# The Rhombic Dodec, Honeycombs, and Microtone

Welcome to this week’s Math Munch! Some cool pictures, videos, and a new game this week.

A couple of week’s ago, Anna wrote about the familiar hexagonal honeycomb that bees make, but that’s not the only sort of honeycomb. Mathematically, a honeycomb is the 3D version of a tessellation. Instead of covering the plane with some kind of polygon, a honeycomb fills space with some polyhedron. The cube works. Do you think tetrahedra would work? Can you think of other shapes that might work. Can you believe this works!?! (Look at the one at the bottom of that page.)

 Inside the cubic honeycomb Truncated Octahedra Tetradecahedra

Rhombic Dodecahedral Honeycomb

I want to introduce you to one of my new favorite “space-filling polyhedra.” Meet, the rhombic dodecahedron, which you can see packed nicely on the right or in crystal form below. (Click the crystal for a really great video by George Hart about crystals and polyhedra.)

Garnet Crystal

I’ll let this video serve as an introduction to the rhombic dodecahedron and some of its features. Plus, it gives you something to make if you’d like. You’ll just need a deck of cards, and maybe a ruler and some tape.

Pretty wonderful, am I right? Here’s a link for a simple paper net you can fold up into a rhombic dodecahedron. For the really adventurous or dexterous, here’s a how-to video for a pretty tricky origami model. And here’s two more related videos showing how one can be built from two cubes.

Stellated rhombic dodecahedral honeycomb

Here’s one final amazing fact about the rhombic dodecahedron. Its first stellation is the star form of the Yoshimoto Cube!!! (background info on stellation here) Perhaps more amazing is the fact that even this shape can stack to fill 3D space!

Microtone

But now, as promised, I present a new game. Microtone is a mindbending pathwinding game played on, you guessed it, rhombic dodecahedra. (I know.) Click to move around the shape and land on all of the X’s. To rotate the dodecahedra, click and drag on the page.

Bon appetit!