Icosahedron

The icosahedron has twenty triangular faces, twelve vertices and thirty edges, with five faces meeting at each vertex. Euler's formula gives 12 - 30 + 20 = 2. Of the five regular solids it has the most faces, the greatest volume for a given surface area, and the dihedral angle closest to flat, about 138.19 degrees, all of which amount to saying that it is the most nearly spherical of them.

It is the dual of the dodecahedron. The icosahedron has twenty faces and twelve vertices; the dodecahedron has twelve faces and twenty vertices; both have thirty edges. Join the centres of the icosahedron's twenty triangles and a dodecahedron appears; join the centres of the dodecahedron's twelve pentagons and the icosahedron returns. They share one symmetry group of 120 operations, 60 of them rotations, the largest of any Platonic solid.

The golden ratio is not attached to the icosahedron by analogy. It is in the coordinates. Take three identical rectangles whose sides are in the ratio 1 to phi, set them mutually perpendicular through a common centre, and their twelve corners are exactly the twelve vertices of a regular icosahedron. Nothing is approximated in that construction, and it can be checked with a ruler and three pieces of card.

There is a further consequence that reaches well outside geometry. The 60 rotations of the icosahedron form a group identical to the alternating group on five letters, which is simple: it has no proper structure inside it to break down into. That fact is the reason there is no general formula in radicals for solving equations of the fifth degree, the result of Abel and Galois that Felix Klein later reworked entirely in terms of this solid. The icosahedron is one of the places where geometry and algebra turn out to be the same subject, and that is a considerably stronger claim about its significance than anything the esoteric literature offers.

In the Timaeus Plato gives the icosahedron to water. The reasoning follows the pattern of the other assignments: among the solids built from triangular faces it is the largest and the most nearly round, so it rolls and flows most easily, and it is matched to the element that flows. Fire gets the sharpest solid, earth the steadiest, air the one in between, and water the one that moves without cutting.

The scheme is more than a table of pairs. Because fire, air and water are all built from the same elementary triangle, they can be broken apart and reassembled into one another, and the dialogue gives the accounting: the twenty triangular faces of a particle of water can be redistributed as two particles of air, at eight faces each, and one of fire, at four. That is a mechanism for change of state, offered with numbers, in the fourth century BC.

It should still be read for what it is. Plato is not describing water molecules, which are not icosahedral and are not made of triangles, and he offers no evidence beyond the fitness of the analogy. The passage is a philosopher's reconstruction of what an intelligible universe would have to be like. Nearly every subsequent association of the icosahedron with water, emotion, flow and adaptability traces back to it, and where a modern source presents that link as ancient universal wisdom, the honest description is that it comes from one dialogue by one author.

What survives the correction is the ambition. Plato asked why substances behave differently and answered in terms of structure rather than of quality, which is the right kind of question to ask. The answers were wrong. The form of the question was not.

The icosahedron is the shape nature builds most insistently at very small scales, and the reason is understood rather than mysterious.

Most viruses have tiny genomes and cannot afford to encode many different coat proteins. Crick and Watson pointed out in the 1950s that a small genome forces a shell built from many copies of a few identical subunits, and identical subunits in a closed shell produce high symmetry. Caspar and Klug worked out the consequences in 1962: a closed shell of identical units contacting one another in nearly but not exactly the same way takes icosahedral symmetry, and only certain sizes are possible. Their triangulation number, T = h squared plus hk plus k squared, permits shells of 60, 180, 240, 420 subunits and so on, and high-resolution structures have confirmed the pattern ever since. The shape is an economy measure, not a symbol.

Other genuine occurrences are worth separating from the folklore. Boron forms clusters of twelve atoms at the vertices of an icosahedron, and those clusters are the building blocks of elemental boron and of boron carbide. Some radiolaria, single-celled marine organisms with mineral skeletons, grow strikingly icosahedral shells; Haeckel's nineteenth-century drawings of them are famous and flatter the geometry somewhat. In 1982 Dan Shechtman observed a metal alloy diffracting with fivefold symmetry, believed impossible in a crystal; the results, published in 1984 and awarded the Nobel Prize in Chemistry in 2011, established icosahedral quasicrystals, ordered solids that never repeat.

Against all that, the modern esoteric assignments deserve a plain label. Placing the icosahedron on the sacral chakra, or with the west, or with the moon, comes from twentieth-century New Age writing and from no ancient source; different authors assign it differently, and the classical Indian diagram for the water element is a flat crescent rather than a solid. The water association itself is Plato's and is genuinely that old. The rest is recent.

Sources Physical Principles in the Construction of Regular VirusesD. L. D. Caspar and A. Klug with Metallic Phase with Long-Range Orientational Order and No Translational SymmetryD. Shechtman, I. Blech, D. Gratias and J. W. Cahn