Cosmos (54 page)
Authors: Carl Sagan
If we divide both sides of this equation by 2 E, we have
We know that n is 3 or more, since the simplest polygon is the triangle, with three sides. We also know that r is 3 or more, since at least 3 faces meet at a given vertex in a polyhedron. If both n and r were
simultaneously
more than 3, the left-hand side of
Equation (5)
would be less than ⅔ and the equation could not be satisfied for any positive value of E. Thus, by another
reductio ad absurdum
argument, either n = 3 and r is 3 or more, or r = 3 and n is 3 or more.
If n = 3,
Equation (5)
becomes (1/3) + (1/r) = (1/2) + (1/E), or
So in this case r can equal 3, 4, or 5 only. (If E were 6 or more, the equation would be violated.) Now n = 3, r = 3 designates a solid in which 3 triangles meet at each vertex. By
Equation (6)
it has 6 edges; by
Equation (3)
it has 4 faces; by
Equation (4)
it has 4 vertices. Clearly it is the pyramid or tetrahedron; n = 3, r = 4 is a solid with 8 faces in which 4 triangles meet at each vertex, the octahedron; and n = 3, r = 5 represents a solid with 20 faces in which 5 triangles meet at each vertex, the icosahedron (see figures on
this page
).
If r = 3,
Equation (5)
becomes
and by similar arguments n can equal 3, 4, or 5 only, n = 3 is the tetrahedron again; n = 4 is a solid whose faces are 6 squares, the cube; and n = 5 corresponds to a solid whose faces are 12 pentagons, the dodecahedron.
There are no other integer values of n and r possible, and therefore there are only 5 regular solids, a conclusion from abstract and beautiful mathematics that has had, as we have seen, the most profound impact on practical human affairs.
(The more technical scientific works are asterisked.)
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