In
five-dimensional Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
(or
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
). It is composed of
5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°.
The ...
,
truncated 5-simplex
In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a Truncation (geometry), truncation of the regular 5-simplex.
There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as p ...
, and
bitruncated 5-simplex facets in a ratio of 1:1:1.
Structure
Its
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is an elongated 5-cell antiprism, two parallel
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional space, four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pe ...
s in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells.
It can be constructed as six sets of parallel
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
s that divide space. The hyperplane intersections generate
cyclotruncated 5-cell honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
See also
Regular and uniform honeycombs in 5-space:
*
5-cubic honeycomb
*
5-demicubic honeycomb
*
5-simplex honeycomb
In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. ...
*
Omnitruncated 5-simplex honeycomb
In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex facets.
The facets of all omn ...
Notes
References
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
* 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,
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. Zeit. 46 (1940) 380-407, MR 2,10(1.9 Uniform space-fillings)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45{{Honeycombs
Honeycombs (geometry)
6-polytopes