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 ? -------------------------------------------------------- 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 ----------------------------------------------------------------- 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). ------------------------------------------------------------------