[seqfan] Re: More [cons] Proposals

[Giovanni Resta]

People looking for real constants should also consider
the huge database of the Inverse Symbolic Calculator

old: http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html
new: https://isc.carma.newcastle.edu.au/

Giovanni

Il 08/09/2018 17:17, Brad Klee ha scritto:
> 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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