What’s an apeirohedron?

Fellow Science Lovers,

Gaming update: No updates this time. I’m absorbed with writing and revising Scribes Emerge.

Geometric Journey

In Scribes’ Descent, we find this headscratcher:

Mallory reclined against a sloped column and took in the surreal maze of her surroundings, her fear tapering off into wonder. She’d studied similar shapes in advanced geometry, but never imagined she’d crawl around in them. “How would we classify these? They look like closely packed polyhedra, tiled in three dimensions… ”

Rain joined in the fun. “…an irregular polytope, self-dual tessellating—”

“Come on guys. It’s an apeirohedron with truncated octahedra and hexagonal prisms,” Boxer said.

Maybe you skimmed over this, taking it for made-up math babble, but all these terms are quite real. Let’s dig in and see what on Daishon these three geeks are talking about.


This comes from the Greek poly, meaning many and hedron, meaning base or seat. A polyhedron is a 3D shape that has flat sides made of polygons. (Polygons are the basic flat shapes we learned in school: squares, pentagons, octagons, etc.)

Tiled polyhedra repeat infinitely in all directions, and resemble lattices that extend in all directions:

Image of tiled polyhedra
By Tomruen – Own work, CC BY-SA 4.0Link

You see–kids make tiled polyhedra with toy blocks all the time. Little geometry geniuses.


This word comes from the Greek ápeiros, which means infinite. This is a polyhedron with an infinite number of faces. This is kind of another way of referring to the tiled polyhedra pictured above, but with some technicalities I’m glossing over to keep things simple.

In nature, honeycombs resemble an apeirohedron, and that was the sort of shape the Scribes were trying to describe using geometry. They’d found themselves in a set of suspended stone structures that resemble the figure below:

an image of an apeirohedron with truncated octahedra and hexagonal prisms
-an apeirohedron with truncated octahedra and hexagonal prisms. By Tomruen – Own work, CC BY-SA 4.0Link

Besides for being a neat concept, the word apeirohedron is such a cool word. If you form a rock band, name it this, and the Scribes will be definitely come from the Scribeverse to attend your concerts and mosh in your moshpits!


To understand the concept of duals, (the geometry type–not pistols at dawn) it’s much easier to watch this video. This would be hard to explain with a bunch of static images: https://www.youtube.com/watch?v=k1QSXoN3uII


This is just a fancy word for tiling: a pattern where shapes fit without gaps or overlaps.


A polyhedron with eight faces:

Image of an octahedron
from Wikipedia by user Cyp

This would hurt if it got into your shoe. To make it hurt less, let’s blunt off all 6 points. (Imagine chopping off each pointy bit with a knife so you’re left with a square face instead.) Now we have a truncated octahedron:

Image of a truncated octahedron
CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=162398

If you think this resembles a modern soccer ball, you’re almost right–that would be a truncated icosahedron. See this short video for a clear visual: https://www.youtube.com/watch?v=PZkblpHOPPA

Looking at our original lattice shape, we find truncated octahedra repeated inside it (the blue and red parts):

But what about the yellow sections that connect them? Those are the hexagonal prisms Boxer referred to. Replace the primary colors with a stony texture, and that’s the odd cave section the Scribes were climbing around on in Chapter 22 of Scribes’ Descent.

That’s it for the geometry terms mentioned in this passage of Scribes’ Descent. Hopefully that helps you visualize that section of the Bioprison. I didn’t explain what self-dual means or what an irregular polytope is, but feel free to look those up on your own.

Why would the Scribes need to learn this obscure stuff? As it turns out, these concepts come in handy when designing intricate nanobots. (Mallory and Leah build those using tiny 3D printers.) And if you make medical nanobots designed to interact with molecules in the human body, you have to understand the 3D shapes those molecules form.

If these concepts still feel fuzzy, this video shows off the basics of tessellation and 3D solids with helpful animations: https://www.youtube.com/watch?v=mLyTevMDJMY

Writing update: So far, I’ve rewritten the first 45 chapters of Scribes Emerge, Scribes Series book 3. My family went to two weeklong national speech and debate tournaments (one for Stoa and one for NCFCA), which kept me from writing for two solid weeks. But now that those are over, I’m back to my 1,000 words-per-day routine.

See you next month,
Dylan West

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