In 4-dimensional
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, there are 7
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. Th ...
s with reflections of D
4 symmetry, all are shared with higher symmetry constructions in the B
4 or F
4 symmetry families. there is also one half symmetry
alternation, the snub 24-cell.
Visualizations
Each can be visualized as symmetric
orthographic projections in
Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ar ...
s of the D
4 Coxeter group, and other subgroups. The B
4 coxeter planes are also displayed, while D
4 polytopes only have half the symmetry. They can also be shown in perspective projections of
Schlegel diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the orig ...
s, centered on different cells.
Coordinates
The ''base point'' can generate the coordinates of the polytope by taking all coordinate permutations and sign combinations. The edges' length will be . Some polytopes have two possible generator points. Points are prefixed by ''Even'' to imply only an even count of sign permutations should be included.
References
*
J.H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
and
M.J.T. Guy: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
*
John H. Conway, Heidi Burgiel,
Chaim Goodman-Strauss
Chaim Goodman-Strauss (born June 22, 1967 in Austin TX) is an American mathematician who works in convex geometry, especially aperiodic tiling. He is on the faculty of the University of Arkansas and is a co-author with John H. Conway of ''The Sym ...
, ''The Symmetries of Things'' 2008, (Chapter 26)
*
H.S.M. Coxeter:
** 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
External links
*
Uniform, convex polytopes in four dimensions: Marco Möller
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{{Demitesseract family
4-polytopes