Stericantic tesseractic honeycomb

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Stericantic tesseractic honeycomb
(No image)
TypeUniform honeycomb
Schläfli symbol h2,4{4,3,3,4}
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
4-face type rr{4,3,3}
t0,1,3{3,3,4}
t{3,3,4}
{3,3}×{}
Cell type rr{4,3}
{3,4}
{4,3}
t{3,3}
t{3}×{}
{3}×{}
Face type{6}
{4}
{3}
Vertex figure
Coxeter group = [4,3,31,1]
Dual?
Properties vertex-transitive

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

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 mathematical system attributed to Euclid

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.

Tessellation tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps

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.

Contents

Alternate names

The [4,3,31,1], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, 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 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.

16-cell honeycomb one of three regular space-filling tessellation in Euclidean 4-space

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.

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, represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets.

24-cell honeycomb

In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol {3,4,3,3}.

In four-dimensional Euclidean geometry, the rectified 24-cell honeycomb is a uniform space-filling honeycomb. It is constructed by a rectification of the regular 24-cell honeycomb, containing tesseract and rectified 24-cell cells.

Notes

    Related Research Articles

    In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.

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

    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.

    Birectified 16-cell honeycomb

    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 steriruncicantic tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.

    References

    International Standard Book Number Unique numeric book identifier

    The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

    Fundamental convex regular and uniform honeycombs in dimensions 2-9
    Space Family / /
    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 133331
    E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
    E9 Uniform 9-honeycomb {3[10]}δ10hδ10qδ10
    En-1Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21