In my previous post, I decided to be judgmental about various 2D and 3D geometric shapes. This post, we’re extending this to 4D geometric shapes. As before, we are judging these shapes emotionally, aesthetically, and morally.

But, how can you judge a 4D shape if you can’t make one in our universe of 3 spatial dimensions?¹ Mathematicians can figure out the properties of 4D shapes by analogy to 2D and 3D shapes. Just like you can describe a point on a 2D plane with 2 coordinates or a point in a 3D volume with 3 coordinates, you can describe a point in a 4D hypervolume with 4 coordinates. And just like you can draw a picture of a cube on a 2D piece of paper, you can draw pictures of 4D shapes as 3D computer models, and even take 2D pictures of those 3D drawings. If you want to learn more about how a four dimensional world would work, you can check out this video by 4D game developer Marc Ten Bosch.²

So here are some 4D shapes that are fine, lame, awesome, underrated, and overrated

Fine: Tesseract

Tesseracts are usually the first 4D shape you learn about. Move a point and the path makes a line segment. Move a line segment the same distance perpendicularly, and the path makes a square. Move a square the same distance perpendicularly, and the path is a cube. So move a cube the same distance perpendicularly, and the path makes a tesseract. It’s simple, it works, it does the job of showing how how 4D shapes work. But it doesn’t show off the coolest properties you get from 4D geometry. Tesseracts are fine.

Lame: Antiprism prisms

Square antiprism prism

OK, so you know how you can take any 2D shape and stretch it into a 3D prism? You can also take a 2D shape and stretch it with a little turn and make a 3D antiprism. If you do this with regular polygons you get prisms and antiprisms with regular faces.

Pentagon prism and pentagon antiprism

Any 3D shape can similarly stretch into a 4D prism. But in 4D, there are also these awesome shapes called duoprisms. You take a 2D shape, but instead of stretching it in a straight line to get a 3D shape, you stretch it around another entire 2D shape to get a 4D shape. The result is this cool hybrid of two different 2D polygons. And you can pick any 2 regular polygons you want!

Pentagon/triangle duoprism

Antiprism prisms don’t hold a candle to duoprisms. They form an infinite family, but only because you can make an antiprism out of any polygon. You only get one choice instead of 2. They’re just the most boring way to extend antiprisms into 4D.

So can you stretch and turn antiprisms to get duoantiprisms? Not really, if you want the faces to be regular. There’s one case with 10 pentagon antiprisms, where it works by coincidence.  That shape (the grand antiprism) is pretty cool I guess. But antiprism prisms are disappointing and lame.

Fold up this mess into 4D and you get a grand antiprism

Awesome: 24-cell

In 3D, there are 5 platonic solids – regular polyhedra that are convex (with no divots or holes). But in 4D there are 6 hyper-platonic solids. 5 of these match up with 3D counterparts. For example, the tesseract matches with the cube. But the 24-cell is special because it doesn’t have a 3D regular partner. It matches most closely with a different 3D shape, the rhombic dodecahedron, which has rhombus faces. In 4D, those irregular rhombuses become regular octahedra, creating the 24-cell. It’s an awesome shape for sure.

Rhombic Dodecahedron

Overrated: Hypersphere

Hyperspheres are extra hard to draw in 3D or 2D

It was hard to pick an overrated 4D shape since most people don’t even know any 4D shapes in the first place. So I feel a little bit bad picking on the hypersphere, but only a little. It’s the second 4D shape you learn about after the tesseract (which was mid-tier). It’s the set of points within a certain radius of a center, but in 4D. It’s mostly just interesting because it’s 4D³, and literally all these shapes are 4D. So since I have to pick one overrated 4D shape, it’s the hypersphere.

Underrated: Clifford Torus

Now, you may be wondering “Hang on, isn’t the torus a 3D shape?”. Well, you can make a 3D torus, but it really wants to be a 4D shape. You can take a rubber band and smoosh or crinkle it down flat into 2D, but it wants to spring back up into 3D so its edges can be even instead of all bent. Similarly, the 3D torus wants to spring back up into 4D so its inner and outer curves can be even instead of all bent.

A crinkled torus and a bent torus

This true, better form of a torus is called the Clifford torus after William Kingdon Clifford who invented geometric algebra. Every part of a Clifford torus can be made by rolling a 2D sheet without creasing or crumpling it. And animations of it look awesome. Totally underrated.

Coming soon: Genghis Khan Day