26 Dec
2002
26 Dec
'02
10:57 a.m.
I asked:
Exercise: what is the black/white (pentagons/hexagons) area ratio of a soccer ball (spherical truncated icosahedron)? I get 720 sqrt(5) + 54887 3 acos(-------------------) 59049 r = -------------------------------- ~ 0.3923179664138, sqrt(3) (7 sqrt(5) + 5) 20 acos(-----------------------) 36
i.e., the ball is only r ----- ~ 28.17732557343% black. r + 1 Though inelegant, this is numerically plausible since it gives a (spherical) hexagon/pentagon area ratio 12/(20 r) ~ 1.52937, compared with the planar area ratio 12 sqrt(6) ---------------- ~ 1.510090, 3/2 (sqrt(5) + 5) given a common side length. --rwg