Nautilus, The Riddler, and Brain Pickings

Welcome to this week’s Math Munch!

Sometimes math pops up in places when you aren’t even looking for it. This week I’d like to share three websites that I enjoy. What they have in common is that they all cover a wide range of subjects—astronomy, politics, pop culture—but also host some great math if you know where to look for it.

nautilusFirst up is a site called Nautilus. In their own words, “We are here to tell you about science and its endless connections to our lives.” Each month they publish articles around a theme. This month’s theme is “Heroes.” Included in Nautilus’s mission is discussing mathematics, and you can find their math articles on this page. Here are a few articles to get you started. Read about how Penrose tiles have made the leap from nonrepeating abstraction to the real world—including to kitchen items. Learn about one of math’s beautiful monsters and how it shook the foundations of calculus. Or you might be interested in learning about how a mathematician is using computers to change the way we write proofs.

riddler_4x3_defaultNext, you might think that, since the presidential election is now over, you won’t be heading to Nate Silver’s FiveThirtyEight quite as often. But do you know about the site’s column called The Riddler? Each week Oliver Roeder shares two puzzles, the newer Riddler Express and the Riddler Classic. Readers can send in their solutions, and some get featured on the website—that could be you! Here are a couple of puzzles to get you started, and you can also check out the full archive. The Puzzle of the Lonesome King asks about the chances that someone will win a prince-or-princess-for-a-day competition. Can You Win This Hot New Game Show? asks you to come up with a winning strategy for a round of Highest Number Wins. And Solve The Puzzle, Stop The Alien Invasion is just what is says on the tin.

brainpickings

The third site I’d like to point you to is Brain Pickings. It’s a wide-ranging buffet of short articles on all kinds of topics, written and curated by Maria Popova. If you search Brain Pickings for math, all kinds of great stuff will pop up. You can read about John ConwayPaul Erdős, Margaret WertheimBlaise Pascal, and more. You’ll find book recommendations, videos, history, and artwork galore. I particularly want to highlight Maria’s article about the trailblazing African American women who helped to put a man on the moon. Their story is told in the book Hidden Figures by Margot Lee Shetterly, and the feature film by the same name is coming soon to a theater near you!

I hope you find lots to dig into on these sites. Bon appetit!

Pixel Art, Gothic Circle Patterns, and First Past the Post

For this week’s Math Munch, we have a re-run from four years ago– which just happened to be our first anniversary on Math Munch and the end of the previous presidential election. What were we thinking about then, and what are we thinking about now?

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Welcome to this week’s Math Munch!

Guess what? Today is Math Munch’s one-year anniversary!

We’re so grateful to everyone who has made this year so much fun: our students and readers; everyone who has spread the word about Math Munch; and especially all the people who do and make the cool mathy things that we so love to find and share.

Speaking of which…

Mathematicians have studied the popular puzzle called Sudoku in numerous ways. They’ve counted the number of solutions. They’ve investigated how few given numbers are required to force a unique solution. But Tiffany C. Inglis came at this puzzle craze from another angle—as a way to encode pixel art!

Tiffany studies computer graphics at the University of Waterloo in Ontario, Canada. She’s a PhD candidate at the Computer Graphics Lab (which seems like an amazing place to work and study—would you check out these mazes!?)

Tiffany C. Inglis, hoisting a buckyball

Tiffany tried to find shading schemes for Sudoku puzzles so that pictures would emerge—like the classic mushroom pictured above. Sudoku puzzles are a pretty restrictive structure, but Tiffany and her collaborators had some success—and even more when they loosened the rules a bit. You can read about (and see!) some of their results on this rad poster and in their paper.

Thinking about making pictures with Sudoku puzzles got Tiffany interested in pixel art more generally. “I did some research on how to create pixel art from generic images such as photographs and realized that it’s an unexplored area of research, which was very exciting!” Soon she started building computer programs—algorithms—to automatically convert smooth line art into blockier pixel art without losing the flavor of the original. You can read more about Tiffany’s pixelization research on this page of her website. You should definitely check out another incredible poster Tiffany made about this research!

To read more of my interview with Tiffany, you can click here.

Cartoon Tiffany explains what makes a good pixelization. Check out the full comic!

I met Tiffany this past summer at Bridges, where she both exhibited her artwork and gave an awesome talk about circle patterns in Gothic architecture. You may be familiar with Apollonian gaskets; Gothic circle patterns have a similar circle-packing feel to them, but they have some different restrictions. Circles don’t just squeeze in one at a time, but come in rings. It’s especially nice when all of the tangencies—the places where the circles touch—coincide throughout the different layers of the pattern. Tiffany worked on the problem of when this happens and discovered that only a small family has this property. Even so, the less regular circle patterns can still produce pleasing effects. She wrote about this and more in her paper on Gothic circle patterns.

I’m really inspired by how Tiffany finds new ideas in so many place, and how she pursues them and then shares them in amazing ways. I hope you’re inspired, too!

A rose window at the Milan Cathedral, with circle designs highlighted.

A mathematical model similar to the window, which Tiffany created.

An original design by Tiffany. All of these images are from her paper.

Here’s another of Tiffany’s designs. Now try making one of your own!

Using the Mathematica code that Tiffany wrote to build her diagrams, I made an applet where you can try making some circle designs of your own. Check it out! If you make one you really like—and maybe color it in—we’d love to see it! You can send it to us at MathMunchTeam@gmail.com.

(You’ll may have to download a plug-in to view the applet; it’s the same plug-in required to use the Wolfram Demonstrations Project.)

Finally, with Election Day right around the corner, how about a dose of the mathematics of voting?

I’m a fan of this series of videos about voting theory by C.G.P. Grey. Who could resist the charm of learning about the alternative vote from a wallaby, or about gerrymandering from a weasel? Below you’ll find the first video in his series, entitled “The Problems with First Past the Post Voting Explained.” Majority rule isn’t as simple of a concept as you might think, and math can help to explain why. As can jungle animals, of course.

Thanks again for being a part of our Math Munch fun this past year. Here’s to a great second course! Bon appetit!


 

 

PS I linked to a bunch of papers in this post. After all, that’s the traditional first anniversary gift!

FIVE, Axiomatic, and Mathekniticians

Welcome to this week’s Math Munch!

It’s time once again for a recap of this month’s post on facebook, and we have some good ones for you. How about a celebration of five and a look at several mathematical artists.

This month marked the five-year anniversary for Math Munch!  Thank you so much to our readers for sticking with us. In honor of the occasion, check out these awesome Numberphile videos, each related to the number 5. There’s the 5 Platonic solids, of course, or Euclid’s 5th postulate, or a fifth-root trick, or even 5 and Penrose tilings. Click on the links to view, or scroll to the bottom of this post.

axiomatic

Up next, meet Timea Tihanyi and Jayadev Athreya. They are a visual artist and mathematics professor, respectively, and the two of them are coming together for a math-art collaboration called Axiomatic.  Geek Wire wrote a nice article about them here. Give it a read. Sadly, there aren’t many images yet, but we hope to see more from this team soon.
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14600930_562808520570937_9124043313080968521_nAlright, a quick break.  How about taking on this little challenge posted by The Dice Lab.  It features their awesome 120-sided isohedral dice, but the question is this, in their words:

“Rack ’em up! How many d120’s are in this tetrahedral pyramid?”

2000Finally, We’ve seen our fair share of mathematical fiber arts here on MM. See these previous posts for some mathematical knitting and crochet. Well I had to share a recent writeup by The Guardian on two mathekniticians, a married couple featured here before: Pat Ashforth and Steve Plummer. Read the article. It’s chock full of great images like the one to the left.

Well that’s it for this week’s Math Munch. See you next time, and bon appetit!

Gödel, Other Crazy Paradoxes, and Math Factor

Welcome to this week’s Math Munch!

Math can be confusing. Everyone knows that. And, actually, that’s what lots of people love about it. Some things in math are more confusing than others. One such thing, in my opinion, is a theorem developed by this kinda creepy-looking guy:

creepy-godelHis name is Kurt Gödel, and he’s responsible for a theorem that basically says: You know how you thought we had rules for arithmetic that work, don’t contradict each other, and can answer all kinds of questions with numbers? Well, there are problems with numbers (really strange problems, granted) that our arithmetic cannot answer. And if you try to fix your system so that it can answer those problems, you’ll have issues with other problems. There’s no way to repair your system so that it stays complete and answers all problems.

If this sounds disturbing to you (math doesn’t work?!?!), you’re not alone. Lots of mathematicians were upset by this. They thought, as lots of us do, that math is supposed to be logical. It’s supposed to give us the answers we need. We’re supposed to be able to rely on it. Gödel arrived at this theorem by playing with paradoxes, or statements that self-contradict. (Such as, “Today is opposite day.”) The statement that he came up with really rocked the world of math.

If you’d like to learn more about Gödel and his disturbing theorem, listen to this podcast episode from Radiolab. It talks about Gödel’s life and what his theorem meant for math, with an appearance by everyone’s favorite mathematician, Steve Strogatz!

Gödel’s confusing theorem is only one in a long string of crazy, confusing math paradoxes. Another of my favorites is the Barber Paradox, which mathematician Bertrand Russell came up with. Here it is, in dry-humor video form:

If you like that paradox, you’ll probably also like the Pinocchio Paradox— which was developed by 11-year-old Veronique Eldridge-Smith:

This video comes from the YouTube channel, SpikedMathGames. I suggest you check it out!

Finally, I thought it would be nice to close off this loopy Math Munch post with a loop back to podcasts– and a link to a very large archive of math podcasts called Math Factor. Math Factor is a podcast produced out of the University of Arkansas about all kinds of interesting math. They even have an episode about the topic of this week’s Math Munch! Give it a listen.

Have a terrible opposite day, and bon appetit!

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.

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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!

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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!

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Rectangles, Explosions, and Surreals

Hi everyone! We’ll be back with a new post next week. Until then, enjoy this “explosive” post from October 2012.

Math Munch

Welcome to this week’s Math Munch!

What is 3 x 4?   3 x 4 is 12.

Well, yes. That’s true. But something that’s wonderful about mathematics is that seemingly simple objects and problems can contain immense and surprising wonders.

How many squares can you find in this diagram?

As I’ve mentioned before, the part of mathematics that works on counting problems is called combinatorics. Here are a few examples for you to chew on: How many ways can you scramble up the letters of SILENT? (LISTEN?) How many ways can you place two rooks on a chessboard so that they don’t attack each other? And how many squares can you count in a 3×4 grid?

Here’s one combinatorics problem that I ran across a while ago that results in some wonderful images. Instead of asking about squares in a 3×4 grid, a team at the Dubberly Design…

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The Dice Lab, Sum of Cubes, and Double Polyhedra

Welcome to this week’s Math Munch!

It’s the final Thursday of September, so it’s time again for a recap of the month’s best from our Facebook page. This month we have a new sort of dice, a beautiful illustration of a numerical fact, and some wonderful new sculpture work from Rinus Roelofs. Let’s dig in.

First, check out this wonderful image. Meditate on it, and see if you can figure out what’s going on, even if you can’t understand the notation.

sum-of-cubes

It’s showing us a simple way to compute a sum of cubes. They can be broken down and reconstructed as a square! Consider the sum of the first 3 cube numbers, for example: 1+8+27=36, and 36 is the square of 6. One step further, 6 is the sum of the first 3 numbers.

So in the picture above, the sum of the first 5 cubes is equal to the square whose side length is the sum of 1 through 5.  AMAZING, and a beautiful illustration. Can you see why it always works, not just for 1 through 5? That’s key! And now test your understanding: What is the sum of the first 100 cube numbers?

dicelablogoUp next, we’ve met Henry Segerman plenty of times on Math Munch, including a look at the project he shares with Robert Fathauer, called The Dice Lab. They make mathematically interesting dice that have, in most cases, never been produced before. There newest creation (also last? see the video to see what I mean) is a 48-sided dice. Very cool. Can you think of a use for a 48-sided die?  It sure looks cool. Reminds me of a rhombic dodecahedron. Do you see the connection?

Finally, another familiar face – the incredible mathematical artist, Rinus Roelofs – has been making incredible things. We met Roelofs in July, but his facebook page has been full of activity since then. His recent work has focused on double-covered polyhedra.  You’ll have to click over and browse to see what I mean. Recently he posted a project I might want to take on. These are fold-up models for his creations. Check out the gallery below.

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I’m not 100% sure how that cube one works, but I think I can figure it out, and I bet some of you can too. Of course, I’m sure we’ll make mistakes, but if we keep on learning, I bet we can get this figured out. If anyone ends up making a template of their own, email it to us and we’ll share it on the site.

Until next time, bon appetit!