Carl Sagan led me to review the existence proof of the 5 perfect Platonic solids as a schoolboy. Beautiful maths but completely useless - this theory caused self-inflicted confusion for thousands of years.

Why is it relevant in a blog about computers and science? Well, the Not Even Wrong Popperian logic brings the relevance to science in an obvious way.

For IT policy makers, it is the sin of starting from a theory and grasping for whatever evidence to back up the theory.

What are they

In order for a solid to be a platonic solid, the figure must use the same regular polygon for all its faces and have the same number of faces meet at each of its vertices. The platonic solids and their regularities were discovered by the Pythagoreans and were initially called the Pythagorean solids.

There are only 5 Platonic solids. They are based on the triangle (tetrahedron), square (hexahedron), triangle again (octahedron), pentagon (dodecahedron) and triangle again again (icosahedron).

Pythagorus (569–475 BC) proved that there were 5 and only 5 perfect solids. This proof was a phenomenal mathematical accomplishment, but was quite a philosophical shock for the ancient Greeks.

The platonic solids are the only convex, three-dimensional geometrical figures to possess the unique qualities of possessing faces, edges which combine to seal the polyhedron, with equal angles at each corner. Euler determined that Number of faces + Number of vertices = Number of edges + 2.

More perfect solids may exist more in higher dimensions.

Neolithic people of Scotland 2000 BC apparently also new about the solids, but had no systematic (mathematical) proof bounding their knowledge. The following stone models were found in neolithic settlements and dated from 10 [Editor - erm, found it on the web so it must be true] centuries before the Greek's proof.

and

Historical Views of the Solids

The existence of these unusual, yet perfectly symmetrical, geometrical figures, led to several different interpretations over the years regarding their meaning (in some of the many attempts, with varying degrees of success, to associate vague mathematical principles with the physical universe over the centuries).

Plato, particularly, devised a clever association with these solids (which, in part, is why they were later named for him). He ingeniously took the five solids and associated them with the four "elements" of the natural world - Earth, Fire, Air, and Water.

Plato argued that each of the elements could be thought of as being composed of the first four solids - the tetrahedron was fire, the cube was earth, the octahedron was air, and the dodecahedron was water. The final solid, the icosahedron, was applied to the "heavenly sphere" upon which rested all of the stars and planets.

When Aristotle (fourth century, B.C.) further looked at the idea of the four elements and added a fifth element, the aether, he very well could have used this to apply the fifth platonic solid, but he did not do so, instead looking at the mater a bit differently.

Even two millenia later, with the rise of "modern" science during the Reneissance period, the platonic solids were still being applied to the universe.

Johannes Kepler (16th century), in particular, found the idea of the regular figures useful in describing the motion of the heavenly bodies. He even created a complicated model of the solar system, where each of the solids was laid into one of the other (like geometrical nesting dolls), separated by spheres circumscribed into each of them (an image of this can be found below).

Kepler associated each of the spheres surrounding solids (giving a total of six, counting the one outside the icosahedron) with the orbits of one of the known planets in the solar system - Mercury, Venus, Earth, Mars, Jupiter and Saturn.

Today it is fairly clear that the planets are not so kind as to follow such perfectly geometrical, spherical paths, but few can doubt that it was a clever association to make. And, of course, Kepler did not stick with this idea for long - his unique perspectives into the motion of heavenly bodies eventually led him to his revolutionary Laws of Planetary Motion.

The Math of the Solids

To date, it has been well proven that the five known solids are the only which can possibly exist in three-dimensional space (this has been proven in multiple ways - Euclid did so geometrically, while modern mathematicians have done so topologically).

They have been analysed in great topological detail and have been completely defined mathematically in far greater detail than most people care to truly examine.

While it has become clear that the platonic solids do not necessarily rest at the heart of the physical universe, either in the construction of matter or in the orbits of heavenly bodies, it has become clear that they do indeed exist in nature.

Living organisms on a microscopic level have been seen to conform, at times, to the unique shapes offered by the platonic solids. In addition, the crystal structures formed by certain atomic bonds are well-known to conform to shapes which echo the platonic solids, thanks to the strength provided by the regular symmetry of the shapes.

Conclusion

Theories may be perfect and beautiful, but they need feedback from the real world.

[1] http://www.kheper.net/topics/cosmology/solids.html
[2] http://mathchaostheory.suite101.com/article.cfm/the_five_platonic_solids...
[3] http://mathworld.wolfram.com/PlatonicSolid.html