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Quoted from Pedersen (1993).

Detailed descriptions of Plato's cosmic scheme can be found in, for example, Knorr (1990),

This example is taken from Dampier (1965), p. 21.

See Waterhouse (1972), in which the historical problems associated with tracing the early history of these shapes are discussed in some detail.

tetrahedron octahedron ^^^^^ icosahedron.

cube dodecahedron

Fig. 1.6. The five Platonic solids.

### Fig. 1.6. The five Platonic solids.

of them (illustrated in Figure 1.6). This is easy to prove.28 We use the fact that the sum of the interior angles of the faces which meet at the vertex of a regular polyhedron must be less than 360°. This condition is fulfilled only by three, four, or five equilateral triangles (since the interior angles of an equilateral triangle are 60° and 6 x 60 = 360), corresponding to a tetrahedron, octahedron and icosahedron, respectively, three squares (the interior angles of a square are 90° and 4 x 90 = 360), corresponding to a cube, and three pentagons (the interior angles of a pentagon are 108° and 4 x 108 > 360), corresponding to a dodecahedron. Three hexagons is already too many (the interior angles of a hexagon are 120° and 3 x 120 = 360).

Plato was searching constantly for a parallel between the hierarchy of material things and that of mathematical objects, and this led him to consider these solids. The fact that there are only five of them (unlike the regular polygons of which there is an infinite number) made them special. Fire, on account of the shape of a flame, was compared with the tetrahedron, and water, the bulkiest of the elements, corresponded to the icosahedron. Air, having an intermediate density, corresponded to the octahedron, which has a number of triangular faces lying between those of the tetrahedron and icosahedron. Earth was likened to a cube since it is the most immobile of bodies and should be represented by the most stable figure. What about the dodecahedron? Plato got round this by saying that the god had used it for the whole Universe, a statement that did not satisfy many of Plato's disciples; laterthe dodecahedron became associated with the ether.

As will become evident, many of the ideas about the Universe promoted by people who were the product of Plato's Academy were extremely influential.

The proof given here (which is probably due to Theaetetus) is given in the concluding proposition of Euclid's Elements, written in about 300 BC.

But some, which in retrospect we can see to have been correct, were not. One of Plato's pupils, Heraclides, assumed a rotating Earth at the centre of the Universe in order to account for the daily motion of the heavens.

Heraclides of Pontus and Ecphantus the Pythagorean move the earth, not however in the sense of translation, but in the sense of rotation, like a wheel fixed on an axis, from west to east, about its own centre.29

Here, we have an example of a modern view, but one that attracted few followers among Heraclides' contemporaries and which was not considered seriously again until the fourteenth century. This may at first seem strange, but, as with other similar examples, the perceived advantages of Heraclides' theory were far outweighed by the obvious disadvantage of requiring the Earth to spin round, something that is quite alien to everyday experience.

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