Harriss Spiral, Math Snacks, and SET

Happy New Year, and welcome to this week’s Math Munch!

The Harriss Spiral

Exciting news, folks! The Golden Ratio curve– that beautiful spiral that everyone adores– has evolved. And not into a freak of nature, either! Into something– dare I say it?– even more beautiful…

Meet the Harriss spiral. It was discovered/invented by mathematician and artist Edmund Harriss (featured before here and here) when he began playing around with golden rectangles. A golden rectangle is a very special rectangle, whose sides are in a particular proportion. You can read more about them here– but what’s most important to this new discovery is what you can do with them. If you make a square inside a golden rectangle you get another golden rectangle– and continuing to make squares and new golden rectangles inside of ever-shrinking golden rectangles, and drawing arcs through the squares, is one way to make the beautiful golden ratio spiral.

Edmund Harriss decided to get creative. What would happen, he wondered, if he cut the golden rectangle into two similar rectangles (same shape, just one is a scaled-down version of the other) and a square? And then what if he did the same thing to the new rectangles, again and again to make a fractal? Edmund’s new golden rectangle fractal makes this pattern, and when you draw a spiral through it, you get a lovely branching shape.

But don’t take my word for it. Math journalist Alex Bellos broke the news just this week in his article in The Guardian. His article explains much, much more than I can here– check it out to learn many more wonderful things about the Harriss spiral (and other spirals that Harriss has created…)!

(Bonus: Here’s a GeoGebra demonstration created by John Golden that builds the Harriss Spiral. It’s awesome!)

Next up is a site that sounds quite a lot like Math Munch. But it’s all games and cartoons, all the time. (Maybe that means you’ll like it better…) Check out Math Snacks, a site developed by a group of math educators at New Mexico State University. They worked hard to create games and animations that are both fun and full of interesting math.

One of my favorite games on Math Snacks is called Game Over Gopher.In this game, you have to save your carrot from an army of gophers by placing little machines that feed the gophers. Where’s the math, you may wonder? Placing the gopher-feeders and the other equipment that can help you save the carrot requires you to think carefully about geometry and coordinates.

Finally, speaking of games, here’s one of my favorites. I love to play SET, and I recently found a way to play online– either against a friend or against the computer. Click on this link to start your own game!

To play SET, you deal 12 cards. Then, you try to find a group of 3 cards that all share and all don’t share the same characteristics. For example, in the picture to the right– do you see the cards with the empty red ovals? They’re all the same shape, shading, and color (oval, empty, red), but they’re all different numbers (1, 2, and 3). Can you find any other sets in the picture? (Hint: One involves purple.)

Want to hone your SET skills without competing? Here’s a daily SET puzzle to challenge you.

Enjoy the games (and maybe invent a spiral of your own) and bon appetit!

Nice Neighbors, Spinning GIFs, and Breakfast

A minimenger.

A minimenger.

Welcome to this week’s Math Munch!

Math projects are exciting—especially when a whole bunch of people work together. One example of big-time collaboration is the GIMPS project, where anyone can use their computer to help find the next large prime number. Another is the recent MegaMenger project, where people from all over the world helped to build a giant 3D fractal.

But what if I told you that you can join up with others on the internet to discover some brand-new math by playing a webgame?

Chris Staecker is a math professor at Fairfield University. This past summer he led a small group of students in a research project. Research Experiences for Undergraduates—or REUs, as they’re called—are summer opportunities for college students to be mentored by professors. Together they work to figure out some brand-new math.

The crew from last summer's REU at Fairfield. Chris is furthest in the back.

The crew from last summer’s REU at Fairfield. Chris is furthest in the back.

The irreducible digital images containing 1, 5, 6, and 7 points.

The irreducible digital images containing 1, 5, 6, and 7 “chunks”.

Chris and his students Jason Haarmann, Meg Murphy, and Casey Peters worked on a topic in graph theory called “digital images”. Computer images are made of discrete chunks, but we often want to make them smaller—like with pixel art. So how can we make sure that we can make them smaller without losing too much information? That’s an important problem.

Now, the pixels on a computer screen are in a nice grid, but we could also wonder about the same question on an arbitrary connected network—and that’s what Chris, Jason, Meg, and Casey did. Some networks can be made smaller through one-step “neighbor” moves while still preserving the correct connection properties. Others can’t. By the end of the summer, the team had come up with enough results about digital images with up to eight chunks to write about them in a paper.

To help push their research further, Chris has made a webgame that takes larger networks and offers them as puzzles to solve. Here’s how I solved one of them:

NiceNeighbors

See how the graph “retracts” onto itself, just by moving some of the nodes on top of their neighbors? That’s the goal. And there are lots of puzzles to work on. For many of them, if you solve them, you’ll be the first person ever to do so! Mathematical breakthrough! Your result will be saved, the number at the bottom of the screen will go up by one, and Chris and his students will be one step closer to classifying unshrinkable digital images.

Starting with the tutorial for Nice Neighbors is a good idea. Then you can try out the unsolved experimental puzzles. If you find success, please let us know about in the comments!

Do you have a question for Chris and his students? Then send it to us and we’ll try to include it in our upcoming Q&A with them.

 

Next up: you probably know by now that at Math Munch, we just can’t get enough of great mathy gifs. Well, Sumit Sijher has us covered this week, with his Tumblr called archery.

Here are four of Sumit’s gifs. There are plenty more where these came from. This is a nice foursome, though, because they all spin. Click to see the images full-sized!

tumblr_mdv99p6WcP1qfjvexo1_500

How many different kinds of cubes can you spot?

This one reminds me of the Whitney Music Box.

This one reminds me of the
Whitney Music Box.

Whoa.

Clockwise or counterclockwise?

Clockwise or counterclockwise?

I really appreciate how Sumit also shares the computer code that he uses to make each image. It gives a whole new meaning to “show your work”!

Through Sumit’s work I discovered that WolframAlpha—an online calculator that is way more than a calculator—has a Tumblr, too. By browsing it you can find some groovy curves and crazy estimations. Sumit won an honorable mention in Wolfram’s One-Liner Competition back in 2012. You can see his entry in this video.

And now for the most important meal of the day: breakfast. Mathematicians eat breakfast, just like everyone else. What do mathematicians eat for breakfast? Just about any kind of breakfast you might name. For some audio-visual evidence, here’s a collection of sound checks by Numberphile.

Sconic sections. Yum!

Sconic sections. Yum!

If that has you hungry for a mathematical breakfast, you might enjoy munching on some sconic sectionsa linked-to-itself bagel, or some spirograph pancakes.

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?

Mars

Mars

The Moon

The Moon

distancetomars.com 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!

Grothendieck, Circle Packing, and String Art

Welcome to this week’s Math Munch!

Grothendieckportra_3107171cThis week brought some sad news to the mathematical world. Alexander Grothendieck, known by many as the greatest mathematician of the past century, passed away on November 13th. You may not have heard of him, but many mathematicians say that the work he did in math was as influential as the work Albert Einstein did in physics.

One of the things that make Grothendieck so interesting is, of course, the math he did. Grothendieck was always very creative. When he was in high school, he preferred to do math problems he made up on his own over the problems assigned by his teacher. “These were the book’s problems, and not my problems,” he said.

When he was young, inspired by some gaps he found in definitions in his geometry book about measuring lengths and areas, Grothendieck re-created some of the most important mathematical ideas of the beginning of the twentieth century. Maybe this sounds silly to you– why re-invent something that’s already been done? But, to Grothendieck, the most important part was that he’d done the whole thing by himself. He’d figured out something in his own way. He later wrote that this experience showed him what being a mathematician was like:

Without having been told, I nevertheless knew ‘in
my gut’ that I was a mathematician: someone who
‘does’ math, in the fullest sense of the word…

During his years as a mathematician, Grothendieck worked on connecting different parts of math (a project requiring a lot of creativity)– algebra, geometry, topology, and calculus, among others.

Grothendieck kid

Alexander Grothendieck as a kid

The other thing that makes Grothendieck so interesting is his life story. As a kid, Grothendieck and his parents fled from Germany to France to escape the Nazis. As an adult, Grothendieck spoke out strongly for peace. He used his fame to take a stand against the wars of the second half of the twentieth century. This eventually led him to step away from the world of mathematicians– which many regretted. But he left behind work that changed all of mathematics for the better.

If you’d like to learn more about Grothendieck’s fascinating life and work, check out this great (but long) article from the American Mathematical Society. This article provides a shorter history, including a great statement Grothendieck made about his feelings on creativity in mathematics. Grothendieck was a very private person, so many of his mathematical writings aren’t available online– but the Grothendieck Circle has done their best to collect everything written about him.

Fractal-Apollonian-Gasket-Variations-02

A pretty circle packing

Next up, a little something for you to play with! We were studying circle packing problems in one of my classes this week. Did you know that you can fit exactly six circles snugly around another circle of the same size? But, if you try to fit circles snugly around a circle twice as large, it doesn’t work? I wonder why that is…

 

I did it!

I did it!

Anyway, my class inspired me to look for a circle packing game– and I found one! In this game, simply called Circle Packing, you have to fit all of the smaller circles into the larger circle– without any of them touching! It’s pretty tricky, and really fun.

String art wall long

String art circleFinally, the Math Munch team got something wonderful in the mail (email, I guess) this week! Math art made by Julia Dweck’s 5th grade math class! Julia’s class has been working hard to make parabolic curve string art– curves made by drawing (or stringing, in this case) many, many straight lines. They plotted each curve precisely before stringing it, to make sure it was both mathematically and artistically perfect. The pieces they made are so creative and beautiful. We’re proud to feature them on our site!

String art parabolaYou can see the whole collection of string art pieces made by Julia’s class on our Readers’ Gallery String Art page. And, want to know more about how the 5th graders made their String Art? Have any questions for Julia and her students about their love of math and the connections they see between math and art? Write your questions here and we’ll send them to Julia’s students!

Have any math art of your own? Send it to mathmunchteam@gmail.com, and we’ll post it in the Readers’ Gallery!

Bon appetit!

George Washington, Tessellation Kit, and Langton’s Ant

Welcome to this week’s Math Munch!

002What will you do with your math notebook at the end of the school year? Keep it as a reference for the future? Save it as a keepsake? Toss it out? Turn it into confetti? Find your favorite math bits and doodles and make a collage?

Lucky for us, our first president kept his math notebooks from when he was a young teenager. And though it’s passed through many hands over the years—including those of Chief Justice John Marshall and the State Department—it has survived to this day. That’s right. You can check out math problems and definitions copied out by George Washington over 250 years ago. They’re all available online at the Library of Congress website.

Or at least most of them. They seem to be out of order, with a few pages missing!

Fred Rickey

That’s what mathematician and math history detective Fred Rickey has figured out. Fred has long been a fan of math history. Since he retired from the US Military Academy in 2011, Fred has been able to pursue his historical interests more actively. Fred is currently studying the Washington cypher books to help prepare a biography about Washington’s boyhood years. You can see two papers that Fred has co-authored about Washington’s mathematics here.

Fred writes:

Washington valued his cyphering books and kept them as a ready source of reference for the rest of his life. This would seem to be particularly true of his surveying studies.

Surveying played a big role in Washington’s career, and math is important for today’s surveyors, too.

Do you have a question for Fred about the math that George Washington learned? Send it to us and we’ll try to include it in our upcoming Q&A with Fred!

A tessellation, by me!

A tessellation, by me!

Next up, check out this Tessellation Kit. It was made by Nico Disseldorp, who also made the geometry construction game we featured recently. The kit is a lot of fun to play with!

One thing I like about this Tessellation Kit is how it’s discrete—it deals with large chunks of the screen at a time. This restriction make me want to explore, because it give me the feeling that there are only so many possible combinations.

I’m also curious about the URL for this applet—the web address for it. Notice how it changes whenever you make a change in your tessellation? What happens when you change some of those letters and numbers—like bababaaaa to bababcccc? Interesting…

For another fun applet, check out this doodling ant:

Langton's Ant.

Langton’s Ant.

Langton’s Ant is following a simple set of rules. In a white square? Turn right. In a black square? Turn left. And switch the color of the square that you leave. This ant is an example of a cellular automaton, and we’ve seen several of these here on Math Munch before. This one is different from others because it changes just one square at a time, and not the whole screen at once.

Breaking out of chaos.

Breaking out of chaos.

There’s a lot that is unknown about Langton’s ant, and it has some mysterious behavior. For example, after thousands of steps of seeming randomness, the ant goes into a steady pattern, paving a highway out to infinity. What gives? Well, you can try out some patterns of your own in the applets on the Serendip website. (previously). And you can read some amusing tales—ant-ecdotes?—about Langton’s ant in this lovely article.

DSC03509I learned about Langton’s Ant from Richard Evan Schwartz in our new Q&A. In the interview, Rich shares his thoughts about computers, art, what to pursue in life, and of course: Really Big Numbers.

Check it out, 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 mathaware.org. 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.

Weights, Crazy Geometry Game, and Pumpkin Polyhedra

Welcome to this week’s Math Munch!

Weighing puzzleHere’s a puzzle for you: You have 12 weights, 11 of which weigh the same amount and 1 of which is different. Luckily you also have a balance, but you’re only allowed to use it three times. Can you figure out which weight is the different weight?

You certainly can! I won’t tell you how, but you can figure it out for yourself while playing this interactive weight game. This puzzle is tricky, but definitely fun. If one weight puzzle isn’t enough for you, you’re in luck– there are many, many variations! Check out this site to try a similar puzzle with nine weights, ten weights, and 27 weights.

Circle two pack

My solution to the Circle Pack 2 challenge. Can you do it in only 5 moves?

Next up, if you like drawing challenges, this is the game for you. Check out this crazy geometry game, in which you have to draw different shapes (like perfect equilateral triangles, squares, pentagons, and groups of circles of particular sizes) using only circles and straight lines! Here’s my solution to one of the challenges, the Circle Pack 2. See the two smaller circles inside of the larger middle circle? That’s what I wanted to draw– but I had to make all of those other circles and lines to get there! I did the Circle Pack 2 challenge in 8 moves, but apparently there’s a way to do it in only 5…

Truncated icosahedron pumpkinFinally, it’s pumpkin season again! Every year I scour the internet for new math-y ways to carve pumpkins. We’re all in luck this year– because I found great instructions for how to carve pumpkin polyhedra from Math Craft!  Check out this site to learn how to carve all the basics– tetrahedra, cubes, octahedra, dodecahedra, and (my favorite) icosahedra– and a bonus polyhedron, the truncated icosahedron (also know as the soccer ball).

Pumpkin polyhedra

Pumpkin Platonic polyhedra!

 

Don’t forget to make pi with the leftover pumpkin! Oh, and, bon appetit!