polyhedra

# Excavated Dodecahedron

Just hearing those two words together, excavated dodecahedron, was enough. No question, had to make one.

A regular dodecahedron is 12 sides, with pentagons on each side. Popularly known these days as D-12 among those who roll dice.

My excavated dodecahedron has the silhouette of a D-12, but the faces are pushed in. The sides of each cave are made of five equilateral triangles.

When I first started building dodecahedrons from paper, I found it really helpful to know that each point of the D-12 was flanked by three and only three pentagons. Armed with, and delighted by, this bit of enlightenment I made my pattern for the excavated dodecahedron from four copies of 15 equilateral triangles arranged together just right – see below.

The above image kind of looks like there is a hexagon in the middle of the pattern, but I’ve colored it to show that said hexagon is three pairs of parts of five-faced arrangements. Here’s a PDF that you can use to make your own, just remember you’ll need four copies of it. There’s no real directions with the PDF, but you should be able to figure it out from the photos as long as you keep in mind that each vertex is surrounded by just three pentagons. I’m calling each caved-in face a pentagon.

I think the top view is my favorite angle.

After I finished up making this D-12- in-grays today, I posted about it on Twitter (yes, I am still there…) https://twitter.com/PaulaKrieg/status/1617249547285151745?s=20&t=4rk94SRppce3l-2wLdrsWA

I actually had a question that I was hoping that someone over there could answer. I was wondering what it would look like if those excavated faces were popped out instead of in. The first thing my wondering made me do was to see if it could even be made. It could. It looks like a dodecahedron that is considering becoming a star.

These are so dimensional that they don’t photograph all that well, but maybe you can get the idea of polyhedra with 60 faces.

Well, my curiosity about the name of this shape was, like, instantaneously satisfied. My friend who goes by the handle @HypercubicPeg sent me a link to this magical looking paper Elevations and Stellations, giving me the confidence to call my shape an elevated dodecahedron. I love that name. I like when things have a name. But that’s not all. @emacdo, Ethan Macdonald, who has a glorious website tab labeled Polyhedra wrote “It has at least one name – hexecontahedron (60-faced polyhedron)” then followed with the tweet below:

All in all, it was a successful Sunday morning.

There’s just one more delightful thing. It seemed to me that my elevated dodecahedron would likely play nicely with my excavated dodecahedrons.

It turns out my hunch was entirely true. My collections of dodecahedron-based polyhedrons fit together like peas in a pod.

One last photo, because I like it so much. Here’s the inside of the excavated D-12 as it’s coming together. Cozy looking in there?

Finally, Happy Birthday to my dear friends Cynthia and Susan. They don’t share a birthday, but close enough.

Yikes! Onw more thing: I’m teaching a three session zoom class at the Center for Book Arts soon, that’s completely focuses on looking at making these kinds of solid forms. It’s a super fun class. Realm of Solids Registration open until February 16, starts February 23.

## 2 thoughts on “Excavated Dodecahedron”

1. OMG! What a fantastic nerd you are, Paula. These are beautiful. I spent much of the day trying to do a Yayoi Kusama black and yellow polka-dotted pumpkin puzzle with amazingly little success, except to know that I had done my due diligence and could put all my sorted pieces back in the box!

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