# Origami Stars, Tessellation Stars, and Chaotic Stars

Welcome to this week’s star-studded Math Munch!

Modular origami stars have taken the school I teach in by storm in recent months! We love making them so much that I thought I’d share some instructional videos with you. My personal favorite is this transforming eight-pointed star. It slides between a disk with a hole the middle (great for throwing) and a gorgeous, pinwheel-like eight-pointed star. Here’s how you make one:

Another favorite is this lovely sixteen-pointed star. You can make it larger or smaller by adding or removing pieces. It’s quite impressive when completed and not that hard to make. Give it a try:

Continuing on our theme of stars, check out these beautiful star tessellations. They come from a site made by Jim McNeil featuring oh-so-many things you can do with polygons and polyhedra. On this page, Jim tells you all about tessellations, focusing on a category of tessellations called star and retrograde tessellations.

Take, for example, this beautiful star tessellation that he calls the Type 3. Jim describes how one way to make this tessellation is to replace the dodecagons in a tessellation called the 12.12.3 tessellation (shown to the left) with twelve-pointed stars. He uses the 12/5 star, which is made by connecting every fifth dot in a ring of twelve dots. Another way to make this tessellation is in the way shown above. In this tessellation, four polygons are arranged around a single point– a 12/5 star, followed by a dodecagon, followed by a 12/7 star (how is this different from a 12/5 star?), and, finally, a 12/11-gon– which is exactly the same as a dodecagon, just drawn in a different way.

I think it’s interesting that the same pattern can be constructed in different ways, and that allowing for cool shapes like stars and different ways of attaching them can open up crazy new worlds of tessellations! Maybe you’ll want to try drawing some star tessellations of your own after seeing some of these.

Finally, to finish off our week of everything stars, check out the star I made with this double pendulum simulator.  What’s so cool about the double pendulum? It’s a pendulum– a weight attached to a string suspended from a point– with a second weight hung off the bottom of the first. Sounds simple, right? Well, the double pendulum actually traces a chaotic path for most sizes of the weights, lengths of the strings, and angles at which you drop them. This means that very small changes in the initial conditions cause enormous changes in the path of the pendulum, and that the path of the pendulum is not a predictable pattern.

Using the simulator, you can set the values of the weights, lengths, and angles and watch the path traced on the screen. If you select “star” under the geometric settings, the simulator will set the parameters so that the pendulum traces this beautiful star pattern. Watch what happens if you wiggle the settings just a little bit from the star parameters– you’ll hardly recognize the path. Chaos at work!

Happy star-gazing, and 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-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.

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.

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 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.

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!

# Partial Cubes, Open Cubes, and Spidrons

Welcome to this week’s Math Munch!

Recently the videos that Paul and I made about the Yoshimoto Cube got shared around a bit on the web. That got me to thinking again about splitting cubes apart, because the Yoshimoto Cube is made up of two pieces that are each half of a cube.

A part of Wall Drawing #601
by Sol LeWitt

A friend of mine once shared with me some drawings of cubes by the artist Sol LeWitt. The cubes were drawn as solid objects, but parts of them were cut away and removed. It was fun trying to figure out what fraction of a cube remained.

On the web, I found a beautiful image that Sol made called Wall Drawing #601. In the clipping of it to the left, I see 7/8 of a cube and 3/4 of a cube. Do you? You can view the whole of this piece by Sol on the website of the Greater Des Moines Public Art Foundation.

The Cube Vinco by Vaclav Obsivac.

There are other kinds of objects that break a cube into pieces in this way, like this tricky puzzle by Vaclav Obsivac and this “shaved” Rubik’s cube modification. Maybe you’ll design a cube dissection of your own!

As I further researched Sol LeWitt’s art, I found that he had investigated partial cubes in other ways, too. My favorite of Sol’s tinkerings is the sculpture installation called “Variations of Incomplete Cubes“. You can check out this piece of artwork on the SFMOMA site, as well as in the video below.

In the video, a diagram appears that Sol made of all of the incomplete open cubes. He carefully listed out and arranged these pictures to make sure that he had found them all—a very mathematical task. It reminds me of the list of rectangle subdivisions I wrote about in this post.

Sol’s diagram got me to thinking and making: what other shapes might have interesting “incomplete open” variations? I started working on tetrahedra. I think I might try to find and make them all. How about you?

Two open tetrahedra I made. Can you find some more?

Finally, as I browsed Google Images for “half cube”, one image in particular jumped out at me.

What are those?!?!

Dániel’s original spidron from 1979

These lovely rose-shaped objects are called spidrons—or more precisely, they appear to be half-cubes built out of fold-up spidrons. What are spidrons? I had never heard of them, but there’s one pictured to the right and they have their own Wikipedia article.

The first person who modeled a spidron was Dániel Erdély, a Hungarian designer and artist. Dániel started to work with spidrons as a part of a homework assignment from Ernő Rubik—that’s right, the man who invented the Rubik’s cube.

 A cube with spidron faces. Two halves of an icosahedron.

A hornflake.

Here are two how-to videos that can help you to make a 3D spidron—the first step to making lovely shapes like those pictured above. The first video shows how to get set up with a template, and the second is brought to you by Dániel himself! Watching these folded spidrons spiral and spring is amazing. There’s more to see and read about spidrons in this Science News article and on Dániel’s website.

And how about a sphidron? Or a hornflake—perhaps a cousin to the flowsnake? So many cool shapes!

To my delight, I found that Dániel has created a video called Yoshimoto Spidronised—bringing my cube splitting adventure back around full circle. You’ll find it below. Bon appetit!

Reflection Sheet – Partial Cubes, Open Cubes, and Spidrons