[seqfan] Re: More [cons] Proposals
Brad Klee
bradklee at gmail.com
Sun Sep 9 21:48:36 CEST 2018
Hi Robert,
Working the analogy between Cuboctahedron and Icosidodecahedron, yes,
we can easily write a zero sum with one free parameter [1],
A_3 = 3*arcsin(z) - Pi, A_4 = 2*Pi - 4*arcsin(z), 8*A_3 + 6*A_4 - 4*Pi
= 0.
The initial configuration is z = 2*sqrt(2)/3 for A_3=A236555 and
A_4=A236556. "More generally", for z in [ sqrt(3)/2 , 1 ] we can yet have a
division of the sphere into 8 regular spherical triangles and their
complement, by
truncation of a cube or by twisting the cuboctahedron into an octahedron
[1,2].
Robert--Unless you have a geometric transformation model for your identity,
we do not agree on the question of generalization. The main difference
between
"y" and "z" parameters seems to be that the "z" domain is more concise, due
to a restriction imposed by octahedral symmetry. Other values of "z" lead
to
measurements arguably worth recording. I doubt that OEIS has facet solid
angles for second tier polyhedra such as truncations of the five regular
solids.
Does the "y" parameter do anything better?
Setting aside solid geometry, the more interesting generalization is to
find
generating functions G(x), which converge over [0,x_0], and then to
calculate
G(x_0), the finite boundary value ( against slow convergence ). This
perspective
includes the Polya Walk constants ( Cf. A039699 / A086232 ) alongside these
solid angles.
Cheers,
Brad
[1] http://mathworld.wolfram.com/SphericalPolygon.html
[2] https://www.youtube.com/watch?v=IQCqaMxnpQA
[3] https://www.youtube.com/watch?v=HekEKdcw5_k
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