SquareRoots, Concave States, and Sea Ice

Welcome to this week’s Math Munch!

The most epic Pi Day of the century will happen in just a few weeks: 3/14/15! I hope you’re getting ready. To help you get into the spirit, check out these quilts.

American Pi.

American Pi.

African American Pi.

African American Pi.

There’s an old joke that “pi is round, not square”—a punchline to the formula for the area of a circle. But in these quilts, we can see that pi really can be square! Each quilt shows the digits of pi in base 3. The quilts are a part of a project called SquareRoots by artist and mathematician John Sims.

John Sims.

John Sims.

There’s lots more to explore and enjoy on John’s website, including a musical interpretation of pi and some fractal trees that he has designed. John studied mathematics as an undergrad at Antioch College and has pursued graduate work at Wesleyan University. He even created a visual math course for artists when he taught at the Ringling College of Art and Design in Florida.

I enjoyed reading several articles (1, 2, 3, 4) about John and his quilts, as well as this interview with John. Here’s one of my favorite quotes from it, in response to “How do you begin a project?”

It can happen in two ways. I usually start with an object, which motivates an idea. That idea connects to other objects and so on, and, at some point, there is a convergence where idea meets form. Or sometimes I am fascinated by an object. Then I will seek to abstract the object into different spatial dimensions.

simstrees

Cellular Forest and Square Root of a Tree, by John Sims.

You can find more of John’s work on his YouTube channel. Check out this video, which features some of John’s music and an art exhibit he curated called Rhythm of Structure.

Next up: Some of our US states are nice and boxy—like Colorado. (Or is it?) Other states have very complicated, very dent-y shapes—way more complicated than the shapes we’re used to seeing in math class.

Which state is the most dent-y? How would you decide?

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West Virginia is pretty dent-y. By driving “across” it, you can pass through many other states along the way.

The mathematical term for dent-y is “concave”. One way you might try to measure the concavity of a state is to see how far outside of the state you can get by moving in a straight line from one point in it to another. For example, you can drive straight from one place in West Virginia to another, and along the way pass through four other states. That’s pretty crazy.

But is it craziest? Is another state even more concave? That’s what this study set out to investigate. Click through to find out their results. And remember that this is just one way to measure how concave a state is. A different way of measuring might give a different answer.

Awesome animal kingdom gerrymandering video!

Awesome animal kingdom gerrymandering video!

This puzzle about the concavity of states is silly and fun, but there’s more here, too. Thinking about the denty-ness of geographic regions is very important to our democracy. After all, someone has to decide where to draw the lines. When regions and districts are carved out in a way that’s unfair to the voters and their interests, that’s called gerrymandering.

Karen Saxe

Karen Saxe.

To find out more about the process of creating congressional districts, you can listen to a talk by Karen Saxe, a math professor at Macalester College. Karen was a part of a committee that worked to draw new congressional districts in Minnesota after the 2010 US Census. (Karen speaks about compactness measures starting here.)

Recently I ran across an announcement for a conference—a conference that was all about the math of sea ice! I never grow tired of learning new and exciting ways that math connects with the world. Check out this video featuring Kenneth Golden, a leading mathematician in the study of sea ice who works at the University of Utah. I love the line from the video: “People don’t usually think about mathematics as a daring occupation.” Ken and his team show that math can take you anywhere that you can imagine.

Bon appetit!

Reflection sheet – SquareRoots, Concave States, and Sea Ice

Braids, Hacktastic, and Rock Climbing

Welcome to this week’s Math Munch!

lym_angel

Math hair braiding art by So Yoon Lym, shown at the 2014 Joint Mathematics Meetings.

First up, a little about one of my favorite things to do (and part of what got me into math in the first place!): hair braiding. If you’ve ever done a complicated braid in someone’s hair before, you might have had an inkling that something mathematical was going on. Well, you’re right! Mathematicians Gloria Ford Gilmer and Ron Eglash have spent much of their careers studying and teaching about the math that goes into hair braiding.

SYL_Diosnedys_new1

See the tessellation?

In their research, Gloria and Ron investigate how math can improve hair braiding, how hair braiding can improve math, and how the overlap between the two can teach us about how different cultures use and understand math. As Gloria shows in her article on math and braids, tessellations are very important to braided designs.

braids

And so are fractals! Ron studies how fractals are used in African and African American designs, including in the layouts of towns, tile patterns, and cornrow braids. (Watch his TED Talk to learn more!) On his beautiful website dedicated to the math of cornrows, Ron shows how braiders use tools essential to making fractals to design their braids.

programmed braid

Just like when making a fractal, braid designers repeat the same shape while shifting, rotating, reflecting, and shrinking it. You can design your own mathematical cornrow braid using Ron’s braid programming app! If you’ve ever used Scratch, this app will look very familiar. I made the spiral braid on the right using the app. Next challenge: try to make your braid on a real head of hair…

trig bracelets Laura Taalman

Next up, a little about something I wish I could do: make awesome 3D-printed art! Here’s a blog that might help me (and you) get started. Mathematician Laura Taalman (who calls herself @mathgrrl on Twitter) writes a blog called Hacktastic all about making math designs, using a 3D-printer and many other tools. She has designs for all kinds of awesome things, from Menger sponges to trigonometric bracelets. One of the best things about Laura’s site is that she tells you the story behind how she came up with her designs, along with all the instructions and code you’ll ever need to make her designs yourself.

Rock climbing Skip

Skip Garibaldi, climbing

Finally, a little about something I’m trying to learn to do better: rock climbing! Mathematician Skip Garibaldi loves both math and rock climbing– so he decided to combine his interests for the better of each. In this video, Skip discusses some of the mathematical ideas important to rock climbing– including some essential to a type of climbing that I find most intimidating, lead climbing. Check it out!

Bon appetit!

Dearing, Edmark, and The Octothorpean Order

Welcome to this week’s Math Munch!

Dearing Wang

Dearing Wang

First up is a wonderful mathematical artist I found on instagram, under the name dearing_draws. Click to see the wonderful work of Dearing Wang. The instagram stream includes lots of timelapse videos showing the creation of the images, which is lovely, but even better is that Dearing has a youtube channel and a website devoted to teaching people how to make their own!! You should click over and follow a tutorial. Make something beautiful and send us a picture.

3 Fish in a Pond

3 Fish in a Pond

Tutorial Video

The Diamond Wedge Pattern

The Diamond Wedge Pattern

Tutorial Video

Impossible Octagon

Impossible Octagon

Tutorial Video

Another great thing about Dearing’s website is that he has a page where you can print out blank sheets to color, if that’s your thing. Not quite as mathematical, maybe, but it is nice. I like to color sometimes, and if you color systematically, maybe symmetrically, then it’s fairly mathematical after all. UPDATE: Dearing has agreed to let us host some some of his coloring sheets on Math Munch.  Click here for easily downloadable sheets to color.

John Edmark

John Edmark

Up next is another mathematical artist, John Edmark, a designer and adjunct professor at Stanford University. I was introduced to John’s incredible work through the following video. Just watch and let your jaw hit the floor in amazement.

This is a video of a zoetrope. The pieces spin and the camera shutter is timed to only show certain points in their rotation. What we see is sort of like a little loop of film showing us several frames of the animation. It’s impressive that John put all those frames together into sculptures that are beautiful, even when they’re not spinning.

PatTurn

PatTurn

But that isn’t all, there’s lots more to see on John’s website. I found his spiral videos pretty mesmerizing and fantastic. I also really like his artist statement, which begins “If change is the only constant in nature, it is written in the language of geometry.” I also just really like hearing artists talk about their work, because it’s a sort of behind the scenes look into their creative process and thinking.

(3D printable files are also available here for the incredibly fortunate among us with access to a 3D printer.)

An octothorpe

An Octothorpe

Finally, if you like solving riddles and puzzles, check out The Octothorpean Order. This is sort of an online puzzle hunt, with clues and tips on the website. You can read about it, but the best thing to do is dive in and start solving puzzles. You probably have to create a user name, but it’s good fun. I recommend it.

By the way,  “octothorpe” is the technical word for the “hashtag” or “pound” or “number sign.” It means eight fields, and I think it represents a farmers house in the middle and eight fields arround it. Cool right?

Here’s to having a mathematical week.  Bon appetit!

Squircles, Coloring Books, and Snowfakes

Welcome to this week’s Math Munch!

Squares and circles are pretty different. Squares are boxy and have their feet firmly on the ground. Circles are round and like to roll all over the place.

Superellipses.

Superellipses.

Since they’re so different, people have long tried to bridge the gap between squares and circles. There’s an ancient problem called “squaring the circle” that went unsolved for thousands of years. In the 1800s, the gap between squares and circles was explored by Gabriel Lamé. Gabriel invented a family of curves that both squares and circles belong to. In the 20th century, Danish designer Piet Hein gave Lamé’s family of curves the name superellipses and used them to lay out parts of cities. One particular superellipse that’s right in the middle is called a squircle. Squircles have been used to design everything from dinner plates to touchpad buttons.

The space of superellipsoids.

The space of superellipsoids.

Piet had the following to say about the gap between squares and circles:

Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. … The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity.

"Squaring the Circle" by Troika.

“Squaring the Circle” by Troika.

These circles aren't what they seem to be.

These circles aren’t what they seem to be.

There’s another kind of squircular object that I ran across recently. It’s a sculpture called “Squaring the Circle”, and it was created by a trio of artists known as Troika. Check out the images on this page, and then watch a video of the incredible transformation. You can find more examples of room-sized perspective-changing objects in this article.

Next up: it’s been a snowy week here on the east coast, so I thought I’d share some ideas for a great indoor activity—coloring!

Marshall and Violet.

Marshall and Violet.

Marshall Hampton is a math professor at University of Minnesota, Duluth. Marshall studies n-body problems—a kind of physics problem that goes all the way back to Isaac Newton and that led to the discovery of chaos. He also uses math to study the genes that cause mammals to hibernate. Marshall made a coloring book full of all kinds of lovely mathematical images for his daughter Violet. He’s also shared it with the world, in both pdf and book form. Check it out!

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Inspired by Mashrall’s coloring book, Alex Raichev made one of his own, called Contours. It features contour plots that you can color. Contour plots are what you get when you make outlines of areas that share the same value for a given function. Versions of contour plots often appear on weather maps, where the functions are temperature, atmospheric pressure, or precipitation levels.

Contour plots are useful. Alex shows that they can be beautiful, too!

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And there are even more mathematical patterns to explore in the coloring sheets at Patterns for Colouring.

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Last up, that’s not a typo in this week’s post title. I really do want to share some snowfakes with you—some artificial snowflake models created with math by Janko Gravner and David Griffeath. You can find out more by reading this paper they authored, or just skim it for the lovely images, some of which I’ve shared below.

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I ran across these snowfakes at the Mathematical Imagery page of the American Mathematical Society. There are lots more great math images to explore there.

Bon appetit!

Reflection sheet – Squircles, Coloring Books, and Snowfakes

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!

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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.

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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!