In
four-dimensional
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...
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 truncated 24-cell honeycomb is a uniform space-filling
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 can be seen as a
truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
of the regular
24-cell honeycomb
In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol .
T ...
, containing
tesseract
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...
and
truncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the Truncation (geometry), truncation of the regular 24-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 24-cell ...
cells.
It has a uniform
alternation, called the
snub 24-cell honeycomb
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 ...
. It is a snub from the
construction. This truncated 24-cell has
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 ...
t, and its
snub
A snub, cut, or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub.
In awards and lists
For awards, the term "snub ...
is represented as s.
Alternate names
* Truncated icositetrachoric tetracomb
* Truncated icositetrachoric honeycomb
* Cantitruncated 16-cell honeycomb
* Bicantitruncated tesseractic honeycomb
Symmetry constructions
There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored
truncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the Truncation (geometry), truncation of the regular 24-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 24-cell ...
facets. In all cases, four truncated 24-cells, and one
tesseract
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...
meet at each vertex, but the vertex figures have different symmetry generators.
See also
Regular and uniform honeycombs in 4-space:
*
Tesseractic honeycomb
In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and consisting of a packing of tesseracts (4-hypercubes).
Its vertex fi ...
*
16-cell honeycomb
In Four-dimensional space, four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycomb (geometry), honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensiona ...
*
24-cell honeycomb
In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol .
T ...
*
Rectified 24-cell honeycomb
*
Snub 24-cell honeycomb
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 ...
*
5-cell honeycomb
In Four-dimensional space, four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb (geometry), honeycomb. It is composed of 5-cells and recti ...
*
Truncated 5-cell honeycomb
*
Omnitruncated 5-cell honeycomb
In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1. Struct ...
References
*
Coxeter, H.S.M. ''
Regular Polytopes
''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...
'', (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs
* 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 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45* George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' Model 99
* o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99
{{Honeycombs
5-polytopes
Honeycombs (geometry)
Truncated tilings