Describing regular tiles in any dimension.
I have for many years been fascinated by polyhedra, tilings, and the like. One of the little tidbits of which I’ve grown fond is Schläfli’s notation.
Ludwig Schläfli invented the notion of higher-dimensional spaces in the 1850s. So did Bernhard Riemann; discoveries often appear in parallel. Before that we knew about points, lines, planes, and volumes, and since that’s where our daily experience stops, that’s where mathematics stopped too. Schläfli literally wrote the bookTheorie der vielfachen Kontinuität, 1852 that introduced much of what is now known as vectors, norms, and linear algebra. As this is the math that underlies pretty much all of modern physics, statistics, differential equations, and every computer simulation of reality I’ve seen, it is hard to imagine how important his work is.
But, for all his immense impact on daily life, it is by eponyms that people get remembered. And Schläfli’s name became attached to his notation for describing polytopes.
A triangle’s important characteristic, compared to, say, a square or nonogon, is just the number 3. So let’s write it that way: {3} means a triangle, {4} a square, {12} a dodecagon, {37} a heptatricontagon, etc.
Now let’s consider a cube and a checkerboard. Both are made out of {4}s, both have every {4} meeting another {4} along every side, etc. So what makes them different? The number of {4}s that meet at a corner! Cubes have 3 squares meeting at a corner while checkerboards have 4. So here’s what we’ll do: we’ll tack that number on after the 4 that describes a square, as {4,3} (a cube) and {4,4} (a checkerboard).
So then, what’s {3,3}? Well, it has triangles (3) and has three of them meeting at each corner. That how to make a 3-sided pyramid, also called a tetrahedron. {3,4} describes an octahedron, {3,5} an icosahedron, and {3,6} a triangular grid. A dodecahedron has three pentagons meeting at each corner, so it’s Schläfli symbol is {5,3}. Honeycombs have hexagons meeting three to a corner, or {6,3}. And that, according to Plato, is where it stops.
But why stop there? If {4,4} means sticking four {4}s together at a corner, couldn’t {4,3,4} mean sticking four {4,3}s together along an edge? That would make {4,3,4} a stack of cubes extending out in every direction.
Or we could ask what would be meant by, say, {3,7}? If you take seven triangles and put them together sharing one corner you’ll find they can’t lie down flat; if you keep going you’ll end up with a wrinkly mess kind of like angular romaine lettuce.
Mathematicians have investigated all kinds of other ideas, like {5,3,3,4} as a tessellation of four-dimensional hyperbolic space. The rules are simple; the challenge is just figuring out what they mean.
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