[seqfan] More [cons] Proposals
Brad Klee
bradklee at gmail.com
Sat Sep 8 17:17:09 CEST 2018
Hi Seqfans,
The OEIS works well as a searchable database of mathematical constants,
so why shouldn't we generate a few ideas for growing this type of content?
Stanislov Sykora has already started on Steradian Solid Angles, nice work!
Through solid geometry of A236555 and A236556, we discovered an identity:
arccos(23/27) + 3*arcsin(1/3) = Pi/2 .
If it seems mysterious, take a look at the vertex figure of the space
tiling by tetrahedra and octahedra. By analogy, we could calculate
and list the face solid angles of an icosidodecahedron:
A_3 = -Pi + 3*arctan(2) = 0.179853499792...
A_5 = 2*Pi - 5*arctan(2) = 0.747441718209...
With characteristic identity 5*A_3 + 3*A_5 = Pi.
We needn't stop short with only areas bounded by great circles. I
calculated
a numerical integral for the vertex solid angle of Klein's quartic [1,2],
A = 3.0203012324...
The algorithm converges on these digits, which seem reasonable. I am not
100% certain of the accuracy. It would be nice to have someone else do
another calculation and compare. I have not found a reference for this
particular number; though, many authors have calculated the period
matrix [3].
I'm currently at my (3/3) limit, but sometime I hope to add at least the
icosahedral measurements, which are easy to calculate.
Cheers,
Brad
[1] http://mathworld.wolfram.com/KleinQuartic.html
[2] https://ptpb.pw/S5JJ.png
[3] https://arxiv.org/abs/0905.4202
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