In
five-dimensional geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a 5-orthoplex, or 5-
cross polytope, is a five-dimensional polytope with 10
vertices, 40
edges, 80 triangle
faces, 80 tetrahedron
cells, 32
5-cell 4-faces.
It has two constructed forms, the first being regular with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol or
Coxeter symbol 2
11.
It is a part of an infinite family of polytopes, called
cross-polytopes or ''orthoplexes''. The
dual polytope is the 5-
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
or
5-cube.
Alternate names
* Pentacross, derived from combining the family name ''cross polytope'' with ''pente'' for five (dimensions) in
Greek.
* Triacontaditeron (or ''triacontakaiditeron'') - as a 32-
facetted 5-polytope (polyteron). Acronym: tac
As a configuration
This
configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
[Coxeter, Complex Regular Polytopes, p.117]
Cartesian coordinates
Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are
: (±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)
Construction
There are three
Coxeter groups associated with the 5-orthoplex, one
regular,
dual of the
penteract with the C
5 or
,3,3,3 Coxeter group, and a lower symmetry with two copies of ''5-cell'' facets, alternating, with the D
5 or
2,1,1Coxeter group, and the final one as a dual 5-
orthotope, called a 5-fusil which can have a variety of subsymmetries.
Other images
Related polytopes and honeycombs
This polytope is one of 31
uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets.
The complete set of convex uniform 5-polytopes ...
s generated from the B
5 Coxeter plane, including the regular
5-cube and 5-orthoplex.
References
*
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.com
*** (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*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* x3o3o3o4o - tac
External links
*
Polytopes of Various Dimensions
{{Polytopes
5-polytopes