I have a custom of sending friends & relatives a "pretty"
arrangement of n points in the plane on their nth birthday.
Usually the arrangement is hastily thought-out.
But what if, for each n -- say 1 <= n <= 100,
we ask specifically for the arrangement(s) of
n points in the plane having the largest-size
symmetry group?
In fact, let S(n) be the size of the largest
isometry group(s) among all subsets of the plane
of size n (with the inherited metric).
Is there a neat formula for S(n) ?
Is there a way to determine the winning arrangment(s) ?
Is S(n) in OEIS ?
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We can ask the same questions of the the sphere S^2, the
hyperbolic plane H^2, square torus T^2 := R^2/Z^2, getting
the sequences SS(n), SH(n) and ST(n), respectively.
--Dan
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P.S.
Even worse, one could generalize S(n) to any higher
dimension d, getting the sequence S_d(n) for R^d.
(Clearly for a given n, S_d(n) reaches its maximum for d >= n-1,
upon which the n vertices of the (n-1)-simplex form the unique
configuration with symmetry group of size n! .)
Likewise, for the d-sphere S^d, hyperbolic d-space H^d,
the cubical d-torus T^d := R^d/Z^d, we'd get sequences
SS_d(n), SH_d(n), and ST_d(n).
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