Tag Archives: building

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


Tangent Spaces, Transplant Matches, and Golyhedra

Welcome to this week’s Math Munch!

You might remember our post on Tilman Zitzmann’s project called Geometry Daily. If you haven’t seen it before, go check it out now! It will help you to appreciate Lawrie Cape’s work, which both celebrates and extends the Geometry Daily project. Lawrie’s project is called Tangent Spaces. He makes Tilman’s geometry sketches move!

A box of rays, by Tilman

A box of rays, by Tilman

A box of rays, by Lawrie.

A box of rays, by Lawrie

409 66 498

Not only do Lawrie’s sketches move, they’re also interactive—you can click on them, and they’ll move in response. All kinds of great mathematical questions can come up when you set a diagram in motion. For instance, I’m wondering what moon patterns are possible to make by dragging my mouse around—and if any are impossible. What questions come up for you as you browse Tangent Spaces?

Next up, Dorry Segev and Sommer Gentry are a doctor and a mathematician. They collaborated on a new system to help sick people get kidney transplants. They are also dance partners and husband and wife. This video shares their amazing, mathematical, and very human story.

Dorry and Sommer’s work involves building graphs, kind of like the game that Paul posted about last week. Thinking about the two of them together has been fun for me. You can read more about the life-saving power of Kidney Paired Donation on optimizedmatch.com.

Last up this week, here’s some very fresh math—discovered in the last 24 hours! Joe O’Rourke is one of my favorite mathematicians. (previously) Joe recently asked whether a golyhedron exists. What’s a golyhedron? It’s the 3D version of a golygon. What’s a golygon? Glad you asked. It’s a grid polygon that has side lengths that grow one by one, from 1 up to some number. Here, a diagram will help:

The smallest golygon. It has sides of lengths 1 through 8.

The smallest golygon. It has sides of lengths 1 through 8.

A golyhedron is like this, but in 3D: a grid shape that has one face of each area from 1 up to some number. After tinkering around some with this new shape idea, Joe conjectured that no golyhedra exist. It’s kind of like coming up with the idea of a unicorn, but then deciding that there aren’t any real ones. But Joseph wasn’t sure, so he shared his golyhedron shape idea on the internet at MathOverflow. Adam P. Goucher read the post, and decided to build a golyhedron himself.

And he found one!

The first ever golyhedron, by Adam P. Goucher

The first ever golyhedron, by Adam P. Goucher

Adam wrote all about the process of discovering his golyhedron in this blog post. I recommend it highly.

And the story and the math don’t stop there! New questions arise—is this the smallest golyhedron? Are there types of sequences of face sizes that can’t be constructed—for instance, what about a sequence of odd numbers? Curious and creative people, new discoveries, and new questions—that’s how math grows.

If this story was up your alley, you might enjoy checking out the story of holyhedra in this previous post.

Bon appetit!

Light Bulbs, Lanterns, and Lights Out

Welcome to this week’s Math Munch!


Edison with his light bulb.

On this day in 1880, Thomas Edison was given a patent for his most famous bright idea—the light bulb.

Edison once said, “Genius is one per cent inspiration, ninety-nine per cent perspiration”—a good reminder that putting in some work is important both in math and in life. He also said, “We don’t know a millionth of one percent about anything.” A humbling thought. Also, based on that quote, it sounds like Edison might have had a use for permilles or even permyraids in addition to percents!

Mike's octahedron.

Mike’s octahedron-in-a-light-buld.

In celebration of this illustrious anniversary, I’d like to share some light mathematical fare relating to, well, light bulbs. For starters, J. Mike Rollins of North Carolina has created each of the Platonic solids inside of light bulbs, ship-in-a-bottle style. Getting just the cube to work took him the better part of twelve hours! Talk about perspiration. Mike has also made a number of lovely Escher-inspired woodcuts. Check ’em out!

Evelyn's Schwartz lantern.

Evelyn’s Schwartz lantern.

Next up is a far-out example from calculus that’s also a good idea for an art project. It’s called the Schwartz lantern. I found out about this amazing object last fall when Evelyn Lamb tweeted and blogged about it.

The big idea of calculus is that we can find exact answers to tough problems by setting up a pattern of approximations that get better and better and then—zoop! take the process to its logical conclusion at infinity. But there’s a catch: you have to be careful about how you set up your pattern!

A "nicely" triangulated cylinder.

A “nicely” triangulated cylinder.

For example, if you take a cylinder and approximate its surface with a bunch of triangles carefully, you’ll end up with a surface that matches the cylinder in shape and size. But if you go about the process in a different way, you can end up with a surface that stays right near the cylinder but that has infinite area. That’s the Schwartz lantern, first proposed by Karl Hermann Amandus Schwarz of Cauchy-Schwartz fame. The infinite area happens because of all the crinkles that this devilish pattern creates. For some delightful technical details about the lantern’s construction, check out Evelyn’s post and this article by Conan Wu.

Maybe you’ll try folding a Schwartz lantern of your own. There’s a template and instructions on Conan’s blog to get you started. You’ll be glowing when you finish it up—especially if you submit a photo of it to our Readers’ Gallery. Even better, how about a video? You could make the internet’s first Schwartz lantern short film!

Robert Torrence and his Lights Out puzzle.

Robert and his Lights Out puzzle.

At the MOVES Conference last fall, Bruce Torrence of Randolf-Macon College gave a talk about the math of Lights Out. Lights Out is a puzzle—a close relative of Ray Ray—that’s played on a square grid. When you push one of the buttons in the grid it switches on or off, and its neighbors do, too. Bruce and his son Robert created an extension of this puzzle to some non-grid graphs. Here’s an article about their work and here’s an applet on the New York Times website where you can play Lights Out on the Peterson graph, among others. You can even create a Lights Out puzzle of your own! If it’s more your style, you can try a version of the original game called All Out on Miniclip.

The original Lights Out handheld game from 1995.

The original Lights Out handheld game from 1995.

There’s a huge collection of Lights Out resources on Jaap’s Puzzle Page (previously), including solution strategies, variations, and some great counting problems. Lights Out and Ray Ray are both examples of what’s called a “sigma-plus game” in the mathematical literature. Just as a bonus, there’s this totally other game called Light Up. I haven’t solved a single puzzle yet, but my limitations shouldn’t stop you from trying. Perspiration!

All this great math work might make you hungry, so…bon appetit!