Building a Concave Pentakis Dodecahedron
A what now? Well, consider the humble dodecahedron. It’s got twelve pentagonal faces. Now, imagine each of those pentagons was replaced with a 5-sided pyramid. The resulting 60-face solid is called a pentakis dodecahedron. If, instead of having the pyramids on each face pointing out from the center of the solid, you have them pointing in, you end up with a different 60-face solid, the concave pentakis dodecahedron.
Of course, upon learning of such a beast, one naturally wants to make one. I started with 60 isosceles triangles. To make things interesting, I wanted a nice pattern on them. This file has half the triangles you’ll need. I used this file twice, to cut the triangles out of chipboard using Crash Space’s Epilog laser cutter.
![](https://blog.crashspace.org/wp-content/uploads/2024/03/Pentakis.png)
Once you have the triangles cut, there’s a tricky process of hot-gluing them together. It’s not easy holding five bits together with one hand in a very precise position, while gluing with the other. I ended up forming a jig from some discarded triangles where the pattern hadn’t cut through. The jig allowed me to place the triangles into a form and glue the point of the pyramid. With this point held in place, it was easier to complete the gluing of the rest of the pentagonal pyramid.
![](https://blog.crashspace.org/wp-content/uploads/2024/03/IMG_3625.jpg)
![](https://blog.crashspace.org/wp-content/uploads/2024/03/IMG_3630.jpg)
![](https://blog.crashspace.org/wp-content/uploads/2024/03/IMG_3633.jpg)
After a few false starts, I got the hang of it. I created the twelve pentagonal pyramids, then glued them together in sets of three. At this point, the task called for hot-glue thread cleanup. Then there was the delicate and finicky process of gluing together the four shapes to form the final “solid.”
![](https://blog.crashspace.org/wp-content/uploads/2024/03/IMG_3635.jpg)
![](https://blog.crashspace.org/wp-content/uploads/2024/03/IMG_3674.jpg)
![](https://blog.crashspace.org/wp-content/uploads/2024/03/IMG_3676.jpg)
![](https://blog.crashspace.org/wp-content/uploads/2024/03/IMG_3679.jpg)
![](https://blog.crashspace.org/wp-content/uploads/2024/03/IMG_3678.jpg)
It turns out that even very small errors when gluing the initial triangles result in big errors when putting the final shape together. As I worked on it, I though the inaccuracies were appalling. But once it was complete and I viewed it from a slight distance, it wasn’t so hard to overlook the gaps and flaws.
This is rad. I’ve always loved dodecahedrons since I saw MC Escher’s “Tetrahedral Planetoid” in high school, and got hooked on these beautiful shapes. Well done, indeed!