By David R. Morrison, Janos Kolla Summer Research Institute on Algebraic Geometry

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But d(p2,p) = 2sin(n/m)d(p,p,) and so d(p,p2) < d(p,p,) if m > 6. per- Pt 2n m>6 P2 A similar argument shows that p3 = R(p2,2ir/m)p, is such that R(p3,2n/m) a S(L). If m = 5 a check shows that d(p,p3) < d(p,p,) and so one concludes that m ic 6 and m * 5. 39 Each of these four remaining cases can occur, as seen by considering one of the following two lattices. , e, then the matrix of f with respect to this basis has integer 2. entries. When n = 2 or 3 and f is a rotation there is an orthonormal basis for R" with respect to which f is represented by the matrix Ae - rcose -sin91 IL sine cose J or by r A, to 0 1 Under our hypotheses this matrix is similar to an integer matrix.

The Platonic Solids We start with some definitions. A subset X of R" is convex if for every pair of points x, y e X the line segment joining them lies entirely within X, that is, for each t with 0 _- t _- 1, the point tx + (1-t)y is in X. That is, X is closed under taking non-negative affine combinations. A subset of R" defined by a finite set of linear inequalities is clearly convex and is called a convex polyhedron. ) So a convex polyhedron X in R3 is bounded by planes, a two-dimensional subset of X that is in such a plane is called a face of X.

E, then the matrix of f with respect to this basis has integer 2. entries. When n = 2 or 3 and f is a rotation there is an orthonormal basis for R" with respect to which f is represented by the matrix Ae - rcose -sin91 IL sine cose J or by r A, to 0 1 Under our hypotheses this matrix is similar to an integer matrix. As the trace of a matrix is invariant under changes of bases it must therefore be an integer. In either case we see that 2cose e Z and the only possibilities are cosO = 0, ± 1/2, ±1 giving that the rotation must have order 2, 3, 4 or 6.

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