¶ Projective Transformation
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a spherical triangle total of 270 degrees, which means an excess of 90 degrees. If the edges are bisected again each of the corners will be 72° 32’ with a total of 221° 36’ so the spherical excess of 221° 36’ minus 180° is equal to 40° 36’ and is being reduced very rapidly. Smaller and smaller spherical triangles give less and less spherical excess. Therefore, I subtriangulate the total of the Earth’s surface in the largest number of identical equi-edged small triangles, i.e., the spherical icosahedron, where the spherical excess is a minimal 36°. . . .
“I have found that if I wanted to take off the total world’s data in the largest number of identical triangles and have them symmetrical, the triangles have to be equilaterals. The maximum number of equilateral triangles into which we can subdivide a spherical unity is twenty: a spherical icosahedron.” You might ask, “Why can’t you have more triangles than twenty?” To explain, I shall first take two triangles and put them together. We see that they hinge and just come to be congruent one to the other.
