In 6-dimensional
geometry, there are 39
uniform polytopes with E
6 symmetry. The two simplest forms are the
221 and
122 polytopes, composed of 27 and 72
vertices respectively.
They can be visualized as symmetric
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
s in
Coxeter planes of the E
6 Coxeter group, and other subgroups.
Graphs
Symmetric
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
s of these 39 polytopes can be made in the E
6, D
5, D
4, D
2, A
5, A
4, A
3 Coxeter planes. A
k has ''k+1'' symmetry, D
k has ''2(k-1)'' symmetry, and E
6 has ''12'' symmetry.
Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E
6 symmetry. The vertices and edges drawn with vertices colored by the number of overlapping vertices in each projective position.
References
*
H.S.M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
:
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
* Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. Zeit. 46 (1940) 380-407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'',
ath. Zeit. 188 (1985) 559-591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45*
N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
*
{{Polytopes
6-polytopes