# Quest for the worst shape

3 Feb 2023, 11:30 PM

In Wednesday’s post, I wrote that I enjoy learning about geometry on Wikipedia, and about a very nice shape that I found there, the rhombic dodecahedron. Another thing I enjoy doing from time to time is hunting for the most messed-up, the most wicked, the most cursed shapes possible. So let’s explore some of the worst shapes I’ve found together.

### Great grand stellated polydodecahedron

Image credit: Robert Webb’s Stella

We’re starting things off gently with a not-so-terrible shape. This is a 4-dimensional polytope made of 120 great stellated dodecahedra (a self-intersecting dodecahedron with pentagrams for faces). This one might give you a start if you bumped into it around a blind corner, but it’s more intimidating than truly depraved. Once you get to know it, you might even regard it as beautiful in its own way.

### 5-orthoplex

Image credit: Claudio Rocchini

The 5-orthoplex is not such a bad shape on its own. It’s actually a fairly simple 5-dimensional shape, being the dual of the 5-cube. It consists of 32 4-simplexes (the 4D extension of a triangle).

What makes this incarnation of it so truly awful is its tortured journey into our 3-dimensional world, not to mention the further bondage it endures to fit on your 2D screen. To put it simply, this is a perspective projection of a stereographic projection of a Schlegel diagram of the 5-orthoplex. I’ll spare you the gory details, but essentially, this shape has been put through a lot, and now it’s hungry for revenge.

### 600-cell, aka hexacosichoron, hexacosihedroid, tetraplex, polytetrahedron, or C600

Image credit: Robert Webb’s Stella

We’re back down to four dimensions again, but this one brings a lot of cells to the table, as well as some freaky images. This first one is the net of the 600-cell, which means it’s unwrapped to display the writhing mass of tetrahedra that make it up. Here’s a projection of it rotating:

Look at it seethe.
Image credit: Jason Hise

And to top it off, here it is forced through an unseemly chain of projections like the 5-orthoplex:

There’s way too much going on in the center of this image.
Image credit: Althepal

### Crossed square antiprism

Image credit: Tomruen

The simplest shape we’ve seen so far, but simply demented. Why would you make a shape like that?

### Boy’s Surface

Image credit: A13ean

This one gets points for the funny name alone. This is the first non-polytope of the batch; it’s actually a result from topology. Werner Boy was told by his teacher David Hilbert to prove that projective plane could not be immersed in 3D space. Instead, he ended up finding a way to do it, and this goofy shape is the result. Strange and lumpy in all the wrong places, Boy’s Surface is more unpleasant and comedic than cursed.

### Schwarz H minimal surface (Triply periodic minimal surface)

Evil baklava. Do not eat.
Image credit: Anders Sandberg

Topology really takes the cake for making awful shapes no one ever wanted to look at. I’m not even sure what goes where here. With the normal geometry stuff, I could at least follow a general idea of what the shape was supposed to be. Now I feel like the shapes are mocking me. Wish I could provide more detail on the math here, but I’m out of my depth.

I could include many more horrible shapes here, but it’s getting late. I’ve already spent way more time on this post than I should have. So get ready, because here comes my pick for the worst shape ever:

## Worst Shape Ever

### Alexander horned sphere

No one deserves credit for this abomination. (just kidding, the image is uncertainly attributed to a user named BernardH but it’s public domain)

Just look at that. Simply criminal. Even by mathematicians, this shape is considered pathological.

The pattern you see here goes on forever. To keep making this disaster of a shape, you have to keep removing sections and replacing it with a recursive pattern, forever. An ouroborous, twisting back to bite its own tail but never reaching, gnashing its teeth and fading away.

Once again, we have topologists to blame for this one, namely James Waddell Alexander II. I bet he’s proud of this nonsense. The really galling thing about it is that topologically, this thing is equivalent to a sphere. The friendly, puffy shape we all know and love, and this wicked, wicked shape claims relation to it. Truly the height of hubris (or perhaps the supremum of hubris for you depraved topologists).

Do you know of any other awful shapes? Have you seen a worse shape than the Alexander horned sphere? (not likely) Do you know of any wholesome shapes that may help me recover from this dark path I’ve lost myself on? Let me know your thoughts at my Ctrl-C email: `gome ​@ ​ctrl-c.club`.