I love geometric shapes. There are shapes you learn about in kindergarten (square, circle, triangle), but there are way more shapes out there. You have Klein bottles and tesseracts and truncated icosidoedecahedra – oh my!

Recently I was rereading my college thesis on geometry, and I thought that it sounded really judgmental of certain shapes. Most shapes have at least some interesting features, and it’s probably mathematically wise to consider them all as objects of study.

But that’s for serious mathematicians doing serious work. On a blog, we can be as judgmental about shapes as we want. It’s not like the shapes are going to judge us back.

I would give a torus more than 0 points, but it might give me 0.

I’m going to judge shapes in four categories: fine, lame, awesome, overrated (not as awesome as they seem), and underrated (more awesome than they seem). For each category, I’m going to include one 2D shape and one 3D shape. We’ll do 4D shapes in another post, since they require more explanation.

Two dimensions

Fine: Equilateral triangle

Equilateral triangles have some things going for them. They’re really symmetrical for a triangle. They’re the simplest regular polygon (other than dumb ones that don’t enclose any area). They can tile the plane, too. But being simple makes them kind of boring. Plus, all that symmetry means they’re bad examples for all the things that make triangles cool. Want to describe the types of triangle centers? Don’t use an equilateral, they’re all the same point. Equilateral triangles are fine.

Lame: Apeirogon

Apeirogons are cheating because they can’t win if they play fair. “Oh, you say regular polygons have equal edges and equal angles? Well, here’s a shape with infinitely many edges and 180 degree angles. Don’t feel so clever now, huh?” It’s rude. Sure, they can technically be regular and even more technically tile the plane (one on the top half, one on the bottom half). But that’s because they’re cheating. Lame.

Awesome: Dodecagram

 

Star polygons are generally cool. Unlike the apeirogon, they actually want you to reckon with the definitions of things you care about, like what counts as a vertex. Pentagrams are the simplest example; they’ve got five equal edges and five vertices with a 36 degree angle on each one. The self intersections aren’t vertices since they don’t cap off the edges.

The 2 heptagrams

The dodecagram is my favorite because it has the highest number of sides while still being unique. There are 2 heptagrams because you can skip 1 vertex or 2 vertices. But 12 just has so many factors that most of those don’t make closed dodecagrams. 12 is an awesome number, and dodecagrams are awesome star polygons.

Overrated: Oval

Ovals are named for being shaped kind of like an egg (Latin “ovum”). And that’s all there is to them. Maybe specific ovals are cool (like an ellipse or a superellipse), but ovals in general are poorly defined. When we teach preschoolers what an oval looks like, we pretty much always use ellipses. So just call it an ellipse! It’s not hard to teach a preschool kid what an ellipse is: it’s a squashed or stretched circle. You could even say it’s like how you get a rectangle from a square. Down with ovals, up with ellipses.

Underrated: 17-gon

17-gons answered a 2,000 year old question posed by the Ancient Greek mathematician Euclid. Euclid did not invent geometry, but he did kind of originate our modern idea of geometry, and from there a lot of the rest of mathematics. His whole thing was using a compass and a straightedge. Using these tools and his rules, Euclid could make regular triangles, squares and pentagons, and he could split a regular shape into one with twice as many sides (so hexagons, octagons, 10-gons, and 12-gons are on the table, too). But Euclid didn’t know if that was all of them.

2,000 years later, along comes Karl Friedrich Gauss with a way to construct a 17-gon. Gauss loved his discovery so much that he wanted a 17-gon inscribed on his grave. More people should know why the 17-gon is cool. Underrated for sure.

Three dimensions:

Fine: Disphenoid

Regular 3D solids have identical-looking vertices, identical-looking edges, and identical-looking faces. The convex ones (that is, the ones with no divots or holes) are the 5 platonic solids, which are pretty famous as e.g. Dungeons & Dragons dice.  If you require just vertices to look identical, you get the Archimedean solids which are fun to look at and study. If you require just faces, you get the Catalan solids which are closely related to the Archimedean solids and all make fair dice.¹

So maybe requiring both faces and vertices to look identical (but not edges) will give you a set of cool shapes? These are called “noble polyhedra”. If you allow concave (non-convex) ones some of them look pretty cool, but if you limit yourself to convex ones you’re stuck with disphenoids – regular tetrahedra distorted in a symmetrical way. It’s fun that they exist, but it’s a little disappointing that they’re not cooler. They’re fine.

Lame: Rectangular Prism

Squishing a cube is less exciting than squishing a tetrahedron, which was already just fine. Rectangular prisms do tile 3D space, but not in a way appreciably different from a cube. If you’re going to give points for cube-like tilings, you should go with parallelepipeds, which let you also tilt the distorted cube. Rectangular prisms are pretty lame if you ask me.

Better option: parallelepiped

Awesome: Great Dodecahedron

Star polygons are great, but star polyhedra are even greater. This one even has great in its name. It consists of 12 regular pentagons, just like a normal regular dodecahedron, but the pentagons pass through each other. Truly a dodecahedron but greater. Regular self-intersecting polyhedra get 4 out of 4 stars.²

Overrated: Square Pyramid

Square pyramids are in so many lists of shapes, but they’re actually kind of lame by themselves. The symmetry group of a square pyramid is just that of a 2D square, and its properties are just properties of squares or properties of pyramids. They’re two great tastes that taste mediocre together. Now, it is easy to make a building in the shape of a square pyramid. Square pyramid buildings can be cool. But this is judging geometric shapes, not judging architecture or engineering! Geometrically, I say square pyramids are overrated.

Underrated: 120-hedron

This is the fair dice with the most symmetries, which means the biggest fair dice that’s roughly spherical and nice to roll. Officially, this is called a “disdyakis triacontahedron”, but tabletop game players might prefer d120. It looks like a disco ball but more geometric, or some fun gem design. Plus, all the symmetries mean that it in a useful sense contains loads of other fair dice inside of it. You can even use it to replace all your standard D&D dice. Definitely underrated.

Coming soon: Judging 4D shapes

EDIT (11/21): Slightly adjusted wording on the disphenoid to be more mathematically accurate.