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 medium.com.

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!

Zentangle, Graph Paper, and Pancake Art

My recent doodling.

Some recent doodling, by me.

Welcome to this week’s Math Munch!

As you start a new school year, you might be looking for some new mathy doodle games to play in the margins of your notebooks. Doodling helps me to listen sometimes, and I love making neat patterns. I especially like seeing what new shapes I can make.

This summer I was very happy to run across Zentangle®—”an easy-to-learn, relaxing, and fun way to create beautiful images by drawing structured patterns.” I’ve learned a lot about Zentangle from a blog called Tangle Bucket by Sandy Hunter. She shares how to doodle snircles, snafoozles, and oodles. There’s a whole dictionary of zentangle shapes over at tanglepatterns.com.

My favorite idea in Zentangle is trying to combine two kinds of designs. Sometimes this is described as one pattern “versus” another one. For instance, check out these:

RPvsA RIvsJ

Maybe you’ll pick some tangle patterns to combine with each other. If you try some, maybe you’ll share them in our Readers’ Gallery.

Sandy writes:

It’s so true that the more I tangle, the more I see the potential in patterns all around me. I catch myself mentally deconstructing them (whether I want to or not) to figure out if they can be broken down into simple steps without too much effort. That’s the trademark of a good tangle pattern.

Try some of Sandy’s weekly challenges, or check out Tiffany Lovering’s time-lapse videos—here’s one with music and one with an interview. Can you learn the names of any of the shapes she creates? I spy a Rick’s Paradox. There are lots of ways to begin zentangling—I hope you enjoy giving it a try.

Squares and dots and crosses, oh my!

Squares & dots & crosses, oh my!

If zentangling is too freeform for your doodling tastes, then let me share with you one of my longtime favorite websites. I’ve used it for years to help me to do math and to teach math, and it’s great for math doodling, too. I might even call it a trusty friend, one that I met one day through the simple online search: “free online graph paper”.

That’s right, it’s Free Online Graph Paper.

Something I love about the site is that it lets you design different aspects of your graph paper. Then you can print it out. First you get to decide what kind of grid you would like: square? triangular? circular? Then you get to tinker with lots of variables, like how big the grid cells are, how dark the lines are, and what color they are. And more!

Free Online Graph Paper was created by Kevin MacLeod, who composes music and shares it for free. That way other people can use it for creative projects. That’s really awesome! I enjoyed listening to Kevin’s “Winner Winner“. It’s always good to be reminded that everything you use or enjoy was almost certainly made by a person—including custom graph paper websites!

A 7/3 star spirocake.

A 7/3 star spirocake.

Last up this week is some doodly math that you can really munch on. Everyone knows that breakfast is the most important meal of the day and that the most important food group is roulette curves.

To get your daily recommended allowance of groovy math, look no further than the edible doodles of Nathan Shields and his family over at Saipancakes.

I can wait until the Shields family tackles the cissoid of Diocles.

Bon appetit!

Fields Medal, Favorite Numbers, and The Grapes of Math

Welcome to this week’s Math Munch! And, if you’re a student or teacher, welcome to a new school year!

fieldsOne of the most exciting events in the world of math happened this August– the awarding of the Fields Medal! This award honors young mathematicians who have already done awesome mathematical work and who show great promise for the future. It also only happens every four years, at the beginning of an important math conference called the International Congress of Mathematicians, so it’s a very special occasion when it does!

 

Maryam Mirzakhani, first woman ever to win a Fields Medal

Maryam Mirzakhani, first woman ever to win a Fields Medal

This year’s award was even more special than usual, though. Not only were there four winners (more than the usual two or three), but one of the winners was a woman!

Now, if you’re like me, you probably heard about the Fields Medal and thought, “There’s no way I’ll understand the math that these Field Medalists do.” But this couldn’t be more wrong! Thanks to these great articles from Quanta Magazine, you can learn a lot about the super-interesting math that the Fields Medalists study– and why they study it.

MB_thumb-125x125

Manjul Bhargava

One thing you’ll immediately notice is that each Fields Medalist has non-math interests that inspire their mathematical work. Take Manjul, for instance. When he was a kid, his grandfather introduced him to Sanskrit poetry. He was fascinated by the patterns in the rhythms of the poems, and the number patterns that he found inspired him to study the mathematics of number patterns– number theory!

But, don’t just take my word for it– you can read all about Manjul and the others in these great articles! And did I mention that they come with videos about each mathematician? 

Want to read more about this year’s Fields Medallists? Check out Alex Bellos’s article in The Guardian. Which brings me to…

download… What’s your favorite number? Is it 7? If it is, then you’re in good company! Alex polled more than 30,000 people about their favorite number, and the most popular was 7. But why? What’s so special about 7? Here’s why Alex thinks 7 is such a favorite:

grapes-of-mathWhy do you like your favorite number? People gave Alex all kinds of different reasons. One woman said about 3, her favorite number, “3 wishes. On the count of 3. 3 little pigs… great triumvirates!” Alex made these questions the topic of the first chapter of his new book, The Grapes of Math. (Get the reference?) In this book, Alex shares many curious ways that math appears in our world. Did you know that a weird pattern in numbers can be used to catch criminals? Or that the Game of Life, a simple computer program, shares surprisingly many characteristics with real life? These are only a few of the hundreds of topics Alex covers in his book. Whether you’re a math whiz or a newbie, you’ll learn something new on every page.

Alex currently writes about math for The Guardian in a blog called, “Alex’s Adventures in Numberland”– but he also loves and writes about soccer (or futbol, as it’s called in his native Brazil)! He even wrote a few articles for his blog about math and soccer. 

Do you have any questions for Alex? (About math, soccer, or their intersection?) Write them here and you might find them featured in our interview with Alex!

Good writing about math is hard to find. If you’ve ever picked up a standard math textbook, you’ll know what I mean. But reading something fascinating, that grabs your interest from the first page and leads you through the most complex ideas like they’re as natural as anything you’ve observed, is a great way to learn. The Grapes of Math and “Alex’s Adventures in Numberland” do just that. Give them a go!

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, mereterasmussen.com.

* * *

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.”

VAX!

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!

Zippergons, High Fashion, and Really Big Numbers

Welcome to this week’s Math Munch!

Bill Thurston

Bill Thurston

Recently I attended a conference in memory of Bill Thurston. Bill was one of the most imaginative and influential mathematicians of the second half of the twentieth century. He worked with many mathematicians on projects and had many students before he passed away in the fall of 2012 at the age of 65. You can read Bill’s obituary in the New York Times here.

Bill worked where geometry and topology meet. In fact, Bill throughout his career showed that there are rich connections between the two fields that no one thought was possible. For instance, it’s an amazing fact that every surface—no matter how bumpy or holey or twisted—can be given a nice, symmetric curvature. A uniform geometry, it’s called. This was proven by Henri Poincaré in 1907. It was thought that 3D spaces would be far too complicated to be behave according to a similar rule. But Bill had a vision and a conjecture—that every 3D space can be divided into parts that can be given uniform geometries. To give you a flavor of these ideas, here’s a video of Bill describing some unusual and fabulous 3D spaces.

Any surface can be given a nice, symmetric geometry.

Any surface can be given a uniform geometry. Even a bunny. Another video.

As you can probably tell, visualizing and experiencing math was very important to Bill. He even taught a course with John Conway called Geometry and the Imagination. Bill often used computers to help himself see the math he was thinking about, and he enjoyed making hands-on models as well. Beginning in spring of 2010, Bill and Kelly Delp of Ithaca College worked out an idea. Usually all of the curving or turning of a polyhedron is concentrated at the vertices. Most of a cube is flat, but there’s a whole lot of pinch at the corners. What if you could spread that pinching out along the edges? And if you could, wouldn’t longer and perhaps wiggly edges help spread it even better? Yes and yes! You can see some examples of these “zippergons” that Bill and Kelly imagined and made in this gallery and read about them in their Bridges article.

A zippergon based on an octahedron.

A paper octahedron zippergon.

Icosadodecahedron.

A foam icosadodecahedron zippergon.

One of Bill’s last collaborations happened not with a mathematician but with a fashion designer. Dai Fujiwara, a noted creator of high fashion in Tokyo, got inspired by some of Bill’s illustrations. In collaboration with Bill, Dai created eight outfits. Each one was based on one of the eight Thurston geometries. You can see the result of their work together in this video and read more about it in this article.

Isn’t it amazing how creative minds in very different fields can learn from each other and create something together?

Richard Evan Schwartz (self-portrait)

Richard Evan Schwartz (self-portrait)

Richard Evan Schwartz was one of the speakers at the conference honoring Bill. Rich studied with Bill at Princeton and now is a math professor at Brown University.

Like Bill, Rich’s work can be highly visual and playful, and he often taps the power of computers to visualize and analyze mathematical structures. There’s lots to explore on Rich’s website. Check out these applets he has made, including ones on Poncelet’s Porism, the Euclidean algorithm (previously), and a game called Lucy & Lily (JAVA required). I love how Rich shares some of his earliest applet-making efforts, like Click On A Triangle To Change Its Color. It’s motivating to see that even an accomplished mathematician like Rich began with the basics of programming—a place where any of us can start!

Screen Shot 2014-07-23 at 2.54.37 AMOn Rich’s site you’ll also find information about his project “Counting on Monsters“. And you should definitely make time to read some of the conversations that Rich has had with his five-year-old daughter Lucy.

Recently Rich published a wonderful new book for kids called “Really Big Numbers“. It is a colorful romp through larger and larger numbers and layers of abstraction, with evocative images to light the way. Check out the trailer for “Really Big Numbers” below!

Do you have a question for Rich—about his book, or about the math that he does, or about his life, or about Bill? Then send it to us in the form below and we’ll try to include it in our interview with him!

Diana and Rich

Diana and Rich

Diana and Bill

Diana and Bill

Bill taught Rich, and Rich in turn taught Diana Davis, whose Dance Your PhD video we featured a while back. In fact, Bill’s influence on mathematics can be seen throughout many of our posts on Math Munch. Bill collaborated with Daina Taimina on hyperbolic crochet projects. He taught Jeff Weeks and helped inspire the games and software Jeff created. Bill oversaw the production of the film Outside In about the eversion of a sphere. He even coined the mathematical term “pair of pants.”

Bill’s vision of mathematics will live on in many people. That could include you, if you’d like. It’s just as Bill wrote:

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new.

Bon appetit!

Stomachion, Toilet Math, and Domino Computer Returns!

Welcome to this week’s Math Munch!

I recently ran across a very ancient puzzle with a very modern solution– and a very funny name. It’s called the Stomachion, and it looks like this:

Stomachion_850So, what do you do? The puzzle is made up of these fourteen pieces carved out of a 12 by 12 square– and the challenge is to make as many different squares as possible using all of the pieces. No one is totally sure who invented the Stomachion puzzle, but it’s definite that Archimedes, one of the most famous Ancient Greek mathematicians, had a lot of fun working on it.

StomaAnimSometimes Archimedes used the Stomachion pieces to make fun shapes, like elephants and flying birds. (If you think that sounds like fun, check out this page of Stomachion critters to try making and this lesson about the Stomachion puzzle from NCTM.) But his favorite thing to do with the Stomachion pieces was to arrange them into squares!

It’s clear that you can arrange the Stomachion pieces into a square in at least one way– because that’s how they start before you cut them out. But is there another way to do it? And, if there’s a second way, is there a third? How about a fourth? Because Archimedes was wondering about how many ways there are to make a square with Stomachion pieces, some mathematicians give him credit for being an inventor of combinatorics, the branch of math that studies counting things.

Ostomachion536Solutions_850It turns out that there are many, many ways to make squares (the picture above shows all of them– click on it for greater detail)– and Archimedes didn’t find them all. But someone else did, over 2,000 years later! He used a computer to solve the problem– something Archimedes could never have done– but mathematician Bill Cutler found that there are 536 ways to make a square with Stomachion pieces! That’s a lot! If you’ve tried to make squares with the pieces, you might be particularly surprised– it’s pretty tricky to arrange them into one unique square, let alone 536. This finding was such a big deal that it made it into the New York Times. (Though you may notice that the number reported in the article is different– that’s how many ways there are to make a square if you include all of the solutions that are symmetrically the same.)

Other mathematicians have worked on finding the number of ways to arrange the Stomachion pieces into other shapes– such as triangles and diamonds. Given that it took until 2003 for someone to find the solution for squares, there are many, many open questions about the Stomachion puzzle just waiting to be solved! Who knows– if you play with the Stomachion long enough, maybe you’ll discover something new!

Next up, the mathematicians over at Numberphile have worked out a solution to a problem that plagued me a few weeks ago while I was camping– choosing the best outdoor toilet to use without checking all of them for grossness first. Is there a way to ensure that you won’t end up using the most disgusting toilet without having to look in every single one of them? Turns out there is! Watch this video to learn how:

http://www.youtube.com/watch?v=ZWib5olGbQ0

Finally, a little blast from the past. Almost two years ago I share with you a video of something really awesome– a computer made entirely out of dominoes! Well, this year, some students and I finally got the chance to make one of our own! It very challenging and completely exhausting, but well worth the effort. Our domino computer recently made its debut on the mathematical internet, so I thought I’d share it with all of you! Enjoy!

Bon appetit!

8-bit, Pixel Art, and Aliens

Hello and welcome to this week’s Math Munch! I’m Mai Li, a current college student, and a former student of your regular team. I’m really happy to be making this week’s guest post! Today we’re going to be talking about one of my favourite topics, pixel art!

When I was a kid, the Game Boy Color came out.

Everything was XTREME in the ninties.

15-bit graphics! No idea why 15.

It had impressive 15-bit color graphics, a huge step up from the 2-bit graphics of the original Game Boy. Just looking at the graphical difference between the original MegaMan on the Game Boy, and MegaMan Xtreme on the Game Boy Color, you can tell that 15-bit offers a much larger color variety than the mere four colors available with 2-bit graphics. But what exactly does it mean to be 15-bit, as opposed to 2-bit?

Kindergarten was an exciting time. Not that I was allowed a GB.

2-bit graphics, in this “pea soup” color scheme.

Well, what’s a bit? A bit is a single piece of information that can be stored by a computer, either a 1 or a 0. A 1-bit system can have up to two whole colors! Either color 1, or color 0. Take Pong, for instance. Let’s say the color scheme is black and white. Now, white can be color 0, and black can be color 1. The pixels making up the paddles, the ball, the board, and the score are color 0, and the background is color 1. Pretty simple! But what if we want more than that?

Pong, the oldest game many people are familiar with. 1-bit colors!

Pong, the oldest game many people are familiar with. 1-bit colors!

In comes 2-bit, to the rescue! The Game Boy had a 2-bit color system, usually four shades of green. As you might have guessed, this mean that each color had two pieces of information, two “bits,” for a total of four possible combinations- 00, 01, 10, 11. And there you go! Four combination, four colors, just like that. For each bit, there are two possibilities, so the number of total colors available is 2^2. That means that for a 15-bit system like the Game Boy Color, there are 2^15 available colors! That comes out to a palette of 32,768 colors! Although the Game Boy Color was only physically capable of displaying 56 different colors simultaneously, you can understand now why 15-bit looks so much nicer than it’s earlier 2-bit counterpart. Now that you know what 8-bit means, you probably want to make your own pixel art, so here are some programs to help you do just that: Piq is a simple program that is available online, without downloading. GraphicsGale is favorite of mine, however it is only for windows. If all else fails. GIMP is a free Photoshop alternative.

One of my all time favourite artists, Fool.

One of my all time favourite artists, Fool.

8-bit is a popular art style these days, and one I often work in myself. 8-bit is 2^8, or 256 different colors. Now days, this rule of 256 colors or less is entirely a stylistic choice, as computers and consoles can work with a much higher color resolution. Many artists, however, will limit themselves to even less than 256 colors, for aesthetic and color theory reasons. In addition, artists might also use a space constraint, like using a canvas that is only 256 pixels high and 256 pixels wide. Besides the limited number of colors, many people consider works to be pixel art only if each of the pixels was hand placed by the artist, read: no Photoshop filters. Because of this, pixel art is often limited in size, simply due to the amount of time it takes to hand place each pixel. Two of my favourite pixel artists are Fool, and Pixelatedcrown. My own artwork can be found here.

By Pixelatedcrown, who’s work I adore. She also does 3D modeling and game dev stuff.

My own work.

My own work.

Retro graphics are making a comeback, and I have to admit I love it. I’m going to shine a spotlight on one of the new games that I think has some of the best retro graphics I’ve seen in a while – Shovel Knight.

Shovel Knight must rescue his friend Shield Knight in a timeless tale of shovelry.

Shovel Knight must rescue his friend Shield Knight in a timeless tale of shovelry.

Looking and playing like something akin to an SNES platformer, Shovel Knight explores an attractive 8-bit world to find his partner, Shield Knight. Although the game itself is ten dollars (and totally worth it), a first impressions video by one of my favorite Youtubers, Rockleesmile, is completely free. The video is part of his Indie Impressions series, which covers a new indie game daily, and which I adore. If that’s not enough, he has a playthrough of the entire game available, if you just need to see all the graphics right now. And if you do, who could blame you?

Finally, an unexpected use of pixel graphics: to contact aliens. No, I’m not kidding.

We sent this into space. No kidding.

We sent this into space. No kidding.

The Arecibo Message was a radio message sent into space in 1974 after the remodeling of the Arecibo Radio Telescope in Puerto Rico. The message, aimed at the globular star cluster M13, which is approximately 25,000 light years away, was mostly sent to prove that we could. Although scientists don’t really expect to hear anything back (and even it we did, it would spend 50,000 years in transit alone!), the message contained information that we thought would be important for aliens to know about us. The entire message is about 210 bits, and is in 1-bit resolution! Try and see if you can figure out what it’s trying to explain. If you can’t, the wiki page explains it all. And check out this Mental Floss article all about it! Speaking of which, if you want to de-code more alien messages, check out the Cosmic Call, a longer message sent by telescopes in Texas in 1999. I was given the message as a sixth grader, and with some friends, was able to decode the first few pages so I highly suggest giving it a try!

The Cosmic Call! Are you smarter than an alien?

The Cosmic Call! Are you smarter than an alien?

It’s generally the same idea as the Arecibo message, but it’s easier to de-code, and they sent it to many more star systems, in hopes of a response. If you were an alien and you received this message, could you understand it? I think so, but you should really see for yourself.

Thanks for reading! Bon appetit!