Monday, February 28, 2011

Mathematical Monday: Platonic Solids

Quick, D&D players--Which of these is most different from the others: the d4, d6, d8, d10, d12, or d20? (For those unfamiliar with such tabletop games, those are dice with the specified number of sides.) The answer is the d10. Why? It's the only one that isn't a Platonic solid!*

A Platonic solid is essentially a regular convex polyhedron. Simply put, polyhedron = 3D shape, convex = has no indents on the outside that could be used to collect water, and regular = same all around (for example, regular polygons--2D shapes--have all side lengths and all angles the same, as in a square or the shape of a stop sign). More specifically, the faces of a Platonic solid are all identical, regular convex polygons, with the same number of faces meeting at each vertex. Thus, all of its edges, faces, and angles are identical.

While there are an infinite number of regular polygons (they start to look more and more like circles as they get more sides, but still), there are exactly five Platonic solids. They all properly have "regular" in front of their names listed below, but usually when the following names are used alone, the regular version of the shape is implied.

Tetrahedron (the d4)
4 triangular faces, 4 vertices, 6 edges
The tetrahedral shape is common in chemistry, formed when one atom shares covalent bonds with four other atoms (with a bunch of caveats that we won't get into here), as in methane, CH4. The hydrogen atoms arrange themselves around the carbon atom such that they would be at the four vertices of a tetrahedron.

Hexahedron, aka Cube (the d6, classic die)
6 square faces, 8 vertices, 12 edges
Everyone's familiar with the cube--it's so common that it has its own special name in addition to the name that specifies its number of sides. The volume of a cube is easy to compute--just take the length of its edge and, well, cube it! Thanks to our perception of 3D space, the cube is intimately tied to how we think about volume and even the term for raising something to the 3rd power.

Octahedron (the d8)
8 triangular faces, 6 vertices, 12 edges
Diamond and fluorite crystals are often octahedrons in nature.

Dodecahedron (the d12)
12 pentagonal faces, 20 vertices, 30 edges
Some quasicrystals, such as Holonium-Magnesium-Zinc crystals, can be dodecahedra.

Icosahedron (the d20)
20 triangular faces, 12 vertices, 30 edges
Viruses often have icosahedral or near icosahedral shapes. This is likely due to the fact that it's a vaguely spherical shape (a convenient shape for maximizing volume for minimal surface area and looking the same from many angles) that can be constructed from a bunch of identical pieces, so it's not too complicated to code for the construction of new viruses within a cell.

These shapes have been known since ancient times. The Platonic solids have been found in carved stone balls made by Scotland's neolithic people from before 1300 B.C. Plato c. 360 B.C. associated each of the classical elements (earth, air, water, fire) with one of the solids, and his contemporary Theaetetus may be responsible for the proof that only five such solids exist. These people recognized the uniqueness of these polyhedra, and hopefully you have a new appreciation for them as well.

* If you answered the d4, because it's the only one that lands with a vertex at the top instead of a face looking up, then I suppose that's a practical answer as well. But it kind of messes with my intro.

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