# 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!

# Linking Newspaper Rings, Pascal’s Colors, and Poetry of Math

Welcome to this week’s Math Munch!

Here’s something that sounds impossible: turn a single newspaper page into two rings, linked together, using only scissors and folding. No tape, no glue– just folding and a few little cuts.

Want to know how to do it? Check out this video by Mariano Tomatis:

On his website, Mariano calls himself the “Wonder Injector,” a “writer of science with the mission of the magician.” And that video certainly looked like magic! I wonder how the trick works…

Mariano’s website is full of fun videos involving mathe-magical tricks. I like watching them, being completely baffled, and then figuring out how the trick works. Here’s another one that I really like, about a fictional plane saved from crashing. It’s a little creepy.

How does this trick work???

Next up is one of my favorite number pattern — Pascal’s Triangle. Pascal’s Triangle appears all over mathematics– from algebra to combinatorics to number theory.

Pascal’s Triangle always starts with a 1 at the top. To make more rows, you add together two numbers next to each other and put their sum between them in the row below. For example, see the two threes beside each other in the fourth row? They add to 6, which is placed between them in the fifth row.

Pascal’s Triangle is full of interesting patterns (what can you find?)– but my favorite patterns appear when you color the numbers according to their factors.

That’s just what Brent Yorgey, computer programmer and author of the blog “The Math Less Travelled,” did! Here’s what you get if you color all of the numbers that are multiples of 2 gray and all of the numbers that aren’t multiples of 2 blue.

Recognize that pattern? It’s a Sierpinski triangle fractal!

If you thought that was cool, check out this one based on what happens if you divide all the numbers in the triangle by 5. The multiples of 5 are gray; the numbers that leave a remainder of 1 when divided by 5 are blue, remainder 2 are red, remainder 3 are yellow, and remainder 4 are green. And here’s one based on what happens if you divide all the numbers in the triangle by 6.

See the yellow Sierpinski triangle below the blue, red, green, and purple pattern? Why might the pattern for multiples of two appear in the triangle colored based on multiples of 6?

If you want to learn more about how Brent made these images and want to see more of them, check out his blog post, “Visualizing Pascal’s Triangle Remainders.”

Finally, I just stumbled across this collection of mathematical poems written by students at Arcadia University, in a class called “Mathematics in Literature.” They’re the result of a workshop led by mathematician and poet Sarah Glaz, who I met this summer at the Bridges Mathematical Art Conference. Sarah gave the students this prompt:

Step1: Brainstorm three recent school or other situations in your

present life – you can just write a few words to reference them.

Step 2: List 10-20 mathematical words you’ve used in class in the
past month.

Step 3: Write about one of the previous situations using as many
of these words as possible. Try to avoid referencing the situation
directly. Write no more than seven words per line.

Here’s one that I like:

ASPARAGUS, by Sarah Goldfarb

An infinity of hunger within me
Dividing a bunch of green
Snap and sizzle,
Green parentheses in a pan
The aromatic property
Simplifying my want
Producing a need
Each fraction of a second
Dragging its feet impatiently as I wait
And when it is distributed on my plate
It is only a moment before zero
Units of nourishment remain.

Maybe you’ll try writing a poem of your own! If you do, we’d love to see it.

Bon appetit!

# Platonic Terrariums, Geometric Decor, and Multiplying Polyhedra

Welcome to this week’s Math Munch! We’ve got some beautiful geometric objects meant to house a plant or decorate your home, as well as a really clever kind of “multiplication chart” relating the Platonic solids to each other.

Icosahedron Terrarium

First up, let’s take a look at some gorgeous glass terrarium models of the Platonic solids. We don’t usually share products here on Math Munch, because we want to make sure you can enjoy the math for free, but these are so beautiful I just had to show you. I’m a sucker for spherical symmetry!

The Turning Triangles Terrarium actually sits on my mantle at home. It’s 20 pieces of triangular glass (with one hinged pane) coming together to make an icosahedron home for a little plant.

Octahedron Terrariums

Above you can see a spread of octahedron terrariums, which will have to be my next purchase. Does \$29 seem like a lot for one of those? I was kind of shocked to  see prices for other ones that are about 4 times that much. Take a look at the dodecahedron and cube terrariums below. They’re over \$100 each, but man are they cool!?

Dodecahedron and Cube Terrariums

I love how they stood the cube up on its corner. Did you ever think about how cutting off the corner of a cube creates a little triangle?

Speaking of cutting off corners, that’s called “truncation.” I bet you never realized the soccer ball pattern is a truncated icosahedron. Well it is! And West Elm is selling a pair of really beautiful truncated polyhedra made of Capiz shells. Below are the corner-cut versions of the icosahedron and dodecahedron.

Capiz Shell Truncated Polyhedra

OK, just a couple more. First, I love the blue and white of these two shapes. One correction: the seller calls them an “octahedron”, but they have more than 8 faces. These are actually cuboctahedra. (Can you figure out how many sides they do have?)

Metal Icosidodecahedra

And lastly, the really cool, metal rhombicosidodecahedron. This is the shape that is used for the Zome construction kit. Check out this video showing a project we did last year. In short, we made a really big version of this out of lots of little ones.

If you end up buying one of these decorative sculptures, let us know. We’d love to see a picture of it in your house.

Finally, this is a really incredible image I found on Pinterest. Can you tell what’s going?

A Platonic solid “multiplication” chart

It’s set up like a multiplication chart, with the Platonic solids along the top and left edges. In the middle, we get a picture showing how the two shapes might be related to each other. I could (and have) stared at this for hours!

A1

In the A1 position, for example, we have a picture showing that the tetrahedron is the dual of the tetrahedron. That means, when you connect the centers of the faces on the tetrahedron, you get another tetrahedron!

 B3 E4 B2 B1

C3

B3 shows that the octhahedron is the dual of the cube. E4 shows that the icosahedron is the dual of the dodecahedron. B2 appears to be a hypercube, and B1 shows the way that a tetrahedron can be made by connecting alternating corners of a cube. It’s a fascinating chart, and I hope you’ll take some time to check it out. Can you figure out what’s going on in C3?

I would love to know where this image came from, but I can’t find anything about it. If you know anything about the origin of the chart, please let us know.

Well that’s it. I hope you found something juicy. Bon appetit!