Games And Mathematics
“It was only after many brilliant mathematicians had struggled with the idea that the dark and obscure landscape of divergent series was illuminated, and mathematicians finally decided what a divergent series ‘really is’ and how they could safely be handled.
Euler's relation for polyhedra A similar confusion arose over another of Euler's interests: polyhedra. He noticed what Descartes had also seen earlier that for the simplest polyhedra, the number of vertices plus the number of faces exceeded... the number of edges by 2: V + F = E + 2.
Thus, in Figure 10.1 a cube has 8 vertices, 6 faces and 12 edges, and 6 + 8 = 14 + 2. The octahedron on the right has 6 vertices, 8 faces and also 12 edges, while the irregular polyhedron below which is a square pyramid face-to-face with a cube, has 9 vertices, 9 faces and 16 edges, and 9 + 9 = 16 + 2.
Figure 10.1 Diagram of cube, octahedron and pyramid ON cube Naturally such simple observations cry out for an explanation, but all the early proofs were defective.MoreLess
Euler's relation for polyhedra A similar confusion arose over another of Euler's interests: polyhedra. He noticed what Descartes had also seen earlier that for the simplest polyhedra, the number of vertices plus the number of faces exceeded... the number of edges by 2: V + F = E + 2.
Thus, in Figure 10.1 a cube has 8 vertices, 6 faces and 12 edges, and 6 + 8 = 14 + 2. The octahedron on the right has 6 vertices, 8 faces and also 12 edges, while the irregular polyhedron below which is a square pyramid face-to-face with a cube, has 9 vertices, 9 faces and 16 edges, and 9 + 9 = 16 + 2.
Figure 10.1 Diagram of cube, octahedron and pyramid ON cube Naturally such simple observations cry out for an explanation, but all the early proofs were defective.MoreLess
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