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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 tesseractic honeycomb is one of the three regular 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 ...
s (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 ...
s), represented by
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 consisting of a packing of
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 ...
s (4-
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 ...
s). 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 a
16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...
. Two tesseracts meet at each cubic
cell Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a de ...
, four meet at each
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
face The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...
, eight meet on each
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
, and sixteen meet at each vertex. It is an analog of the
square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tili ...
, , of the plane and the
cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each verte ...
, , of 3-space. These are all part of the
hypercubic honeycomb In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for . The tessellation is constructed from 4 -hypercube ...
family of tessellations of the form . Tessellations in this family are self-dual.


Coordinates

Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).


Sphere packing

Like all regular hypercubic honeycombs, the tesseractic honeycomb corresponds to a
sphere packing In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing p ...
of edge-length-diameter spheres centered on each vertex, or (dually) inscribed in each cell instead. In the hypercubic honeycomb of 4 dimensions, vertex-centered 3-spheres and cell-inscribed 3-spheres will both fit at once, forming the unique regular
body-centered cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the Crystal structure#Unit cell, unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There ...
lattice of equal-sized spheres (in any number of dimensions). Since the tesseract is radially equilateral, there is exactly enough space in the hole between the 16 vertex-centered 3-spheres for another edge-length-diameter 3-sphere. (This 4-dimensional body centered cubic lattice is actually the union of two tesseractic honeycombs, in dual positions.) This is the same densest known regular 3-sphere packing, with kissing number 24, that is also seen in the other two regular tessellations of 4-space, the 16-cell honeycomb and the 24-cell-honeycomb. Each tesseract-inscribed 3-sphere kisses a surrounding shell of 24 3-spheres, 16 at the vertices of the tesseract and 8 inscribed in the adjacent tesseracts. These 24 kissing points are the vertices of a 24-cell of radius (and edge length) 1/2.


Constructions

There are many different
Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...
s of this honeycomb. The most symmetric form is 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 ...
. Another form has two alternating
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 ...
facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 16 types of facets around each vertex and a prismatic product Schläfli symbol 4. One can be made by stericating another.


Related polytopes and tessellations

The 24-cell honeycomb is similar, but in addition to the vertices at integers (i,j,k,l), it has vertices at half integers (i+1/2,j+1/2,k+1/2,l+1/2) of odd integers only. It is a half-filled body centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not). The
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 ...
can make a regular tessellation of the 4-sphere, with three tesseracts per face, with Schläfli symbol , called an ''order-3 tesseractic honeycomb''. It is topologically equivalent to the regular polytope
penteract In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts ...
in 5-space. The tesseract can make a regular tessellation of 4-dimensional
hyperbolic space In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symme ...
, with 5 tesseracts around each face, with Schläfli symbol , called an
order-5 tesseractic honeycomb In the geometry of hyperbolic 4-space, the order-5 tesseractic honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol , it has five 8-cells (also known as tesseracts) around each face. Its du ...
. The
Ammann–Beenker tiling In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker. ...
is an
aperiodic tiling An aperiodic tiling is a non-periodic Tessellation, tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic set of prototiles, aperiodic if copie ...
in 2 dimensions obtained by cut-and-project on the tesseractic honeycomb along an eightfold rotational axis of symmetry.Beenker FPM, Algebraic theory of non periodic tilings of the plane by two simple building blocks: a square and a rhombus, TH Report 82-WSK-04 (1982), Technische Hogeschool, Eindhoven


Birectified tesseractic honeycomb

A birectified tesseractic honeycomb, , contains all rectified 16-cell (
24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...
) facets and is the Voronoi tessellation of the D4* lattice. Facets can be identically colored from a doubled _4×2, 4,3,3,4 symmetry, alternately colored from _4, ,3,3,4symmetry, three colors from _4, ,3,31,1symmetry, and 4 colors from _4, 1,1,1,1symmetry.


See also

Regular and uniform honeycombs in 4-space: * 16-cell honeycomb * 24-cell honeycomb * 5-cell honeycomb * Truncated 5-cell honeycomb * Omnitruncated 5-cell honeycomb


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 1 * x∞o x∞o x∞o x∞o, x∞x x∞o x∞o x∞o, x∞x x∞x x∞o x∞o, x∞x x∞x x∞x x∞o,x∞x x∞x x∞x x∞x, x∞o x∞o x4o4o, x∞o x∞o o4x4o, x∞x x∞o x4o4o, x∞x x∞o o4x4o, x∞o x∞o x4o4x, x∞x x∞x x4o4o, x∞x x∞x o4x4o, x∞x x∞o x4o4x, x∞x x∞x x4o4x, x4o4x x4o4x, x4o4x o4x4o, x4o4x x4o4o, o4x4o o4x4o, x4o4o o4x4o, x4o4o x4o4o, x∞x o3o3o *d4x, x∞o o3o3o *d4x, x∞x x4o3o4x, x∞o x4o3o4x, x∞x x4o3o4o, x∞o x4o3o4o, o3o3o *b3o4x, x4o3o3o4x, x4o3o3o4o - test - O1 {{Honeycombs Honeycombs (geometry) 5-polytopes Regular tessellations