What we like about the Cosmic PDS Hypothesis

We must recognize that this whole PDS business may turn out to be moonshine - without basis in fact. If that's the case, then, with heavy heart, we will have to turn away from it, and look to pastures new. Meanwhile, there's no reason why we shouldn't take a look at some of the reasons we would like it to be true.

Firstly, there is the long-standing 'mysticism of the dodecahedron'.

***Plato, in his Timaeus, says of it (after describing the other four so-called 'Platonic' solids):

There still remained a fifth construction, which the God used for embroidering the constellations on the whole heaven.

The other four solids were assigned to the four terrestrial elements (fire, air, water and earth). The dodecahedron was set apart and associated with the heaven, i.e. with the sphere of the fixed stars, thought of as an outermost skin or shell of the (physical) world.

***Proposition 17 of book 13 of Euclid's Elements, the second last proposition of the whole book, explains how to construct a dodecahedron inside a sphere (touching it in 20 points). Its position in the book makes it hard not to think of this construction as a sort of grand finale of the whole work.

***Johannes Kepler was extremely susceptible to this sort of mysticism. He said that the greatest moment of his life was when he realized, in a moment of dazzling illumination , that the orbits of the planets were determined by celestial polyhedra, nested inside one another, as detailed in Kepler's famous diagram:

***Maybe Salvador Dali had the last word on dodecahedron-mysticism, in the shape of his painting of 1955, entitled 'Sacrament of the Last Supper'. The painter described it as an "arithmetic and philosophical cosmogony based on the paranoiac sublimity of the number twelve".

Secondly, there is the curious coincidence that if Roukema is right, our sphere of visibility, here and now (limited by our 'cosmological horizon') has almost the same diameter as the cosmic dodecahedron. If we had occurred much earlier in the history of the cosmos, our visibility sphere would have been much smaller than the dodecahedron; much later, and it would have been much greater than the dodecahedron - big enough, in fact, to contain many copies of the dodecahedron.

To some minds, this coincidence is so absurd that it seems to be a reason to reject the hypothesis. Myself, I relish the absurdity, the unlikeliness of it. It's as good as the fact that we live in an era when the moon is just big enough in the sky to blot out the sun when she stands in front of him. In times to come, the moon will have receeded still further from the earth and will be too small to blot out the sun, and there will no total eclipses (just annular ones).

Thirdly, there is the close link between Poincaré Dodecahedral Space and a beautiful mathematical object which exists in four dimensional space: the 120-cell or dodecaplex. In four dimensions there exist highly symmetrical solids (or rather, hypersolids) which are the analogues in 4 dimensions of the regular polyhedra in 3 dimensions. Their king, the 120-cell, will be the subject of future posts.

The 120-cell is linked in turn to the remarkable sphere-packing lattice in 8 dimensions known as E8, which is linked in turn to the biggest of the exceptional Lie groups, also called E8. Some people think that this group may hold the key to particle physics (see, for example, Garrett Lisi's Exceptionally Simple Theory of Everything). Clearly, it would be 'kinda neat' if the particles and forces of the universe turned out to be modelled on the 'ultimate' Lie group - and even better if this was somehow tied in with the topology of the universe.

The first link in this chain (120-cell to E8) will be the subject of my next post.