The Platonic Solids
Take your romantic liquids somewhere else
It’s hard to produce a meaningful, interesting definition of a thing in mathematics without it applying to an infinite number of things,1 applying to only one or a few exceptions,2 or being tautological.3 The sweet spot (especially if you want the thing to be named after you) is, it appears, about a handful of things. Let’s look at the five convex regular polyhedra - or more commonly - the Platonic solids. I’ll spend some time talking about this meaningful, interesting definition before looking at each of our heroes in more depth. Let’s begin at the end and go on till we come to the beginning.
The End: Polyhedra
Polyhedron is a fancy word4 for what amounts to “three-dimensional shape”, as long as there are no curved faces or edges (e.g. a sphere is not a polyhedron). You can have any number of faces, edges, and vertices. You can have holes, caves, or worse - irregular polygons for faces. There are a lot of nuances5 in here that really only topologists6 really need to be careful of, so let’s move on.
The Middle: Regular
This is up there with “normal” on the list of Mathematical Terms That Are More Narrowly Precise Than They Sound.7 This one doesn’t hurt too much, though. It just says all of our polyhedron’s faces must be the same regular polygon. So we can use equilateral triangles, squares, etc. to make our 3-D shape. A ton of polygons are irregular and a lot of them are even interesting. But we’re looking at the regular ones for our Platonic faces.
The Beginning: Convex
The opposite of concave. A convex polyhedron (or polygon) is its own convex hull. One way to think of this8 is to imagine the polyhedron in the gravity-less vacuum of space. The polyhedron is convex if there’s no way for an astronaut to jump (externally) from one face to another. Every conceivable line between every conceivable pair of points in a convex polyhedron is contained within the polyhedron. A cube is convex.9 A cup is not.
Our Handful of Heroes
So we understand our thing definition. What are the things that qualify?10 A lot of smarter people than me tell me these are the only five that can exist. For no particular reason besides familiarity, let’s start with the lone square-faced Platonic solid:
The Cube
Perhaps the most telling thing about the cube is that it’s the only Platonic solid that isn’t a mouthful. No -hedron in sight.11 To those of us most attuned to reality in three dimensions, this solid is probably the most natural, most obvious. This solid’s regular face shape is a square. One more face per vertex and you’d have a tessellation. One fewer and the whole shape collapses into a single plane. The cube is intrinsically tied to another Platonic solid through dueling dualing. If we take a cube (with its six faces and eight vertices) and make vertex points at each face’s center, we can make a completely new polyhedron (one with eight faces and six vertices). That polyhedron happens12 to be…
The Octahedron
This double-pyramid swapped the cube’s faces and vertices and has the same number of edges (twelve) while changing the fundamental face shape to an equilateral triangle. Now we have four faces meeting at every vertex, but have more space to work with before triangles tesselate (at six) or collapse (like in squares, at two). First, let’s decrease this to three faces per vertex and visit the dual dead-end that is…
The Tetrahedron
You might know this shape from methane or caltrops.13 You can’t have a polyhedron with fewer faces or vertices (both four). Dualing this shape leaves a shrunken copy with identical properties. This makes the tetrahedron the only Platonic solid that is its own dual. It also happens to be the only of the five that is unsatisfying to roll.14 So, having completed our out-and-back, let’s now add another equilateral triangle per vertex to bring it up to five-per. The senior superlative for most spherical goes to…
The Icosahedron
If the Platonic solids were a band, some of the fans at live shows would be math dorks15, but most of the modern crowd would probably be RPG dorks.16 Those RPG dorks are heading straight for the exits after d20/Icosahedron, but the Platonic solids are old heads17 and I think would agree with my decision not to put it last on the setlist. Too obvious.18 The icosahedron is locked in an eternal battle for geodesic dome contracts with its own dual…
The Dodecahedron
The only pentagonal-faced solid, swapping the icosahedron’s twenty and twelve. A black sheep in a family of black sheep.19 Maybe I’ve been compromised by my doubter’s curiosity in numerology,20 but there’s an undeniable aesthetic pleasure in each of these. To hold them in your hands, or look through a wireframe, or spin one on the table. The symmetry is magnetic. If you have a hard time memorizing all the something-hedrons, just remember: Jesus, Pharaoh, Zelda, Herpes, Daisies.21 Where is that shirt?
Like say, prime numbers. Meaningful, interesting, infinite set.
Like say, even primes. Meaningful, interesting, one member (2).
Like say, one-digit primes. Finite set with an arbitrary boundary.
This is not the last time I’ll poke at jargon. Jargon is just a fancy word for “fancy words that people use to gatekeep knowledge from other people”. Sometimes it’s necessary to communicate the full density of an idea compactly (like with convex regular polyhedra for example), and sometimes efforts to do so achieve the opposite. And sometimes we just tack on a specific definition to a word that’s already in heavy use elsewhere (like with regular).
Straight edges, sharp corners, non-zero volume. These sound like something between “duh” “according to who” but have strict formal definitions.
Those powerful few.
Also on this list: real, friendly, power, odd (who got away with that one?).
Among many, surely much better ways to think about it.
Spoilers.
Look me in the face and tell me you say hexahedron.
There are no coincidences in mathematics.
Hello chemist/cloak-and-dagger spy readership?
I have justification for this. It can’t land face-up, also a unique trait among this cohort.
Like me.
Also like me.
At least as old as the universe, arguably older.
What am I even talking about anymore?
Tetrahedron: only to not land face up, only self-dual. Cube: only square-faced (and even-faced!), only solo honeycomber. Octahedron: only four-faces-to-a-vertex. Dodecahedron: only pentagon-faced. Icosahedron: only five-faces-to-a-vertex.
Either all the geometries are sacred or none of them are. I lean towards the former.
Since writing this I have infected myself with it. An oddly brainworm-y list.