User:Tomruen/Thorold Gosset

Thorold Gosset (1869-1962)

polytopes


 N Lines            Polytope         Dim  Symmetry Group 9   0                 0              0     A0 U(1) 8   1 or 0       line segment        1     A1 SU(2) 7   3              triangle          2     A2 SU(3) 6   6        half-cuboctahedron      3     A3=D3 SU(4)=Spin(6) 5  10        6+4 faces of 4cube      4     D4 Spin(8) 4  16     Gosset 1_21 (half-5cube)   5     D5 Spin(10) 3  27             Gosset 2_21        6     E6 2   56=28+28       Gosset 3_21        7     E7 1  240        Witting = Gosset 4_21   8     E8 0  infinite        Gosset 5_21        9     E9 = affine extension of E8 

 The real 4_21 Witting polytope of the E8 lattice in R8 has

240 vertices;

6,720 edges;

60,480 triangular faces;

241,920 tetrahedra;

483,840 4-simplexes;

483,840 5-simplexes 4_00;

138,240 + 69,120 6-simplexes 4_10 and 4_01; and

17,280 7-simplexes 4_20 and 2,160 7-cross-polytopes 4_11.



http://www.liga.ens.fr/~dutour/Regular/  Semi-regular polytopes All regular polytopes 0_21 also called hypersimplex 1_21, half-5-cube 2_21, Delaunay polytope of the root lattice E6 3_21, Delaunay polytope of the root lattice E7 4_21, Voronoi polytope of the root lattice E8 snub 24-cell octicosahedric polytope, i.e. the medial of 600-cell. 