Truncated 16-cell honeycomb

Last updated March 22, 2019

Truncated 16-cell honeycomb
(No image)
Type	Uniform honeycomb
Schläfli symbols	t{3,3,4,3} h₂{4,3,3,4} t{3,3^1,1,1}
Coxeter diagrams	=
4-face type	{3,4,3} t{3,3,4}
Cell type	{3,3} t{3,3}
Face type	{3} {6}
Vertex figure	cubic pyramid
Coxeter group	${\tilde {F}}_{4}$ = [3,3,4,3] ${\tilde {B}}_{4}$ = [4,3,3^1,1] ${\tilde {D}}_{4}$ = [3^1,1,1,1]
Dual	?
Properties	vertex-transitive

In four-dimensional Euclidean geometry, the truncated 16-cell honeycomb (or cantic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by 24-cell and truncated 16-cell facets.

Four-dimensional space geometric space with four dimensions

A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible generalization of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring its length, width, and height.

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

Alternate names

Truncated hexadecachoric tetracomb / Truncated hexadecachoric honeycomb

Related honeycombs

The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,3^1,1] families. The alternation (13) is also repeated in other families.

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

F4 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[3,3,4,3]		×1	₁, ₃, ₅, ₆, ₈, ₉, ₁₀, ₁₁, ₁₂
[3,4,3,3]		×1	₂, ₄, ₇, ₁₃, ₁₄, ₁₅, ₁₆, ₁₇, ₁₈, ₁₉, ₂₀, ₂₁, ₂₂₂₃, ₂₄, ₂₅, ₂₆, ₂₇, ₂₈, ₂₉
[(3,3)[3,3,4,3^*]] =[(3,3)[3^1,1,1,1]] =[3,4,3,3]	= =	×4	₍₂₎, ₍₄₎, ₍₇₎, ₍₁₃₎

The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.

In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements. Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.

C4 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[4,3,3,4]:		×1	₁, ₂, ₃, ₄, ₅, ₆, ₇, ₈, ₉, ₁₀, ₁₁, ₁₂, ₁₃
[[4,3,3,4]]		×2	₍₁₎, ₍₂₎, ₍₁₃₎, ₁₈ ₍₆₎, ₁₉, ₂₀
[(3,3)[1⁺,4,3,3,4,1⁺]] ↔ [(3,3)[3^1,1,1,1]] ↔ [3,4,3,3]	↔ ↔	×6	₁₄, ₁₅, ₁₆, ₁₇

The [4,3,3^1,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations, represented by Schläfli symbol {3,3,4,3}, and constructed by a 4-dimensional packing of 16-cell facets, three around every face.

In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra.

B4 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[4,3,3^1,1]:		×1	₅, ₆, ₇, ₈
<[4,3,3^1,1]>: ↔[4,3,3,4]	↔	×2	₉, ₁₀, ₁₁, ₁₂, ₁₃, ₁₄, ₍₁₀₎, ₁₅, ₁₆, ₍₁₃₎, ₁₇, ₁₈, ₁₉
[3[1⁺,4,3,3^1,1]] ↔ [3[3,3^1,1,1]] ↔ [3,3,4,3]	↔ ↔	×3	₁, ₂, ₃, ₄
[(3,3)[1⁺,4,3,3^1,1]] ↔ [(3,3)[3^1,1,1,1]] ↔ [3,4,3,3]	↔ ↔	×12	₂₀, ₂₁, ₂₂, ₂₃

There are ten uniform honeycombs constructed by the ${\tilde {D}}_{4}$ Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)^*] (index 24), [3,3,4,3^*] (index 6), [1⁺,4,3,3,4,1⁺] (index 4), [3^1,1,3,4,1⁺] (index 2) are all isomorphic to [3^1,1,1,1].

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors. An unlabeled branch implicitly represents order-3.

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

The ten permutations are listed with its highest extended symmetry relation:

D4 honeycombs
Extended symmetry	Extended diagram	Extended group	Honeycombs
[3^1,1,1,1]		${\tilde {D}}_{4}$	(none)
<[3^1,1,1,1]> ↔ [3^1,1,3,4]	↔	${\tilde {D}}_{4}$ ×2 = ${\tilde {B}}_{4}$	(none)
<2[^1,13^1,1]> ↔ [4,3,3,4]	↔	${\tilde {D}}_{4}$ ×4 = ${\tilde {C}}_{4}$	₁, ₂
[3[3,3^1,1,1]] ↔ [3,3,4,3]	↔	${\tilde {D}}_{4}$ ×6 = ${\tilde {F}}_{4}$	₃, ₄, ₅, ₆
[4[^1,13^1,1]] ↔ [[4,3,3,4]]	↔	${\tilde {D}}_{4}$ ×8 = ${\tilde {C}}_{4}$ ×2	₇, ₈, ₉
[(3,3)[3^1,1,1,1]] ↔ [3,4,3,3]	↔	${\tilde {D}}_{4}$ ×24 = ${\tilde {F}}_{4}$	₇, ₈, ₉
[(3,3)[3^1,1,1,1]]⁺ ↔ [3⁺,4,3,3]	↔	½ ${\tilde {D}}_{4}$ ×24 = ½ ${\tilde {F}}_{4}$	₁₀

Notes

Related Research Articles

In four-dimensional Euclidean geometry, the rectified tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space. It is constructed by a rectification of a tesseractic honeycomb which creates new vertices on the middle of all the original edges, rectifying the cells into rectified tesseracts, and adding new 16-cell facets at the original vertices. Its vertex figure is an octahedral prism, {3,4}×{}.

In four-dimensional Euclidean geometry, the bitruncated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space. It is constructed by a bitruncation of a tesseractic honeycomb. It is also called a cantic quarter tesseractic honeycomb from its q₂{4,3,3,4} construction.

In four-dimensional Euclidean geometry, the birectified 16-cell honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the steriruncitruncated tesseractic honeycomb is a uniform space-filling honeycomb.

In four-dimensional Euclidean geometry, the stericantellated tesseractic honeycomb is a uniform space-filling honeycomb.

In four-dimensional Euclidean geometry, the omnitruncated tesseractic honeycomb is a uniform space-filling honeycomb. It has omnitruncated tesseract, truncated cuboctahedral prism, and 8-8 duoprism facets in an irregular 5-cell vertex figure.

In four-dimensional Euclidean geometry, the truncated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space. It is constructed by a truncation of a tesseractic honeycomb creating truncated tesseracts, and adding new 16-cell facets at the original vertices.

In four-dimensional Euclidean geometry, the cantellated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space. It is constructed by a cantellation of a tesseractic honeycomb creating cantellated tesseracts, and new 24-cell and octahedral prism facets at the original vertices.

In four-dimensional Euclidean geometry, the runcinated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space. It is constructed by a runcination of a tesseractic honeycomb creating runcinated tesseracts, and new tesseract, rectified tesseract and cuboctahedral prism facets.

In four-dimensional Euclidean geometry, the cantitruncated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the runcitruncated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the steritruncated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the runcicantitruncated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the runcicantellated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the stericantitruncated tesseractic honeycomb is a uniform space-filling honeycomb. It is composed of runcitruncated 16-cell, cantitruncated tesseract, rhombicuboctahedral prism, truncated cuboctahedral prism, and 4-8 duoprism facets, arranged around an irregular 5-cell vertex figure.

In four-dimensional Euclidean geometry, the steric tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the steriruncic tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the stericantic tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the steriruncicantic tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

In four-dimensional Euclidean geometry, the runcinated 16-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a runcination of the regular 16-cell honeycomb, containing Rectified 24-cell, runcinated tesseract, cuboctahedral prism, and 3-3 duoprism cells.

References

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, ISBN 978-0-471-01003-6
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. 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)
Klitzing, Richard. "4D Euclidean tesselations". (x3x3o *b3o4o), (x3x3o *b3o *b3o), x3x3o4o3o - thext - O105

v t e Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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