Tag Archives: geometry

Functionized Photos, Projective Games, and Traffic

Welcome to this week’s Math Munch!

Have you ever looked in a distorted mirror– one that stretched and squeezed your face so that you looked very, very silly? If you like that, check out this program called the Function Explorer that distorts your picture according to different functions!

Crazy Mikos

My cat under the “fraction” function

To use the program, you’ll have to turn on your webcam. Then, select one of the functions listed– maybe similarity, log, or fraction. Then, watch as the image in front of your webcam twists, expands, and repeats as the function distorts the picture!

What’s going on here? The program treats your picture like it’s on something called the complex plane— which is kind of like the regular two-dimensional plane we’re used to, except that some of the numbers multiply strangely. One of the dimensions on the complex plane is made of regular, normal numbers– which, in this situation, are called the “real numbers”– while the other dimension is made of different numbers, called “imaginary numbers.” These are the numbers that do weird things when you multiply them together. Maybe you’ve heard that you can’t take the square-root of a negative number. Well, on the complex plane you can. And when you do, you get an imaginary number!


Windows, under 1/z

If you’re curious about these crazy creatures called imaginary numbers and how they work to make images go wild on the complex plane, I recommend you check out this site. It gives a great interactive explanation of imaginary numbers (and teaches you about fractals, too!). But I also wouldn’t blame you if you wanted to spend a few hours holding things in front of your webcam and seeing what happens to them under different function transformations!

Gummy bears

Gummy bears! Which function did this?


Meet Donna

Next up, I’d like to share a fun collection of games with you. They’re all made by mathematician Donna Dietz, and they all have to do with a particular kind of math that I find very interesting– projective geometry! You can still enjoy the games even if you know nothing about projective geometry (and you might learn something at the same time).

screen-shot-2016-09-14-at-9-19-29-pmThe rules are pretty simple: Donna gives you a bunch of cards with symbols on them. For example, in the version shown here, you get 13 cards with 4 symbols on them each. There are a bunch of different symbols. Your task is to pick four cards to discard and arrange the remaining nine so that the cards in each row, column, and diagonal share exactly one symbol.

Donna’s projective geometry games page has links to lots more games (if you think the game with cards in three rows and columns is too easy, try one with five) and information about them.

“What does this have to do with geometry?” you might be wondering. These games show a very important property of points and lines in projective geometry. In regular geometry (which you could also call Euclidean geometry), you can have two lines that don’t share any points– meaning that they’d be parallel. But this isn’t possible in projective geometry. All pairs of lines share exactly one point. How is this related to Donna’s games? If lines are rows, columns, and diagonals of cards, and points the symbols on them…

If you’d like to learn more about how and why Donna developed these games, check out this page!

Finally, I’ve been driving a lot lately. I live in the Bay Area, and there is SO MUCH TRAFFIC AAAAAAAA!!! I went searching for solutions, and I came across this great video by our friend CGP Grey (who also made these great videos about voting theory). There’s a lot of math going on here, even if it isn’t immediately apparent. Can you find the math? (Oh, and can you stop causing traffic jams? Thanks.)

Don’t Math Munch and drive, and bon appetit!

Squricangle, Magic Angle Sculpture, and …

Welcome to this week’s Math Munch!

There’s a neat old problem/puzzle that goes like this: make a 3-D shape that could fit snugly through each of three holes—one a square, one a circle, and one a triangle. To make a shape that works for just two holes isn’t so tricky. For example, a cylinder that is just as tall as it is across would fit snugly through a circle hole and a square hole. Can you think of what would work for each of the other two shape combos? What about all three?


Three holes, three shapes…and what’s that over in the corner??

If you’re curious about the answer, you might enjoy this post by Kit Wallace or this page by George Hart or—believe it or not—roundsquaretriangle.com. I don’t know the origin of this puzzle and would love to. I haven’t found any info about it after to poking around the internet for a while. So if you locate any information about the backstory of the squircangle—which is not its real name, just one that I made up—please let us know!

Even though I knew about the square-circle-triangle problem, I was not at all prepared to encounter the solution to the jet-butterfly-dragon problem!


Dragon Butterfly Jet is just one of several “magic angle sculptures” created by artist, chemist, and PhD, and high school dropout John V. Muntean. John writes the following in his Artist Statement:

As a scientist and artist, I am interested in the how perception influences our theory of the universe. … Every 120º of rotation, the amorphous shadows evolve into independent forms. Our scientific interpretation of nature often depends upon our point of view. Perspective matters.

There’s much more to see on John’s website. And you can check out Dragon Butterfly Jet in action in the video below, along with Knight Mermaid Pirate-Ship. I also recommend this video made by John where he demonstrates how his sculpture works himself. It also includes a stop-frame animation of the sculpture being built! So cool.


No, not ellipses…

And finally, what you’ve all been waiting for…


That’s right! My final share of the week is that most outspoken of punctuation marks, the ellipsis. Because often what you don’t say says a whole lot! That’s true when writing a story or some dialogue, and it’s also true in mathematics. Watch: 1+2+3+…+100. See? Pretty neat! Those three dots sure say a mouthful…

The ellipsis is probably my second favorite punctuation mark—after the em dash, of course. But don’t take my word for it. Instead, check out this article about the history and uses—mathematical and otherwise—of the humble ellipsis. Author Cameron Hunt McNabb writes:

Thus the ellipsis has been used to indicate anything from the erroneous to the irrational, and its intrigue lies in resistance to meaning. As long as we have things to say, we will have things to omit.


The very first equals sign, in 1557.

I could go on and on about the ellipsis, just like pi does: 3.1415… But anyway, while we’re on the subject of punctuation, let me point you to one of my favorite sites on the mathematical internet: the Earliest Uses of Various Mathematical Symbols page, maintained by Jeff Miller. Jeff teaches high school math in Florida and also has some other great pages, too, including this one about mathematicians featured on stamps.




A nice visualization of the squircangle by Matt Henderson


Wild Maths, Ambiguous Cylinders, and 228 Women

Welcome to this week’s Math Munch!

You should definitely take some time to explore Wild Maths, a site dedicated to the creative aspects of mathematics. Wild Maths is produced by the Millennium Mathematics Project, which also makes NRICH and Plus.


I won!

One fun things you’ll find on Wild Maths is a game called Square It! You can play it with a friend or against the computer. The goal is to color dots on a square grid so that you are the first to make a square in your color. It is quite challenging! To the left you’ll find my first victory against the computer after losing the first several matches.

You’ll find lots more on Wild Maths, including an equal averages challenge, a number grid journey, and some video interviews with mathematicians Katie Steckles and Nira Chamberlain. Wild Maths also has a Showcase of work that has been submitted by their readers, much like our own Readers’ Gallery. (We love hearing from you and seeing your creations!)

Next up is a video of an amazing illusion:

Now, I am as big of a fan of squircles as anyone, but this video really threw me for a loop. The illusion just gets crazier and crazier! The illusion was designed by Kokichi Sugihara of Meiji University in Japan. It recently won second place in the Best Illusion of the Year Contest.

We are fortunate that Dave Richeson has hit it out of the park again, this time sharing both an explanation of the mathematics behind the illusion and a paper template you can use to make your own ambiguous cylinder!

PWinmathFinally this week, I’d like to share a fascinating document with you. It is a supplement to a book called Pioneering Women in American Mathematics: The Pre-1940s PhD’s by Judy Green and Jeanne LaDuke.

The supplement gives biographies of all 228 American women who earned their PhD’s in mathematics during the first four decades of the 20th century. You might enjoy checking out this page from the National Museum of American History, which describes some about the origin of the book project.


Judy Green, Jeanne LaDuke, and fifteen women who received their PhD’s in math before 1940.

I hope you will find both pleasure and inspiration in reading the stories of these pioneers in American mathematics. I have found them to be a lot of fun to read.

Bon appetit!

Web Applets, Space Fillers, and Sisters

Welcome to this week’s Math Munch!

Recently I’ve been running across tons of neat, slick math applets. I feel like they all go together. What do they have in common? Maybe you’ll be able to tell me.

First up, you can tinker with some planetary gears. Then try out these chorded polygons. And then how about some threaded lines?

plantearygears chords shapes

Ready for some more? Because with these sorts of visualizations, Dan Anderson has been on fire lately. Dan is a high school math teacher in New York state. He and his students had fifteen minutes of fame last year when they investigated whether or not Double Stuf Oreos really have double the stuf.

Here is Dan’s page on OpenProcessing. (Processing is the computer language in which Dan programs his applets.) And check out the images and gifs on Dan’s Tumblr. Here’s a sampling!

tumblr_nm56rdMlvl1uppablo1_r3_400 tumblr_noqxoi8EsC1uppablo1_400 tumblr_nolvf9dSt61uppablo1_400

Dan also coordinates Daily Desmos, which we’ve feature previously. Check out the latest periodic and “obfuscation” challenges!

That’s a chunk of math to chew on already, but we’re just getting started! Next up, check out the space-filling artwork of John Shier.

doublecircles eyes
 fish  hearts

John’s artwork places onto the canvas shapes of smaller and smaller sizes. Notice that the circles below fill in gaps, but they don’t touch each other, they way circles do in an Apollonian gasket.

circle_prog_1B_AnimeYou can learn more about John’s space-filling shapes on this page and find further details in this paper.

Thanks for making us this sweet banner, John!

Thanks for making us this sweet banner, John!

Last up this week, head to this site to watch an awesome trailer of a film about Julia Robinson. The short clip focuses on Julia’s work on Hilbert’s tenth problem. It includes interviews with a number of people who knew Julia, including her sister Constance Reid. Constance wrote extensively about mathematics and mathematicians. I’ve read her biography of Hilbert and can highly recommend it. You can read more about Julia and Constance here and here.

Julia Robinson

Julia Robinson

Julia's sister, Constance Reid

Julia’s sister, Constance Reid

Julia and Constance as young girls.

Julia and Constance as young girls.

You might enjoy visiting the site of the Julia Robinson Mathematics Festival. Check to see if a festival will be hosted in your area sometime soon, or find out how you can run one yourself!

With May wrapped up and June getting started, I hope you have a lot of math to look forward to this summer. Bon appetit!

Sphericon, National Curve Bank, and Cardioid String Art

Welcome to this week’s Math Munch!

Behold the Sphericon!

What is that? Well, it rolls like a sphere, but is made of two cones attached with a twist– hence, the spheri-con! The one in the video is made out of pie (not sure why…), but you can make sphericons out of all kinds of materials.

Wooden sphericonIt was developed by a few people at different times– like many brilliant new objects. But it entered the world of math when mathematician Ian Stewart wrote about it in his column in Scientific American. The wooden sphericon was made by Steve Mathias, an engineer from Sacramento, California, who read Ian’s article and thought sphericons would be fun to make. To learn more about how Steve made those beautiful wooden sphericons, check out his site!

Even if you’re not a woodworker, like Steve, you can still make your own sphericon. You can start with two cones and make one this way, by attaching the cones at their bases, slicing the whole thing in half, rotating one of the halves 90 degrees, and attaching again:How to make a sphericon

Or you can print out this image, cut it out, fold it up, and glue (click on the image for a larger printable size):

Sphericon pattern

If you do make your own sphericon (which I recommend, because they’re really cool), watch the path it makes as it rolls. See how it wiggles? What shape do you think the path is?

ncbmastertitleI found out about the sphericon while browsing through an awesome website– the National Curve Bank. It’s just what it sounds like– an online bank full of curves! You can even make a deposit– though, unlike a real bank, you can take out as many curves as you like. The goal of the National Curve Bank is to provide great pictures and animations of curves that you’d never find in a normal math book. Think of how hard it would be to understand how a sphericon works if you couldn’t watch a video of it rolling?

epicycloidaThere are lots of great animations of curves and other shapes in the National Curve Bank– like the sphericon! Another of my favorites is the “cycloid family.” A cycloid is the curve traced by a point on a circle as the circle rolls– like if you attached a pen to the wheel of your bike and rode it next to a wall, so that the pen drew on the wall. It’s a pretty cool curve– but there are lots of other related curves that are even cooler. The epicycloid (image on the right) is the curve made by the pen on your bike wheel if you rode the bike around a circle. Nice!

You should explore the National Curve Bank yourself, and find your own favorite curve! Let us know in the comments if you find one you like.

String cardioid

String art cardioid

Finally, to round out this week’s post on circle-y curves (pun intended), check out another of my favorite curves– the cardioid. A cardioid looks like a heart (hence the name). There are lots of ways to make a cardioid (some of which we posted about for Valentine’s Day a few years ago). But my favorite way is to make it out of string!

String art is really fun. If you’ve never done any string art, check out the images made by Julia Dweck’s class that we posted last year. Or, try making your own string art cardioid! This site shows you how to draw circles, ovals, cardioids, and spirals using just straight lines– you could follow the same instructions, replacing the straight lines you’d draw with pieces of string attached to tacks! If you’re not sure how the string part would work, check out this site for basic string art instructions.

Bon appetit!

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.


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?


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

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.



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!

Screen Shot 2015-01-27 at 1.19.18 PM Screen Shot 2015-01-27 at 1.19.48 PM Screen Shot 2015-01-27 at 1.20.13 PM

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.

Screen Shot 2015-01-27 at 2.05.49 PM Screen Shot 2015-01-27 at 2.05.28 PM Screen Shot 2015-01-27 at 2.06.04 PM

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