O platos description at 54e and 55b tells us that

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o Plato’s description at 54e and 55b tells us that each t is made of 6 a ’s, and each s is made of 4 b ’s. (See diagrams, RAGP 640.) But 57c-d makes clear that he envisages other ways of constructing these faces out of primitive a ’s and b ’s.
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3. Construction of solid particles out of the faces The construction of the particles is described at 54d-55c. The particles are identified with the four elements at 55d-56b. Click on the names of the elements to see a diagram of a particle of that element: o Fire : a particle of fire is a tetrahedron (4-sided solid), made of 4 t ’s consisting of 24 a ’s altogether. o Air : a particle of air is an octahedron (8-sided solid), made of 8 t ’s consisting of 48 a ’s altogether. o Water : a particle of water is an icosahedron (20-sided solid), made of 20 t ’s consisting of 120 a ’s altogether. o Earth : a particle of earth is a cube (6-sided solid), made of 6 s ’s consisting of 24 b ’s altogether. 4. Transformation of elements (described at 56c-57c) Inter-elemental transformations are among fire, air, and water only. Earth cannot be transformed into any of the others (54c, 56d). Transformations can be described at the level of equilateral triangles (that are the faces of the three solids). Since a fire molecule has 4 faces (one F is made up of 4 t ), an air molecule 8 (one A is made up of 8 t ), and a water molecule 20 (one W is made up of 20 t ), any of the following transformations (for example) are possible. (Each transformation is represented by an equation on the left; its geometrical basis is shown by the equation on the right.): 1 A = 2 F 8 t = 2 × 4 t 1 W = 5 F 20 t = 5 × 4 t 2 W = 5 A 2 × 20 t = 5 × 8 t 1 W = 2 A + 1 F 20 t = (2 × 8 t ) + 4 t 1 W = 3 F + 1 A 20 t = (3 × 4 t ) + 8 t
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