimensional.  A plane is 2D, and you can find many fractals with dimension in between!!  Weird, right?

I was also really pleased to find Andrew’s Instagram feed, which features some of his beautiful origami creations.  Andrew’s agreed to answer your questions for an upcoming Q&A, so ask away!

Compound of 5 tetrahedra Truncated Icosahedron Cube Dodecahedron
Cameron Browne

Cameron Browne

Up next, meet Cameron Browne. He’s an accomplished researcher who designs and studies games. Take a look at the many many games Cameron has created. The rules and descriptions are there, and Cameron sent along links to playable versions of a few, which you can find by clicking the pictures below.  For the third one, you’ll need to search for “Margo” or “Spargo.” For his research, Cameron investigates the possibilities of artificial intelligence, and how a computer can be used to generate games and puzzles.



Margo and Spargo

Margo and Spargo

Cameron is also an artist, and he has a page full of his graphic designs.  I found Cameron through his page of Truchet curves.  I love the way his pages are full of diagrams and just enough information to start making sense of things, even if it’s not perfectly clear.  Cameron also has MANY pages of wonderful fractal-ish graphics: Impossible Fractals, Cantor Knots, Fractal Board Games, Woven Horns, Efficient Trees, and on and on…  And he has agreed to do a Q&A with us, so please, submit a question. What are you wondering?

A Cantor Knot

A Cantor Knot

A Truchet curve "Mona Lisa"

A Truchet curve “Mona Lisa”

An "impossible" fractal

An “impossible” fractal

And, as if that wasn’t enough mathy awesomeness, check out this video about turning a sphere inside out.  A bit of personal history, I actually used this video  (though it was only on VHS back then, checked out from the library) as part of the research for my independent research project during my senior year of college.  It gets pretty tricky, but if you watch it all the way through it starts to make some sense.

Have a great week.  Bon appetit!

Reflection sheet – Andrew Hoyer, Cameron Browne, & Sphere Inversion

Coasts, Clueless Puzzles, and Beach Math Art

summerAh, summertime. If it’s as hot where you are as it is here in New York, I bet this beach looks great to you, too. A huge expanse of beach all to myself sounds wonderful… And that makes me wonder – how much coastline is there in the whole world?

Interestingly, the length of the world’s coastline is very much up for debate. Just check out this Wikipedia page on coastlines, and you’ll notice that while the CIA calculates the total coastline of the world to be 356,000 kilometers, the World Resources Institute measures it to be 1,634,701! What???

Measuring the length of a coastline isn’t as simple as it might seem, because of something called the Coastline Paradox. This paradox states that as the ruler you use to measure a coastline gets shorter, the length of the coastline gets longer – so that if you used very, very tiny ruler, a coastline could be infinitely long! This excellent video by Veritasium explains the problem very well:

2000px-KochFlakeAs Vertitasium says, many coastlines are fractals, like the Koch snowflake shown at left – never-ending, infinitely complex patterns that are created by repeating a simple process over and over again. In this case, that simple process is the waves crashing against the shore and wearing away the sand and rock. If coastlines can be infinitely long when you measure them with the tiniest of rulers, how to geographers measure coastline? By choosing a unit of measurement, making some approximations, and deciding what is worth ignoring! And, sometimes, agreeing to disagree.

Need something to read at the beach, and maybe something puzzle-y to ponder? Check out this interesting article by four mathematicians and computer scientists, including James Henle, a professor in Massachusetts. They’ve invented a Sudoku-like puzzle they call a “Clueless Puzzle,” because, unlike Sudoku, their puzzle never gives any number clues.

Clueless puzzleHow does this work? These puzzles use shapes instead of numbers to provide clues. Here’s an example from the paper: Place the numbers 1 through 6 in the cells of the figure at right so that no digit appears more than once in a row or column AND so that the numbers in each region add to the same sum. The paper not only walks you through the solution to this problem, but also talks about how the mathematicians came up with the idea for the puzzles and studied them mathematically. It’s very interesting – I recommend you read it!

Finally, if you’re not much of a beach reader, maybe you’d like to make some geometrically-inspired beach art! Check out this land art by artist Andy Goldsworthy:

Andy Goldsworthy 1
Andy Goldsworthy 2

Or make one of these!

Happy summer, and bon appetit!

The Numbers Project, Epidemics, and Cut ‘n Slide

Welcome to this week’s Math Munch!

It’s an end-of-the-year group post!

Brandon Todd WilsonPaul: This week I found Brandon Todd Wilson, a graphic artist who lives in Kansas City. He started a new and ambitious project. He wants to make a design for each of the numbers 0 through 365, making a new one each day of the year. That’s tough, but he’s done some amazing things so far. Check them out over at the numbers project. I’m amazed by the sneaky, clever ways he comes up with to showcase the numbers. Can you tell what numbers these three are below? Click to find out.

40 Screen Shot 2013-06-07 at 12.22.20 AM 118

Maybe you could try a numeric design of your own. Perhaps for your favorite number or your birthday. If you make something your proud of, email us at mathmunchteam@gmail.com, and we could feature your work on Math Munch!

[Here are some numeric creations inspired by Brandon’s!]

ninaAnna: Next up, it’s probably the end of the school year for most of you readers out there. Our school year is wrapping up, too. It’s sad, but also exciting, because we’re looking forward to what comes in the future. Recently, some of my students, looking to their futures, have been wondering what many students wonder: If I like math, what are some things I can do with it after I leave school? (We’ve posted about this question before – check out this post on the site We Use Math and any of the interviews on our Q&A page.) We here at Math Munch had the honor last week to meet an awesome woman who uses math all the time in her work as a scientist – Nina Fefferman!

green_virus_tNina works mainly as a biologist at Rutgers University in New Jersey researching all kinds of cool and interesting things relating to epidemiology, or the study of infectious diseases and how they spread into epidemics in groups of people. How does she use math? In everything! Since dealing with infectious diseases is best done before they become epidemics, scientists like Nina make mathematical models to predict how a disease will spread before it hits. These models are really important for governments and hospitals, who use them to figure out how they can prepare for possible epidemics.

Nina loves math and her work – and you can hear all about it in this TEDx talk she did in 2010.

Justin: Finally, check out this short video by Sander Huisman, of mathematical pasta fame:

Sander has some more great videos, too. The shape that Sander’s cut and slide pattern gets closer and closer to is called the twindragon. It’s related to the more famous dragon fractal. Notice how the area of the shape stays the same throughout the video. Thanks to the kind folks at math.stackexchange for helping me to identify this fractal so quickly!

An earlier stage and a later stage of my cut & slide exploration.

An earlier stage and a later stage of my cut & slide exploration.

In searching about this geometry idea of “cut and slide”, I ran across some great stuff. One thing I found was this neat applet by Frederik Vanhoutte. (Warning: JAVA required.) Frederik is a med­ical radi­a­tion physi­cist who lives in Belgium and who likes to make wonderful graphics in his spare time. Frederik has shared many of these on his site—check out his portfolio.

On his About page, Frederik says this about why he makes his generative graphics:

“When rain hits the wind­screen, I see tracks alpha par­ti­cles trace in cells. When I pull the plug in the bath tub, I stay to watch the lit­tle whirlpool. When I sit at the kitchen table, I play with the glasses to see the caus­tics. At a can­dle light din­ner, I stare into the flame. Sometimes at night, I find myself behind the com­puter. When I finally blink, a mess of code is draw­ing ran­dom struc­tures on the screen. I spend the rest of the night staring.”

Bon appetit!

TED, Bridges, and Silk

Welcome to this week’s Math Munch!


The Math Munch team at TEDxNYED

Marjorie Rice

Marjorie Rice | click to watch her interview video

On Saturday, the Math Munch team gave a 16-minute presentation at TEDxNYED about Math Munch!  (Eventually there will be a video, and we’ll be sure to share it with you right away, but you’ll have to wait a month, maybe.)

We started with the story of Marjorie Rice, and in searching for a good picture of her, we came across this wonderful interview in a documentary about Martin Gardner.  It’s so neat to hear her speak about her discoveries.  You can see how proud she is and how much she truly loves math.  Feel free to watch the whole documentary if you like.  I haven’t gotten a chance yet, but I know it’s full of incredible stuff.

In the spirit of TED, I decided to share a few mathematical TED talks.  This one is absolutely fascinating.  In it, mathematician Ron Eglash describes how fractals underly the african designs.  You know how we love fractals.

If you’re hungry for another TED talk, here’s one about connections between music, mathematics, and sonar.

Up next, remember when we wrote about attending last year’s Bridges conference?  Well it happens every year, of course, and this year’s gallery of mathematical art is available online!  Click on one of those images and you get to more of the artists work.  I could easily spend hours staring at this art, trying to understand them, and reading the descriptions and artist statements.  Seriously, there is just way too much cool stuff there, so I’ve picked out a few of my favorites.  Also, I have great news to announce: Chloé Worthington (previously featured) had some of her art accepted to the exhibition!  Congratulations, Chloe!  If you look closely, you’ll see some of my art in there too.  🙂

Bjarne Jespersen

Bjarne Jespersen

Marc Chamberland

Marc Chamberland

Bob Rollings

Bob Rollings

Chloe Worthington

Chloé Worthington

Mehrdad Garousi

Mehrdad Garousi

By the way, if you ever create any mathematical art of your own, we’d love to see it!  Send us an email at mathmunchteam@gmail.com, and maybe we’ll feature your work in an upcoming Math Munch. (Only if you want us too, of course.)

Yuri Vishnevsky

Silk creator Yuri Vishnevsky

Finally, I know many of you like playing around with Symmetry Artist, which can be found on our page of Math Art Tools.  If you like that, then you’ll love Silk!  It’s much the same, but generates a certain kind of whispiness as you draw that looks really cool.  It also lets you spiral your designs toward the center, a feature which Symmetry Artist lacks.  You can download the Silk app for iPad or iPhone, if you like.  Silk was designed by Yuri Vishnevsky, with sound design by Mat Jarvis.  Yuri has agreed to do a Q&A for us, but we haven’t quite finished it just yet.  I’ll upload it as soon as possible, but for now, you can read an interview Mat and Yuri did with a website called Giant Fire Breathing Dragon.

Bon appetit!

Silk1 Silk4 Silk2


Maths Ninja, Folding Fractals, and Pi Fun

Welcome to this week’s Math Munch!

ninjaFirst up, have you ever been stuck on a gnarly math problem and wished that a math ninja would swoop in and solve the problem before it knew what hit it?  Have you ever wished that you had a math dojo who would impart wisdom to you in cryptic but, ultimately, extremely timely and useful ways?  Well, meet Colin Beverige, a math (or, as he would say, maths) tutor from England who writes a fun blog called Flying Colours Maths.  On his blog, he publishes a weekly series called, “Secrets of the Mathematical Ninja,” in which the mathematical ninja (maybe Colin himself?  He’s too stealthy to tell)  imparts nuggets of sneaky wisdom to help you take down your staunchest math opponent.

colin_bridgeFor example, you probably know the trick for multiplying by 9 using your fingers – but did you know that there’s a simple trick for dividing by 9, too?  Ever wondered how to express thirteenths as decimals, in your head?  (Probably not, but maybe you’re wondering now!)  Want to know how to simplify fractions like a ninja?  Well, the mathematical ninja has the answers – and some cute stories, too.  Check it out!

A picture of the Julia set.

A picture of a Julia set.

Next, I find fractals fascinating, but – I’ll admit it – I don’t know much about them.  I do know a little about the number line and graphing, though.  And that was enough to learn a lot more about fractals from this excellent post on the blog Hackery, Math, and Design by Steven Wittens.  In the post How to Fold a Julia Fractal, Steven describes how the key to understanding fractals is understanding complex numbers, which are the numbers we get when we combine our normal numbers with imaginary numbers.

complex multiplicationNow, I think imaginary numbers are some of the most interesting numbers in mathematics – not only because they have the enticing name “imaginary,” but because they do really cool things and have some fascinating history behind them.  Steven does a really great job of telling their history and showing the cool things they do in this post.  One of the awesome things that imaginary numbers do is rotate.  Normal numbers can be drawn on a line – and multiplying by a negative number can be thought of as changing directions along the number line.  Steven uses pictures and videos to show how multiplying by an imaginary number can be thought of as rotating around a point on a plane.

here comes the julia set

A Julia set in the making.

The Julia set fractal is generated by taking complex number points and applying a function to them that squares each point and adds some number to it.  The fractal is the set of points that don’t get infinitely larger and larger as the function is applied again and again.  Steven shows how this works in a series of images.  You can watch the complex plane twist around on itself to make the cool curves and figures of the Julia set fractal.

Steven’s blog has many more interesting posts.  Check out another of my favorites, To Infinity… and Beyond! for an exploration of another fascinating, but confusing, topic – infinity.

Finally, a Pi Day doesn’t go by without the mathematicians and mathematical artists of the world putting out some new Pi Day videos!  Pi Day was last Thursday (3/14, of course).  Here’s a video from Numberphile in which Matt Parker calculates pi using pies!

In this video, also from Numberphile, shows how you only need 39 digits of pi to make really, really accurate measurements for the circumference of the observable universe:

Finally, it wouldn’t be Pi Day without a pi video from Vi Hart.  Here’s her contribution for this year:

Bon appetit!