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

Inside the cubic honeycomb

Truncated Octahedra

Truncated Octahedra



Rhombic Dodecahedral Honeycomb

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

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.

Yoshimoto Stack

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!



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!

20 responses »

  1. Oh, math goodness! This is fantastic!

    On a related note, honeycomb cells terminate in rhombic pyramids, and the layers nest back to back. I can’t wait to build a few rhombic dodecahedra!


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  4. Wow! That is so Cool!!!! Can you make them with different cards or random numbers like 247?? And can you make it with regular paper with numbers on it or does it have to be cards?

  5. Wow the Rhombic Dodec was sooo cool when I saw the video I thought it had nothing to do with math but as I kept going and saw everything I saw the geometric shape in the square model it got me confused like I didn’t no what was the guy from the video trying to say so I saw it twice and I got what he was trying to say what got me confused was when he explained the math in the card model I will like to try this but with different cards a and I was wondering how many rhombic dodec are in the rhombic dodecahendral honeycomb well again thay was so cool I will like to try this later on.

  6. the game microtone was sooo cool I was kind confused what did you have to do but then I figured it out it was confusing why you had to go to the X but it was soo cool I was wondering if the game was based on the Rhombic Dodec because it kinda looked like it.

  7. this video is great even with random card would it still shuffle them into the position where they all have no repeats and still be in order?

  8. This is amazing. This is way more complicated than a rubix cube. I would like to try this. With using the step of making the cards into a rhombus, can you make a different shape or do you have to fold the cards in a different way?

  9. That was a really cool video! When I was close to finished making the rhombic dodec, it was interesting to realize that the 3 card and 4 card corners all matched perfectly (the jack queen king and 4 suites matched in each corner.) The last few cards fit perfectly with those I had already placed.
    I liked that the video didn’t show the whole putting together process step by step. That way I had to think about what would fit in where. It’s also nice that the video showed the math behind the artwork. The patterns used in creating the figure made sense.
    If card decks had 5 suites and 4 types of face cards and you assembled a figure similar to the way you did in the video, would it be possible to form a closed 20 sided figure?

  10. This video that is saw was really hard because I don’t get how you fold them and put them together. I also didn’t get this because the man was saying J for jack, Q for Queen, and K for king and he was saying that over and over until he stop. I also like this video because it was really fun seeing him build it and folding it also the others project he was doing!!!!

  11. I think this video is very confusing because he does not show all the steps. He could also show you a little better on where to make the folds. Overall this video was very complex but you can learn that if you take a cube and add more sides you come up with almost a completely different shape.

  12. Watching the card trick video was very amazing. I wish I can make one of those but it seems complicated for me. The most thing I like is how the three cards (Kings, Queens, and Jacks) were together in all the corners.

    • Well, maybe you could put it together. You should get some cards and give it a try. I’m sure it will take some time to figure it out, but imagine how good it will feel when you finally do. Also you might need some tape to help while you build it.

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