A133682 - OEIS

%I #7 Jan 28 2023 12:19:50

%S 1,22,8,7,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%T 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%U 3,3

%N Number of regular complex polytopes in n-dimensional unitary complex space.

%C In each dimension there are infinite families which we count as a single polytope: the generalized complex n-cube with generalized Schlaefli symbol m(4)2(3)2...2(3)2 with m^n vertices and its dual, the generalized complex n-cross-polytope.

%D H. S. M. Coxeter, Regular complex polytopes, Cambridge University Press, 1974.

%D E. Schulte, Symmetry of Polytopes and Polyhedra, in J. E. Goodman and J. O'Rourke, Handbook of discrete and computational geometry, 2nd edition, Chapman & Hall / CRC, 2004.

%H G. C. Shephard, <a href="https://doi.org/10.1112/plms/s3-2.1.82">Regular complex polytopes</a>, Proc. Lond. Math. Soc. (3), Vol. 2 (1952), pp. 82-97.

%e a(3) = 8 because in C^3 the regular complex polytopes correspond to the following generalized Schlaefli symbols: m(4)2(3)2 (generalized complex cube), 2(3)2(4)m (generalized complex octahedron), 2(6)2(6)2 (tetrahedron), 2(6)2(10)2 (icosahedron), 2(10)2(6)2 (dodecahedron), 3(3)3(3)3, 3(3)3(4)2, 2(4)2(3)3.

%Y Cf. A060296.

%K nonn

%O 1,2

%A _Brian Hopkins_, Jan 03 2008