Dodecahedron

The dodecahedron has twelve regular pentagonal faces, twenty vertices and thirty edges, with three faces meeting at each vertex. Euler's formula gives 20 - 30 + 12 = 2. Its dihedral angle is about 116.57 degrees, the angle whose cosine is minus one over the square root of five.

It is the dual of the icosahedron: twelve faces and twenty vertices against twenty faces and twelve vertices, with thirty edges in common. Joining the centres of one solid's faces produces the other, in either direction, and the two share a symmetry group of 120 operations, 60 of which are rotations.

The golden ratio enters through the pentagon. In a regular pentagon the diagonal is longer than the side by exactly that ratio, so any solid with regular pentagonal faces carries it throughout. A less obvious consequence is that a cube can be inscribed in a dodecahedron with its twelve edges lying as diagonals across the twelve pentagonal faces, and there are five distinct ways to do it. Those five inscribed cubes are the reason the rotation group of the dodecahedron is the group of even permutations of five objects: the rotations shuffle the cubes.

Euclid constructs the dodecahedron in Book XIII of the Elements, and it is the hardest of the five to build with compass and straightedge, which is part of why it comes last. The book then compares the edges of all five solids and closes by arguing that no further regular solid can exist.

A caution about crystals. Pyrite grows twelve-sided crystals that are often called dodecahedra, and they are not regular ones. The pyritohedron has twelve pentagonal faces, but the pentagons are irregular and the solid has no fivefold axis of symmetry. It cannot have one: a true fivefold axis is impossible in any periodically repeating crystal lattice, which is a theorem of crystallography rather than an accident of growth. Photographs captioned as natural Platonic dodecahedra are almost always pyrite.

Four of the five solids are given elements in the Timaeus. The dodecahedron is not. Plato writes that there remained a fifth construction, and that the god used it for the whole, embroidering figures upon it, which in context means that the dodecahedron is employed for the cosmos itself rather than for anything within it. The pentagonal solid is the roundest and most sky-like of the five, and it takes the sky.

That is all he says, and the point matters because of what was later added. The dodecahedron is constantly described as Plato's aether or quintessence. Plato does not use those terms of it. A fifth element for the heavens is Aristotle's contribution, and the joining of that element to the fifth solid is a later development within the Platonic tradition, taken up in the Renaissance and repeated ever since as though it were in the dialogue. Reporting it as Plato's word is the commonest single error made about the five solids.

The most ambitious attempt to find the cosmos in them came nineteen centuries later. In Mysterium Cosmographicum of 1596 Johannes Kepler proposed that the six known planets are spaced as they are because the five regular solids are nested between their spheres: an octahedron between Mercury and Venus, an icosahedron between Venus and Earth, a dodecahedron between Earth and Mars, a tetrahedron between Mars and Jupiter, and a cube between Jupiter and Saturn. Five solids allow exactly six spheres, so the model appeared to explain why there were six planets, a question nobody else had thought to ask.

It is wrong. The fit was never good, the better observations he inherited from Tycho Brahe made it worse, and the later discovery of further planets removed its central appeal. It is also frequently misreported that Kepler abandoned it. He did not: he issued a second edition in 1621, long after his own laws of planetary motion had superseded it. The same conviction that the heavens were built on ratio is what carried him to the laws that actually worked.

More dubious claims attach to this solid than to the other four together. They are not all of one kind, and they are worth separating.

Disputed, and largely rejected. Several hundred carved stone balls survive from Neolithic Scotland, dating to around the third millennium BC, and it is often asserted that they show knowledge of the five Platonic solids a thousand years before the Greeks. Mathematicians who have examined the claim reject it. The knob counts vary enormously, from three to more than a hundred and sixty, with six much the commonest; the photograph used to support the claim has bands added that do not follow the carving, and the objects shown are not the Ashmolean Museum balls they are said to be; no securely icosahedral example is known. The balls are genuine artefacts of unknown purpose. They are not a set of regular solids.

Genuinely unknown. Roughly 130 hollow bronze dodecahedra survive from the Roman provinces, from Britain to Hungary, dated between the second and fourth centuries AD, each face pierced by a hole of a different size. No ancient text mentions them and no ancient image shows one. Proposed uses run from surveying instruments and knitting frames to gaming pieces and ritual objects, with no present way to decide between them.

Serious, and superseded. In 2003 cosmologists led by Jean-Pierre Luminet proposed that space might be finite and shaped as a Poincare dodecahedral space, opposite faces identified after a twist, which would account for an observed shortage of large-angle correlations in the cosmic microwave background. This was legitimate physics and was tested as such: the Planck mission's analysis of cosmic topology found none of the matched circles the model requires, and excluded a fundamental domain small enough to be detectable. A dodecahedral universe much larger than the observable region cannot be ruled out, but neither can it be tested.

Modern attribution. Assignments of the dodecahedron to the third eye or crown chakra, to aether, to Jupiter or to the pineal gland come from twentieth-century esoteric writing and have no classical source. So does the claim that all five solids can be read out of Metatron's Cube: the thirteen-circle figure it is drawn from is old, the name and the extraction are recent, and the projections offered for the dodecahedron and the icosahedron are not exact.

Sources Neolithic Carved Stone PolyhedraGeorge W. Hart with A Functional Reassessment of Roman Dodecahedra as Tools for Forming Standardised Wax ObjectsGreg Lamb, Planck 2013 results. XXVI. Background geometry and topology of the UniversePlanck Collaboration, Rainbow Body: A History of the Western Chakra System from Blavatsky to BrennanKurt Leland, The Ancient Secret of the Flower of Life, Volume 1Drunvalo Melchizedek