Tag Archives: geometry

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.

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.

And 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.

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.

I 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!

Posted by Justin Lanier. Categories: Math Munch. Tags: applet, applied math, art, board games, cellular automata, comics, dimensions, games, geometry, puzzles, tessellation, video. 27 Comments

Jan. ’13

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…

Well, 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.

Higher 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.

Finally, 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!

Posted by Anna Weltman. Categories: Math Munch. Tags: art, chess, dimensions, doodling, geometry, puzzles, tessellation, video. 10 Comments

Dec. ’12

The Museum of Math, Shapes That Roll, and Mime-matics

Welcome to this week’s Math Munch! We have so many exciting things to share with you this week – so let’s get started!

Something very exciting to math lovers all over the world happened on Saturday. The Museum of Mathematics opened its doors to the public!

The Museum of Mathematics (affectionately called MoMath – and that’s certainly what you’ll get if you go there) is in the Math Munch team’s hometown, New York City.

There are so many awesome exhibits that I hardly know where to start. But if you go, be sure to check out one of my favorite exhibits, Twist ‘n Roll. In this exhibit, you roll some very interestingly shaped objects along a slanted table – and investigate the twisty paths that they take. And you can’t leave without seeing the Human Tree, where you turn yourself into a fractal tree.

Or going for a ride on Coaster Rollers, one of the most surprising exhibits of all. In this exhibit, you ride in a cart over a track covered with shapes that MoMath calls “acorns.” The “acorns” aren’t spheres – and yet your ride over them is completely smooth! That’s because these acorns, like spheres, are surfaces of constant width. That means that if you pick two points on opposite ends of the acorn – with “opposite” meaning points that you could hold between your hands while your hands are parallel to each other – the distance between those points is the same regardless of the points you choose. See some surfaces of constant width in action in this video:

One such surface of constant width is the shape swept out by rotating a shape called a Reuleaux triangle about one of its axes of symmetry. Much as an acorn is similar to a sphere, a Reuleaux triangle is similar to a circle. It has constant diameter, and therefore rolls nicely inside of a square. The cart that you ride in on Coaster Rollers has the shape of a Reuleaux triangle – so you can spin around as you coast over the rollers!

Maybe you don’t live in New York, so you won’t be able to visit the museum anytime soon. Or maybe you want a little sneak-peek of what you’ll see when you get there. In any case, watch this video made by mathematician, artist, and video-maker George Hart on his first visit to the museum. George also worked on planning and designing the exhibits in the museum.

We got the chance to interview Emily Vanderpol, the Outreach Exhibits coordinator for MoMath, and Melissa Budinic, the Assistant Exhibit Designer for MoMath. As Cindy Lawrence, the Associate Director for MoMath says, “MoMath would not be open today if it were not for the efforts” of Emily and Melissa. Check out Melissa and Emily‘s interviews to read about their favorite exhibits, how they use math in their jobs for MoMath, and what they’re most excited about now that the museum is open!

Finally, meet Tim and Tanya Chartier. Tim is a math professor at Davidson College in North Carolina, and Tanya is a language and literacy educator. Even better, Tim and Tanya have combined their passion for math and teaching with their love of mime to create the art of Mime-matics! Tim and Tanya have developed a mime show in which they mime about some important concepts in mathematics. Tim says about their mime-matics, “Mime and math are a natural combination. Many mathematical ideas fold into the arts like shape and space. Further, other ideas in math are abstract themselves. Mime visualizes the invisible world of math which is why I think math professor can sit next to a child and both get excited!”

One of my favorite skits, in which the mime really does help you to visualize the invisible world of math, is the Infinite Rope. Check it out:

In another of my favorite skits, Tanya interacts with a giant tube that twists itself in interesting topological ways. Watch these videos and maybe you’ll see, as Tanya says, how a short time “of positive experiences with math, playing with abstract concepts, or seeing real live application of math in our world (like Google, soccer, music, NASCAR, or the movies) can change the attitude of an audience member who previously identified him/herself as a “math-hater.”” You can also check out Tim’s blog, Math Movement.

Tim and Tanya kindly answered some questions we asked them about their mime-matics. Check out their interview by following this link, or visit the Q&A page.

Bon appetit!

Posted by Anna Weltman. Categories: Math Munch. Tags: geometry, infinity, interview, mime, museum of math, shapes, topology, video. 10 Comments