13 Jul
2007
13 Jul
'07
8:44 p.m.
On 7/13/07, Bill Gosper <gosper@alum.mit.edu> wrote:
... So, fresh start: the solid angle at the apex of a regular n-gonal pyramid with slant edges 1 and height h ... = 2 pi - 2 n atan(h t),
where t = tan pi/n, but explement the middle term (2 pi - acos) when x< 1/(1+sec pi/n). There must be some obvious insight for the incredibly simple rhs.
This looks suspiciously like the area formula for the dual spherical polygon: this has angles and sides corresponding to the sides and angles of the of the original polygon (whose angles corrspond to dihedral angles, and sides to apical angles). See Todhunter. WFL