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- W1974464424 abstract "We have colored one point from each pair of antipodal points; give the other point the same color. One checks that this coloring is proper. To show that four colors are needed, suppose that a proper 3-coloring exists. Antipodal points must have the same color, and only two colors can appear on a great circle. Let A, B, C be three pairwise orth-ogonal points. Let their colors be red, blue, green, respectively. Let D be the point midway between B and C on the open quarter circle between them; we may assume by symmetry that D is blue. Let E be a point orthogonal to D on the great circle spanned by B and C; E must be green. Let F and G be the points on the great circle spanned by A and D that have spherical distance Tr/4 from A; note that F and G are orthogonal. Hence one of {F, GI is red and the other is blue; we may assume that F is red. The great circle determined by ( A, E } is colored red and green, the great circle determined by { B, F} is colored blue and red, and the great circle determined by {C, G} is colored green and blue. By the symmetry in the construction of F and G, the great circles determined by {B, F} and by (C, G} intersect at two points on the great circle determined by {E, A}. There is no color available for the common points." @default.
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- W1974464424 date "2001-10-01" @default.
- W1974464424 modified "2023-10-16" @default.
- W1974464424 title "Convergence of a Series: 10808" @default.
- W1974464424 doi "https://doi.org/10.2307/2695637" @default.
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