Author Archives: Paul Salomon

2016, ScienTile, and a New Algorithm

Welcome to this week’s Math Munch!

In this week’s post we check out a tile designing contest from 2010, learn about some breakthrough news in computational algorithms, and get a DIY project to ring in the new year.

Speaking of the new year… welcome to the new year!! 2016 is 11111100000 in binary, by the way. Pretty cool right!? The five 0’s at the end tell you that 2016 has five 2’s in its prime factorization. That is, you can divide 2016 by 2 five times and still get a whole number. The big bunch of 1’s at the start means its also divisible by a number that is one less than a power of 2. 63, basically.

That is to say, 2016 = (26–1)(25). I think that signals a promising year. Bring it on.

2016 card project

DIY Möbius strip project to ring in the new year

cord project template

template here

Here’s a great way to start your year off. How about a paper folding project from mathematical artist Rinus Roelofs? I found the project posted by on their facebook page. According to the post, “this card is the representation of a Möbius strip.”

Click here for the downloadable, printable, cuttable, foldable, template and make your own!


A tile pattern I designed. Anyone else thinking bee hive?

Up next, I’ve been playing a lot of board games that use square or hexagon tiles, and I’ve been thinking about what other tiles might make for a cool game. First off, here’s a little tile I came up with that always leaves hexagons in the places where they meet. Might make a neat game  where you build a bee colony. Who knows.  But in my searching for groovy tiles, I found ScienTile.

ScienTile was an “open tile design competition” initiated by Dániel Erdély, a Hungarian mathematician and mathematical artist featured previously on MM for his spidrons.  In fact, ScienTile was meant to commemorate the 2010 Bridges Conference, which was in Hungary. Sadly, I don’t think the ScienTile competition was repeated in later years, but the results from 2010 are quite beautiful. I was most struck by the picture below, a tile designed by Gabor Gondos. I also really liked this one by the wonderful Craig Kaplan (featured previously here), but all the submissions can be found here.

A gorgeous and flexible tile design by Gabor Gandos

A gorgeous and flexible tile design by Gabor Gondos

All these graphs are isomorphic, and the new algorithm could tell you that... really fast!

All these graphs are isomorphic, and the new algorithm could tell you that… really fast!

Finally, some breakthrough math news from the computational world. Computer scientists have develop a new fast algorithm for solving” the graph isomorphism” problem, which simply checks whether or not two graphs (think connect-the-dots pictures) are really the same. All the graphs in the gif on the right are isomorphic, because they can be morphed into each other without changing the connectivity of the dots.


The 5,2 Johnson Graph

The new algorithm breaks a computational record that was unbroken for the last 30 years, which is a crazy long time in computer terms. Congratulations László Babai, who can be seen below presenting his breakthrough paper at the University of Chicago. His algorithm actually doesn’t cover all types of graphs, but Babai was able to show that the only type of graph not covered were the highly symmetric Johnson Graphs. You can see one of these on the right.

You can find more info on the record-breaking algorithm in this article from or this write up from Quanta Magazine.


László Babai presenting his record-breaking algorithm

Have a great week, and bon appetit!

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!

Dearing, Edmark, and The Octothorpean Order

Welcome to this week’s Math Munch!

Dearing Wang

Dearing Wang

First up is a wonderful mathematical artist I found on instagram, under the name dearing_draws. Click to see the wonderful work of Dearing Wang. The instagram stream includes lots of timelapse videos showing the creation of the images, which is lovely, but even better is that Dearing has a youtube channel and a website devoted to teaching people how to make their own!! You should click over and follow a tutorial. Make something beautiful and send us a picture.

3 Fish in a Pond

3 Fish in a Pond

Tutorial Video

The Diamond Wedge Pattern

The Diamond Wedge Pattern

Tutorial Video

Impossible Octagon

Impossible Octagon

Tutorial Video

Another great thing about Dearing’s website is that he has a page where you can print out blank sheets to color, if that’s your thing. Not quite as mathematical, maybe, but it is nice. I like to color sometimes, and if you color systematically, maybe symmetrically, then it’s fairly mathematical after all. UPDATE: Dearing has agreed to let us host some some of his coloring sheets on Math Munch.  Click here for easily downloadable sheets to color.

John Edmark

John Edmark

Up next is another mathematical artist, John Edmark, a designer and adjunct professor at Stanford University. I was introduced to John’s incredible work through the following video. Just watch and let your jaw hit the floor in amazement.

This is a video of a zoetrope. The pieces spin and the camera shutter is timed to only show certain points in their rotation. What we see is sort of like a little loop of film showing us several frames of the animation. It’s impressive that John put all those frames together into sculptures that are beautiful, even when they’re not spinning.



But that isn’t all, there’s lots more to see on John’s website. I found his spiral videos pretty mesmerizing and fantastic. I also really like his artist statement, which begins “If change is the only constant in nature, it is written in the language of geometry.” I also just really like hearing artists talk about their work, because it’s a sort of behind the scenes look into their creative process and thinking.

(3D printable files are also available here for the incredibly fortunate among us with access to a 3D printer.)

An octothorpe

An Octothorpe

Finally, if you like solving riddles and puzzles, check out The Octothorpean Order. This is sort of an online puzzle hunt, with clues and tips on the website. You can read about it, but the best thing to do is dive in and start solving puzzles. You probably have to create a user name, but it’s good fun. I recommend it.

By the way,  “octothorpe” is the technical word for the “hashtag” or “pound” or “number sign.” It means eight fields, and I think it represents a farmers house in the middle and eight fields arround it. Cool right?

Here’s to having a mathematical week.  Bon appetit!

Mars, Triangulation, and LOMINOES.

Welcome to this week’s Math Munch!

First things first, I simply must mention a video that one of our readers sent us. Lily Ross was inspired by a recent post and created this amazing fake movie trailer!!! WOW! Thank you, Lily!

The video has been added to our Readers’ Gallery. Send us your creations and we’ll add them too.

Did you know that NASA is planning to send people to Mars around the year 2030? How far away would they be going? Click the picture to find out. It’s incredibly cool.

How Far is Mars?

How far is it to Mars?



The Moon

The Moon is an interactive website that answers the question, “how far is it to Mars?” It was created by a pair of designers, David Paliwoda and Jesse Williams. Think of how long that took to get there, and now realize that it takes light 3 times longer (since we were traveling impossibly fast, at 3 times the speed of light). That’s 3 light-minutes, so when we look at “the red planet,” we are seeing light that took more than 3 minutes to make the trip from Mars to our eye. We’re seeing what Mars looked like 3 minutes in the past!!! That’s pretty cool, I’d say.

Triangulation #9

Triangulation #9

Up next, another interactive website experience. This one is a series of interactive digital art — a sort of meditation on the essence of the triangle. Check out Triangulation.  Can you imagine adding a page to this? What would you design? Maybe you could use Scratch to actually make it!

Thanks to our friend, Malke Rosenfeld, for sending us this.

Screen Shot 2014-12-05 at 10.21.18 PM Screen Shot 2014-12-05 at 10.23.09 PM Screen Shot 2014-12-05 at 10.24.40 PM

Before we get to our last item this week, a couple of important announcements. As in prior years, Plus Magazine is hosting a mathematical advent calendar. Each day, a new number becomes clickable, linking to a page about nifty math stuff.

The 2014 Plus Magazine Mathematical Advent Calendar

The 2014 Plus Magazine Mathematical Advent Calendar

I also want to mention that The Aperiodical (an awesome (fairly advanced) math blog) is hosting a Math Pun Conmpetition!!! Here’s my submission, for those with a little bit of plane geometric knowledge:

Q: Why was it so hard for the equilateral quadrilateral to get home after school?

A: It got on the rhom BUS!

Rggie Rhombus

OK, now on to our last item of the week. Here it is…

A Pot-Pourri of People, Pictures, Places, Penrose Patterns, Polyhedra, Polyominoes, Posters, Posies, and Puzzles! (How about that?)

Alan Schoen with a model of a gyroid

Alan Schoen with a model of a gyroid

I don’t know a whole lot about Alan Schoen, but his website has some pretty enticing images on it. Really, all I know about Schoen is that he discovered the Gyroid when he worked at NASA in 1970. He also created The Geometry Garret, a website full of cool stuff.

The thing that I want to share is something I’ve never seen before – LOMINOES. These are polyominoes, like the ones we’ve featured at least twice before, but they are simply in the shape of an L. Alan wrote a 10-page booklet on the subject as well as a much longer book. (147 pages!)

They’re both worth poking through. If an image grabs your fancy, start reading and see what you can learn.

Screen Shot 2014-12-06 at 1.52.16 PMHave a great week and bon appetit!

Scary-o-graphic Projection, Thinky the Dragon, and Martin Gardner

Welcome to this week’s Math Munch!

Halloween is quickly approaching, which is why last week, Anna shared some pumpkin polyhedra. It just so happens that Justin did some pumpkin-y math of his own last year. He created a must-watch video called “Scary-o’-graphic Projection,” which was shown in the 2014 Bridges Short Film Festival. Enjoy, but don’t get too scared.

A stereographic projection sculpture by Henry Segerman.

A stereographic projection sculpture by Henry Segerman.

To learn some more about stereographic projection, watch one of Henry Segerman’s videos. You’ll also get to see some of his 3D printed sculptures.  (1 2)

In other news, Oct. 21 marked the 100th anniversary of the birthday of Martin Gardner!! (previously featured here, here, and here, among others) Around this time every year people get together to do math in his honor as part of Celebration of Mind.

This year we’re featuring one of Gardner’s optical illusions. Let’s begin with a video. Meet Thinky the Dragon.

You can find printable make-your-own templates here. (There are other colors as well.) Thinky is an example of a “hollow face” illusion, many more of which can be found on There you can also find this video explaining the geometry behind this illusion.

And look at this AMAZING movie trailer that one of our readers made.  Thank you Lily!!!

Can you fold this strip of 7 squares into a cube?

Can you fold this strip of 7 squares into a cube?

Whats special about this square?  More than you think!

Whats special about this square? More than you think!

Thank you to Colm Mulcahy for his recent post on the BBC website, where Colm put together a list of 10 really wonderful problems from the hundreds that Gardner wrote about and popularized during his career. Gardner helped show the world that thinking about problems and mathematics was a really fun way to spend time. Watch the video below to learn more about Celebration of Mind events, and click here to see if there’s an event near you.  Note: You can even host an event of your own.

BONUS: I just have to mention MoSAIC for any math art enthusiasts in our audience. Around the country, small mathematical art conferences and exhibitions will go on this year. Click to learn more or find an event near you.

Munch in honor of Martin Gardner. Bon appetit!

Picture from the MoSAIC website.

Picture from the MoSAIC website.

Marc Chamberland, Math Fonts, and Congruent Triangles

Welcome to this week’s Math Munch!

Marc Chamberland

Marc Chamberland

Up first, a follow up to our post about the World Cup a while back. We received an email from Marc Chamberland linking us to a nice little video (below) about World Cup Balls and their various properties. You may remember seeing Marc’s mathematical art in this post. Below you can see another nice piece that was included in the mathematical art exhibit at the 2013 Joint Mathematics Meetings. Click for a nice description of the math puzzle it solves.  (in short: What’s the area of the red square?)

"Inner Square" by Marc Chamberland

“Inner Square” by Marc Chamberland

Marc is a math professor at Grinnell College. In March of 2014 (3-14?) he began working on Tipping Point Math, a youtube channel full of videos showing “math as you never imagined.” I encourage you to find something nice there. For now, here’s that video about World Cup Balls I promised you.

A font based on glass bending

A font based on glass bending

Up next are some nifty, fun fonts based on mathematics. Erik Demaine is no stranger to Math Munch readers, and it’s no wonder why. His stuff is clever and downright intriguing. He and his father Martin published a very interesting paper last April about a series of mathematical typefaces they’ve created over the course of their last decade of research and play.

A Conveyor Belt Font by Erik and Martin Demaine

A conveyor belt typeface by Erik and Martin Demaine

Their paper was published to the arXiv (pronounced “archive”) where it is publicly available.  You can read it here. Or, if you like something slightly more plain-language, here’s a nice review over on

Screen Shot 2014-09-15 at 9.05.54 PM Screen Shot 2014-09-15 at 9.07.28 PM
The 4051 Tektronix Graphics Terminal

The 4051 Tektronix Graphics Terminal

Finally, I want to share a sleepy little video called “Congruent Triangles.”  I like to think of it as a slice of mathematical cultural history. This film was made in 1977 on an early computer called a Tektronix 4051 Graphics Terminal. It was made by Bruce and Katherine Cornwell as part of a series of mathematical videos. The way the shapes move and deform to present the ideas and connect the pieces together is so very cool. I also love the choice of music. It tells you something about what math was like for people then. I’d say sort of “groovy.”

Screen Shot 2014-09-15 at 9.22.09 PMThere’s more to the story and many more cool videos to enjoy.  You can look forward to seeing more from the Cornwells, but for now, enjoy this one video and do some hunting on your own if you’re interested.  That’s called “research.”

Bon appetit!

The Art of Merete Rasmussen, a Game About Squares, and VAX!

Welcome to this week’s Math Munch! We’ve got a pair of new games for you to play later, but first I want to share something beautiful and impressive.

Hyperseeing Summer '14

Ready for some mathematical art? The new issue of Hyperseeing begins with a review of Merete Rasmussen’s ceramic sculpture. Merete is a Danish artist who lives in London, and her recent work features complex and beautiful, smooth two-dimensional surfaces.

Editor Nat Friedman’s writeup begins with this wonderful quote by Rasmussen:


“I want to create a form that you can’t understand until you see the other side. You have to look at it for a while to realize how it is connected.”

Merete Rasmussen at work

Merete Rasmussen at work

A lot of mathematical work is done just trying to describe and understand the ideas or pictures in our head. Merete’s sculpture get us to do math as we try to understand the nature of her sculptural surfaces. How many sides do they have? How many edges? How many holes? I just love that.

Blue Gray

The article is very enjoyable, and I encourage you to read the entire text, but what got me hooked, what completely mesmerized and inspired me, was a video about Merete’s work and process that I found referenced at the end of the article. The video is presented in dual screen, which is really fantastic, because just like Merete’s sculptures, you may need to view it a couple of times to catch all that’s going on.

I recommend the full video. I recommend full screen.

You can learn more about Merete Rasmussen and view more of her work at her website,

* * *

OK, now on to a couple of new games.

Game About Squares

Game About Squares“Right from the start I was thinking about creating a simple game, with simple graphics and simple game design.” That’s what 26-yr old Andrey Shevchuk said about his recent creation, “Game About Squares.” You’ll find as you play, however, that these little puzzles can get oh so complicated, despite their simple presentation.

I love imagining how Andrey must have had to think creatively to keep developing his simple idea in new ways, and I love the way that the puzzles get us to think in new ways. All in all, this is just a wonderful game.

Oh, and thinking about the very viral 2048, Andrey had this to say,

“Squares are trendy.  Hexagons aren’t even close, let alone triangles.”


Screen Shot 2014-08-08 at 9.29.36 PMThat’s short for vaccine, in case you don’t know.  The Salathé Group recently released a game about vaccinations and fighting the spread of epidemics (previously). The game is called VAX!, and it’s based on a graph theory representation for the spread of disease. Take the tour and you’ll learn everything you need to play.

There’s also a module that explains herd immunity. That’s where random vaccines are used to isolate the potentially infected from potential carriers of the disease.

Bon appetit.  Dig in!