# Tessellating building units in Minecraft

1 Feb 2023, 11:00 PM

Geometry is one of those subjects that’s a ton of fun to read about on Wikipedia. With most other mathematical subjects, the material is so in-depth that most of it goes above the head of a layperson. That’s also true with geometry, but at least there are fun pictures to look at.

One of my favorite shapes I ever discovered on Wikipedia is the rhombic dodecahedron. Twelve was my favorite number as a kid, and it has twelve faces. The faces are all identical but not made of regular polygons, which I believe makes it unique among convex polyhedra (but I’m not 100% sure on that). It has a satisfying level of complexity but it’s easily recognizable. And best of all, it tessellates in three dimensions.

Image credit: Andrew Kepert

At some point when playing Minecraft, I had the idea to try building using repeating tessellating units. The rhombic dodecahedron was a perfect candidate, because it tessellates in line with a cubic grid, which is the whole basis of Minecraft. It’s actually a very closely related shape to a cube, because you can create one from a cube by popping its faces out into pyramids.

Image credit: TED-43

So I went into Minecraft and built out a few examples, trying to get a version of the shape with a good relative size and ease of building. I settled on a slightly modified design that looks like this:

The cubic frame is in red & white to show the side length of five, while the popped out faces are in yellow.

The faces don’t terminate in points, because in Minecraft it would just end up being a single block floating there on each face, and that didn’t look very good. This is what three of them tessellating together looks like:

You can just keep extending the frame of 5×5 cubes and building the appropriate yellow faces where needed.

I love the look of these units. They take advantage of Minecraft’s classic blocky look while including some rounding, so they don’t feel utterly boxy. They’re versatile and well-sized for building a useful room. Also, they are pretty easy to modify or augment as needed, so they make a great starting point for a new build.

A quick example home I threw together to demonstrate how the units can be built into the landscape nicely

I came up with these units years ago, but I keep coming back to them whenever I want to play Minecraft again. Because they’re so simple, it’s easy to try a lot of new things with them, and I never run out of ideas for new ways to use them. But despite their simplicity, they’re distinctive and (as far as I know) unique to me, so I consider them a signature of my Minecraft building style.

Do you play Minecraft? Do you have any signature building patterns or techniques? Do you like to read about geometry on Wikipedia? Let me know your thoughts at my Ctrl-C email: `gome ​@ ​ctrl-c.club`.

Bonus image: I stumbled on this while writing Friday’s post. It’s a much better demonstration of how rhombic dodecahedra tessellate, so I thought I’d include it here after the fact.
Image credit: TED-43