Tag Archives: dimensions

roTopo, de Gua, and Bibi-binary

Welcome to this week’s Math Munch!

Today we’re going to look at a few examples of going “up a dimension”. Our first example is what got me thinking about this theme. It’s a game called roTopo. (If you have trouble getting it to load, try using a different browser.)

 rotopo1.png  rotopo2

Maybe you have played the game B-Cubed. RoTopo is similar—trace through a sequence of squares as they get eliminated one by one. I like B-Cubed because it combines spatial thinking with strategic thinking—planning ahead. Rotopo, with its twists and turns in 3D, stretches a player’s spatial thinking even further. I hope you enjoy giving it a try! Maybe you could design a roTopo level of your own with a drawing or with some blocks.

What else can we find when we look “up a dimension”? Maybe the most famous theorem in all of mathematics is the Pythagorean theorem. There are several ways we might try to take a^2+b^2=c^2 up a dimension. If we start to increase the numbers in the exponents, like a^3+b^3=c^3, we head in the direction of Fermat’s Last Theorem. If we add more terms, like a^2+b^2+c^2=d^2, we can find distances in 3D instead of 2D.


A right tetrahedron—the kind needed for de Gua’s Theorem.

And if those aren’t enough to make you go “wow”, then you need to hear about De Gua’s Theorem. The Pythagorean Theorem relates the sides of a right triangle. De Gua’s Theorem relates the faces of a right tetrahedron. The sum of the squares of the areas of the the three “leg” faces is equal to the square of the area of the “hypotenuse” face. So wild! You can read a proof de Gua’s Theorem here. The theorem is named for the 18th-century French mathematician who presented it to the Paris Academy of Sciences in 1783 (although it was known to others before him). De Gua’s Theorem in turn is a special case of a still more general theorem. Once mathematicians start upping dimensions, the sky is the limit!

Last up: Bibi-binary. No, that’s not the way that Justin Timberlake counts—although that funny thought is why I Googled “bibibinary” in the first place. But when I did, this totally silly number system popped up!


How to count in Bibi-binary.

Well, I guess it’s not the number system that’s silly so much, since it’s actually just hexadecimal. Hexadecimal is like binary, but up a couple of dimensions. The system uses sixteen symbols to represent numbers, just as the decimal system uses ten symbols and binary uses two. What makes Bibi-binary silly, then, is not its logical structure but how it sounds.

There are sixteen syllables in Bibi-binary, which are made from combinations of four consonants and four vowels. Three is “hi” and eight is “ko”. If you want to have three 16’s and eight more—56—that would be “hiko”. As another example, 66319344 is “hidihidihidiho”. Bibi-binary was invented in 1971 by a French singer and actor named Boby Lapointe.

I think it would be fun to learn to count in Bibi-binary. Can you believe that I could find zero (“ho”) videos online of people counting in Bibi-binary? I wonder if any of our readers might enjoy making one…

img_colormapHexadecimal is not just fun and games. It’s also used for making codes to stand for colors, especially in making webpages. Most of Math Munch is either 683D29 or 6AB690, would you believe. You can explore using hexadecimal to name colors in this applet.

You can learn lots more about Bibi-binary on the great website dCode, and you’ll also find an applet there that can convert between decimal and Bibi-binary. DCode has lots of tools related to cryptography (get it?) and other math topics, too.

Do you have any favorite examples of math that goes “up a dimension”? We’d love to hear about them in the comments.

Bibi-bi for now! Bon appetit!


Digital Art, Mastermind, and Pythagoras

Welcome to this week’s Math Munch… on (approximately) Math Munch’s second birthday! Hooray!

Check out this video of mathematical art made by artist Nathan Selikoff:

Cool, right? This piece is called “Beautiful Chaos.” The curves on the screen are made from equations (if you’ve ever graphed a line or a parabola you’ll know what I mean). As the viewer waves her hands around, the equations change– and as the equations change, so do the curves! The result is something that might remind you of the images your computer makes when you play music on it or maybe of something you’d make using a spirograph. All in all, a beautiful and interactive piece of mathematical art.

nathanNathan lives and works as a mathematical artist in Orlando, Florida. As he writes on his website, Nathan uses computer code along with other materials to make art that plays with the mathematical ideas of space, motion, and interaction between objects. To see more of how Nathan does this, check out his giant, interactive marionette or this song that explores the first, second, third, and fourth dimensions:

My school is really lucky to be hosting Nathan this week! We didn’t want any of you, dear readers, to miss out on the excitement, though– so Nathan has kindly agreed to answer your interview questions! Got a question for Nathan? Write it in the box below. He’ll answer seven of your best questions in two weeks!

565px-MastermindNext up, who doesn’t love to play Mastermind? It’s a great combination of logic, patterns, and trickery… but I just hate having to use all those tiny pegs. Well, guess what? You can play it online— no pegs (or opponent) necessary!

As I was playing Mastermind, I started wondering about strategy. What’s the best first guess to make? If I were as smart as a computer, is there a number of guesses in which I could guess any Mastermind code? (This kind of question reminds me of God’s Number and the Rubik’s cube…)

Well, it turns out there is a God’s Number for Mastermind – and that number is five. Just five. If you played perfectly and followed the strategy demonstrated by recreational mathematician Toby Nelson on his website, you could guess ANY Mastermind code in five guesses or less. Toby shares many more interesting questions about Mastermind on his website— I suggest you check it out.

What ARE those irrational numbers, so weird that they get their own bubble??

What ARE those irrational numbers, so weird that they get their own bubble??

Finally, sometime in your mathematical past you may have heard of irrational numbers. These are numbers like the square-root of 2 or pi or e that can’t be written as a fraction– or so people claim. When you start thinking about this claim, however, it may seem strange. There are A LOT of fractions– and none of them equal the square-root of 2? Really? What kind of number is that? It seems like only an irrational person would believe that, at least without proof.

Vi Hart to the rescue! Irrational numbers were encountered long, long ago by the ancient Greek mathematician (and cult leader) Pythagoras– and he didn’t like them much. In this great video, Vi tells all about Pythagoras and the controversial discovery of numbers that aren’t fractions.

If you didn’t follow her explanation of why the square-root of 2 is irrational on your first watch, don’t worry– it’s a complicated idea that’s worth a second (or third or fourth) run-through.

Thanks for a great two years of Math Munch! Bon appetit!

MoMA, Pop-Up Books, and A Game of Numbers

Welcome to this week’s Math Munch!

Thank you so much to everyone who participated in our Math Munch “share campaign” over the past two weeks. Over 200 shares were reported and we know that even more sharing happened “under the radar”. Thanks for being our partners in sharing great math experiences and curating the mathematical internet.

Of course, we know that the sharing will continue, even without a “campaign”. Thanks for that, too.

All right, time to share some math. On to the post!

N_JoshiTo kick things off, you might like to check out our brand-new Q&A with Nalini Joshi. A choice quote from Nalini:

In contrast, doing math was entirely different. After trying it for a while, I realized that I could take my time, try alternative beginnings, do one step after another, and get to glimpse all kinds of possibilities along the way.

By Philippe Decrauzat.

By Philippe Decrauzat.

I hope the math munches I share with you this week will help you to “glimpse all kinds of possibilities,” too!

Recently I went to the Museum of Modern Art (MoMA) in New York City. (Warning: don’t confuse MoMA with MoMath!) On display was an exhibit called Abstract Generation. You can view the pieces of art in the exhibit online.

As I browsed the galley, the sculptures by Tauba Auerbach particularly caught my eye. Here are two of the sculptures she had on display at MoMA:

CRI_244599 CRI_244605

Just looking at them, these sculptures are definitely cool. However, they become even cooler when you realize that they are pop-up sculptures! Can you see how the platforms that the sculptures sit on are actually the covers of a book? Neat!

Here’s a video that showcases all of Tauba’s pop-ups in their unfolding glory. Why do you think this series of sculptures is called [2,3]?

This idea of pop-up book math intrigued me, so I started searching around for some more examples. Below you’ll find a video that shows off some incredible geometric pop-ups in action. To see how you can make a pop-up sculpture of your own, check out this how-to video. Both of these videos were created by paper engineer Peter Dahmen.

Taura Auerbach.

Tauba Auerbach.

Tauba got me thinking about math and pop-up books, but there’s even more to see and enjoy on her website! Tauba’s art gives me new ways to connect with and reimagine familiar structures. Remember our post about the six dimensions of color? Tauba created a book that’s a color space atlas! The way that Tauba plays with words in these pieces reminds me both of the word art of Scott Kim and the word puzzles of Douglas Hofstadter. Some of Tauba’s ink-on-paper designs remind me of the work of Chloé Worthington. And Tauba’s piece Componants, Numbers gives me some new insight into Brandon Todd Wilson’s numbers project.

0108 MM MM-Tauba-Auerbach-large

This piece by Tauba is a Math Munch fave!

For me, both math and art are all about playing with patterns, images, structures, and ideas. Maybe that’s why math and art make such a great combo—because they “play” well together!

Speaking of playing, I’d like to wrap up this week’s post by sharing a game about numbers I ran across recently. It’s called . . . A Game of Numbers! I really like how it combines the structure of arithmetic operations with the strategy of an escape game. A Game of Numbers was designed by a software developer named Joseph Michels for a “rapid” game competition called Ludum Dare. Here’s a Q&A Joseph did about the game.

A Game of Numbers.

A Game of Numbers.

If you enjoy A Game of Numbers, maybe you’ll leave Joseph a comment on his post about the game’s release or drop him an email. And if you enjoy A Game of Numbers, then you’d probably enjoy checking out some of the other games on our games page.

Bon appetit!

PS Tauba also created a musical instrument called an auerglass that requires two people to play. Whooooooa!

Reflection Sheet – MoMA, Pop-Up Books, and A Game of Numbers

Yang Hui, Pascal, and Eusebeia

Yanghui_triangleWelcome to this week’s Math Munch! I’ve got some mathematical history, an interactive visualization site, some math art, and a mathematical story from the fourth dimension for you.

Yang Hui's Triangle animated

First, take a look at the animation and picture above. What do you notice? This is sometimes called Pascal’s Triangle (click for background info and cool properties of the triangle.) It’s named for Blaise Pascal, the mathematician who published a treatise on its properties in 1653. (Click here for some history of Pascal’s life and work.)

Yang Hui

Yang Hui

BUT actually, Pascal wasn’t the first to play with the triangle. Yang Hui, a 13th century Chinese mathematician, published writings about the triangle more than 500 years earlier! Maybe we ought to be calling it Yang Hui’s Triangle! The picture above is the original image from Yang Hui’s 13th century book. (Also look at the way the Chinese did numbers at that time. Can you see out how it works at all?)  Edit: David Masunaga sent us an email telling us about an error in Yang Hui’s chart.  He says some editors will even correct the error before publishing.  Can you find the mistake?

I bring this all up, because I found a neat website that illustrates patterns in this beautiful triangle. Justin posted before on the subject, including this wonderful link to a page of visual patterns in Yang Hui’s triangle. But I found a website that lets you explore the patterns on your own! The website lets you pick a number and then it colors all of its multiples in the triangle. Below you can see the first 128 lines of the triangle with different multiples colored. NOW YOU TRY!



Multiples of 4

Multiples of 4

Ends in 5 or 0

Ends in 5 or 0

* * *

Recently, I’ve been working on a series of artworks based on the Platonic and Archimedean solids. You can see three below, but I’ll share many more in the future. These are compass and straight-edge constructions of the solids, viewed along various axes of rotational symmetry.

All of these drawings were done without “measuring” with a ruler, but I still had to get all of the sizes right for the lines and angles, which meant a lot of research and working things out. Along the way, I found eusebeia, a brilliant site that shows off some beautiful geometric objects in 3D and 4D. There’s a rather large section of articles (almost a book’s worth) describing 4D visualization. This includes sections on vision, cross-sections, projections, and anything you need to understand how to visualize the 4th dimension.

Uniform Polyhedra

A few uniform solids

The 5-cell and a story about it called "Legend of the Pyramid"

The 5-cell, setting for the short story, “Legend of the Pyramid

The site goes through all of the regular and uniform polyhedra, also known as the Platonic and Archimedean solids, and shows their analogs in 4D, the regular and uniform polychora. You may know the hypercube, but it’s just one of the 6 regular polychora.

I got excited to share eusebeia with you  when I found this “4D short story” at the bottom of the index. “Legend of the Pyramid” gives us a sense of what it would be like to live inside of the 5-cell, the 4D analog of the tetrahedron.

Well there you have it. Dig in. Bon appetit!

Yanghui_magic_squareBonus: Yang Hui also spent time studying magic squares.  (Remember this?)  In the animation to the right, you can see a clever way in which Yang Hui constructed a 3 by 3 magic square.

Mathematical Impressions, Modular Origami, and the Tenth Dimension

Welcome to this week’s Math Munch!

First up, check out the latest video in George Hart‘s series called “Mathematical Impressions.”  George has been making videos for “Mathematical Impressions,” which is sponsored by the Simons Foundation, since summer, when he made his video debut – so there are many videos to watch!  Here’s his newest video, called, “Attesting to Atoms,” about how the geometric structure of crystals gives clues to the existence of atoms.  (Click on the picture below to watch the video.)

Atoms video

I love how this video shows a real way in which knowledge of mathematics – which can seem very abstract at times – can help us to understand the structure of the world, which is very concrete.  In this second video, one of my favorites, George talks about the reverse of that – allowing our knowledge of something concrete to help us understand abstract mathematics.  This video is called, “Knot Possible.”  (Again, click on the picture to watch the video!)

Knot video

I could have used these words of wisdom from George when I was thinking about the problem he poses in this video: “Don’t let your knowledge of mathematics artificially limit what you think is physically possible.  Quite to the contrary!  Mathematics is a tool which can empower us to do amazing things that no one has ever done before.”  Well said, George!

sierpinski-tetrahedron-tri-2Speaking of using mathematics to do and make amazing things, check out this website of modular origami models and patterns!

This site was put together by Michal Kosmulski, who lives in Poland and works in information technology.  In addition, however, he folds these amazing modular origami polyhedra, fractals, and other awesome mathematical objects!  Michal’s site is full of pictures of his modular origami creations and links to patterns for how to make them yourself as well as information about the mathematics behind the objects.  He has also included some useful tips on how to make the more challenging shapes.

fit-five-intersecting-tetrahedra-60deg-2One of my favorites is the object to the left, “Five Intersecting Tetrahedra.”  I think that this structure is both beautiful and very interesting.  It can be made by intersecting five tetrahedra, or triangular-based pyramids, as shown, or by making a stellation of an icosahedron.  What does that mean?  Well, an icosahedron is a polyhedron with twenty equilateral triangular faces.  To stellate a polyhedron, you extend some element of the polyhedron – such as the faces or edges – in a symmetric way until they meet to form a new polyhedron.  There are 59 possible stellations of the icosahedron!  Michal has models of several of them, including the Five Intersecting Tetrahedra and the great stellated icosahedron shown below on the left.  The figure on the right is called “Cube.”


Finally, all the talking about dimensions that we’ve been doing for the past few weeks reminded me of my favorite video about higher dimensions.  It’s called, “Imagining the Tenth Dimension,” and it shows a way of thinking about dimensions, from the zero dimension all the way up to the tenth.  I can watch this video again and again and still find it mind-blowing and fascinating.

Bon appetit!

Ghost Diagrams, Three New Games, and Scrabble Tiles

Welcome to this week’s Math Munch!


A ghost diagram composed of two different tiles.

An organism is more than the sum of its organs. When the organs are fitted together, the organism becomes something more. This surprising something more we call “spirit” or “ghost”. Ghost Diagrams finds the ghosts implicit in simple sets of tiles.

So writes Paul Harrison, creator of the amazing Ghost Diagram applet. Paul creates all kinds of free software and has his Ph.D. in Computer Science. I found his Ghost Diagram applet through this huge list of links about generative art.

A '111-' tile connected to a '1aA1' tile.

A ‘111-‘ tile connected to a ‘1aA1’ tile.

Given a collection of tile types, the applet tries to find a way to connect them so that no tile has any loose ends. A tile type is specified through a string of letters, numbers, and dashes. Each of these specifies an edge. You can think of a four-character tile as being a modified square and a six-character tile as being a modified hexagon. Two tiles can connect if they have edges that match. Number edges match with themselves—1 matches with 1—while letter edges match with the same letter with opposite capitalization—a matches with A.

It’s amazing the variety of patterns that can emerge out of a few simple tiles. Here are a couple of ghost diagrams that I created. You can click them to see live versions in the applet. There are many other nice ghost diagrams that Paul has compiled on the site. Also, be sure to check out the random button—it’s a great way to get started on making a pattern of your own. I hope you enjoy tinkering with the ghost diagram applet as much as I have.

gd1 gd3
gd4 gd2

loops-of-zenAnd now for some more fun: three new games! When I ran across Loops of Zen, I had ghost diagrams on my mind. I think they have a similar feel to them. The goal in each level of Loops of Zen is to orient the paths and loops so that they connect up without any loose edges. I feel like this game—like good mathematics—requires both a big-picture, intuitive grasp of the playing field and detailed, logical thinking. Put another way, you need both global strategy  and local tactics. Also, if you like playing Entanglement, then I bet you’ll like Loops of Zen, too.


Last week we wrote about Flatland. This book and the movies it inspired describe what it might be like if creatures of different dimensionality were to meet each other. The game Z-Rox puts you in the shoes of a Flatlander. Mystery shapes pass through your field of vision a slice at a time, and it’s up to you to identify what they are. It’s a tricky task that requires a good imagination.

Hat tip to Casual Girl Gamer for both of these great mathy games.


Steppin’ Stones

Steppin’ Stones is a fun little spatial puzzle game I recently came across. You should definitely check it out. It also provides a nice segue to our last mathy item for the week, because a Steppin’ Stones board looks a lot like a Scrabble board. Scrabble, of course, is a word game. Aside from the arithmetic of keeping score, there isn’t much mathematics involved in playing it. In addition, the universe of Scrabble—the English dictionary—is not particularly elegant from a math standpoint. However, it’s the amazing truth that even in arenas that don’t seem very mathematical, math can often be applied in useful ways.

From a comic about Prime Scrabble on Spiked Math.

From a comic about Prime Scrabble on Spiked Math.

In Re-evaluating the values of the tiles in Scrabble™, the author—who goes by DTC and is a physics graduate student at Cornell—wonders whether the point values assigned to letters in Scrabble are correctly balanced. The basic premise is that the harder a letter is to play, the more it should be worth. DTC does what any good mathematician does—lays out assumptions clearly, reasons from them to make a model, critiques the arguments of others, and of course makes lots of useful calculations. One tool DTC uses is the Monte Carlo method. In the end, DTC finds that the current Scrabble point values are very close to what the model would assign.

MATHI really enjoyed the article, and I hope you will, too. And since Scrabble is a “crossword game”, I think I’ll leave you with a couple of “crossnumber” puzzles. Here are some straightforward ones, while these require a little more thinking.

Have a great week, and bon appetit!

P.S. I can’t resist sharing this video as a bonus: a cellular automaton of rock-paper-scissors! Blue beats green, green beats red, and red beats blue. Hooray for non-transitive swirls!

A Closet Full of Puzzles, Sphereland, and Math Doodles

Welcome to this week’s Math Munch!

After a few weeks off, we’re back with some exciting things to share.  First up is Futility Closet, a blog featuring “an idler’s miscellany of compendious amusements.”  The blog is full of big-worded phrases like that, but I most love the puzzles they often post – everything from chess to numbers, codes, and devilish word play.  I also love that the name of the person who wrote each puzzle accompanies it.  Take a look at the few I’ve posted below and click here for the full list of puzzles.

Here’s a puzzle called Swine Wave, by Lewis Carroll. The puzzle: Lace 24 pigs in these sties so that, no matter how many times one circles the sties, he always find that the number in each sty is closer to 10 than the number in the previous one. Want to know the solution? Click on the image above to visit Futility Closet.
This puzzle is called Project Management, by Paul Vaderlind. The question: If a blacksmith requires five minutes to put on a horseshoe, can eight blacksmiths shoe 10 horses in less than half an hour? The catch: A horse can stand on three legs, but not on two. Click on the image to visit Futility Closet for the solution!

Next, have you ever wondered what it would be like to visit another dimension?   In 1884, Edwin A. Abbott wrote about life in the second dimension, in a nice little book called Flatland: A Romance of Many Dimesnions.  (Fun fact: the “A” in Edwin’s name stands for Abbott.  So his name is Edwin Abbott Abbott.)  Click on that link and you can read the whole book, if you like.  The book is about a world of flat beings who have no idea that the third dimension exists.  In the book, the main character, A Square, is visited by a sphere from the unknown world “above” him.  Kind of makes me wonder whether we’re just like the characters in Flatland, three-dimensional creatures ignorant of the fourth dimension that exists “above” us…

spherelandWell, the recently released movie Flatland 2: Sphereland deals with precisely that issue.  The Math Munch team had the opportunity to preview this movie, and we loved it.  In Sphereland, the granddaughter of the Square from Flatland, Hex, and her friend Puncto try to understand some mysterious triangles that Puncto thinks will cause the disastrous end of a space exploration mission and go on an adventure to help their three-dimensional friend Spherius with a problem he brought back from the fourth dimension.

portfolio-TorusHigher dimensions can be very difficult to wrap your head around.  This movie does a great job of helping the movie-watcher to understand how higher and lower dimensions relate to each other through the plot twists and challenges that the characters face.  You can really learn a lot about dimensions and the shape of space by watching this movie.  Plus, the characters are engaging and the images are fun.  Sphereland features the voices of a number of really great actors, including Kristen Bell, Danny Pudi, Michael York, and Danica McKellar.

Want to learn more about Sphereland?  Check out the trailer:

And, here’s an interview with Danny Pudi, the voice of Puncto, and Tony Hale, who does a fantastic job as the King of Pointland:

By the way, the makers of Sphereland also made a movie of Flatland!  The Math Munch team loved that one, too.  Here’s a link to the trailer.

tumblr_mgw2ainZDX1s0payeo1_1280Finally, check out this beautiful blog of mathematical doodles by high school math student and artist Chloé Worthington!  Chloé started mathematically doodling a few years ago in… well, in class.  When she doodles in class, Chloé is better able to focus on what’s going on and makes beautiful art.   (We at Math Munch encourage you to pay attention in class while you doodle.)

Chloé does all of her doodles by hand with ink pens.  She does a lot of work with triangles, as shown here.  One of her signature doodles is this nested puzzle piece doodle:


Doodling mathematically is one of the ways that Chloé does math and shares what she loves about it with the world.  She’s a trigonometry student, too.  How do you share what you love about math – or any other subject?

Bon appetit!