Tesseract

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Tesseract
8-cell
4-cube
Schlegel wireframe 8-cell.png
Type Convex regular 4-polytope
Schläfli symbol {4,3,3}
t0,3{4,3,2} or {4,3}×{ }
t0,2{4,2,4} or {4}×{4}
t0,2,3{4,2,2} or {4}×{ }×{ }
t0,1,2,3{2,2,2} or { }×{ }×{ }×{ }
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Cells 8 (4.4.4) Hexahedron.png
Faces 24 {4}
Edges 32
Vertices 16
Vertex figure 8-cell verf.png
Tetrahedron
Petrie polygon octagon
Coxeter group C4, [3,3,4]
Dual 16-cell
Properties convex, isogonal, isotoxal, isohedral
Uniform index 10

In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes.

A generalization of the cube to dimensions greater than three is called a "hypercube", "n-cube" or "measure polytope".1 The tesseract is the four-dimensional hypercube, or 4-cube.

According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek τέσσερεις ακτίνες ("four rays"), referring to the four lines from each vertex to other vertices.2 In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract." Some peoplecitation needed have called the same figure a tetracube, and also simply a hypercube (although a tetracube can also mean a polycube made of four cubes, and the term hypercube is also used with dimensions greater than 4).

Geometry

The tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol {4,3,3} with hyperoctahedral symmetry of order 384. Constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol {4,3} × { }, with symmetry order 96. As a duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol {4}×{4}, with symmetry order 64. As an orthotope it can be represented by composite Schläfli symbol { } × { } × { } × { } or { }4, with symmetry order 16.

Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4}.

The standard tesseract in Euclidean 4-space is given as the convex hull of the points (±1, ±1, ±1, ±1). That is, it consists of the points:

\{(x_1,x_2,x_3,x_4) \in \mathbb R^4 \,:\, -1 \leq x_i \leq 1 \}

A tesseract is bounded by eight hyperplanes (xi = ±1). Each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.

Projections to 2 dimensions

A diagram showing how to create a tesseract from a point

The construction of a hypercube can be imagined the following way:

  • 1-dimensional: Two points A and B can be connected to a line, giving a new line segment AB.
  • 2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.
  • 3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
  • 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.
A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom

This structure is not easily imagined, but it is possible to project tesseracts into three- or two-dimensional spaces. Furthermore, projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. In this fashion, one can obtain pictures that no longer reflect the spatial relationships within the tesseract, but which illustrate the connection structure of the vertices, such as in the following examples:

A tesseract is in principle obtained by combining two cubes. The scheme is similar to the construction of a cube from two squares: juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length. This view is of interest when using tesseracts as the basis for a network topology to link multiple processors in parallel computing: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.

Tesseracts are also bipartite graphs, just as a path, square, cube and tree are.

Parallel projections to 3 dimensions

The rhombic dodecahedron forms the convex hull of the tesseract's vertex-first parallel-projection. The number of vertices in the layers of this projection is 1 4 6 4 1 - the fourth row in Pascal's triangle.
Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)

The cell-first parallel projection of the tesseract into 3-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining 6 cells are projected onto the 6 square faces of the cube.

The face-first parallel projection of the tesseract into 3-dimensional space has a cuboidal envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the 4 remaining cells project to the side faces.

The edge-first parallel projection of the tesseract into 3-dimensional space has an envelope in the shape of a hexagonal prism. Six cells project onto rhombic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto 6 rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases.

The vertex-first parallel projection of the tesseract into 3-dimensional space has a rhombic dodecahedral envelope. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving a total of 8 possible parallelepipeds. The images of the tesseract's cells under this projection are precisely these 8 parallelepipeds. This projection is also the one with maximal volume.

Image gallery

3-D net of a tesseract

The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space (view animation). An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract.3 The unfoldings of the tesseract can be counted by mapping the nets to paired trees (a tree together with a perfect matching in its complement).

3D stereographic projection tesseract.PNG
Stereoscopic 3D projection of a tesseract (parallel view Stereogram guide parallel.png)

Alternative projections

Tesseract.gif
A 3D projection of an 8-cell performing a double rotation about two orthogonal planes
Tesseract-perspective-vertex-first-PSPclarify.png
Perspective with hidden volume elimination. The red corner is the nearest in 4D and has 4 cubical cells meeting around it.
Tesseract tetrahedron shadow matrices.svg

The tetrahedron forms the convex hull of the tesseract's vertex-centered central projection. Four of 8 cubic cells are shown. The 16th vertex is projected to infinity and the four edges to it are not shown.

Stereographic polytope 8cell.png
Stereographic projection

(Edges are projected onto the 3-sphere)

2D orthographic projections

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t0.svg 4-cube t0 B3.svg 4-cube t0 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane Other F4 A3
Graph 4-cube column graph.svg 4-cube t0 F4.svg 4-cube t0 A3.svg
Dihedral symmetry [2] [12/3] [4]

Related uniform polytopes

Convex p-gonal prismatic prisms
Name {3}×{}×{} {4}×{}×{} {5}×{}×{} {6}×{}×{} {7}×{}×{} {8}×{}×{} {p}×{}×{}
Coxeter
diagrams
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Image 3-4 duoprism.png
4-3 duoprism.png
4-4 duoprism.png 4-5 duoprism.png
5-4 duoprism.png
4-6 duoprism.png
6-4 duoprism.png
4-7 duoprism.png
7-4 duoprism.png
4-8 duoprism.png
8-4 duoprism.png
Cells 3 {4}×{} Hexahedron.png
4 {3}×{} Triangular prism.png
4 {4}×{} Hexahedron.png
4 {4}×{} Tetragonal prism.png
5 {4}×{} Hexahedron.png
4 {5}×{} Pentagonal prism.png
6 {4}×{} Hexahedron.png
4 {6}×{} Hexagonal prism.png
7 {4}×{} Hexahedron.png
4 {7}×{} Prism 7.png
8 {4}×{} Hexahedron.png
4 {8}×{} Octagonal prism.png
p {4}×{} Hexahedron.png
4 {p}×{}
Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter
diagram
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schläfli
symbol
{4,3,3} t1{4,3,3}
r{4,3,3}
t0,1{4,3,3}
t{4,3,3}
t0,2{4,3,3}
rr{4,3,3}
t0,3{4,3,3} t1,2{4,3,3}
2t{4,3,3}
t0,1,2{4,3,3}
tr{4,3,3}
t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
Schlegel wireframe 8-cell.png Schlegel half-solid rectified 8-cell.png Schlegel half-solid truncated tesseract.png Schlegel half-solid cantellated 8-cell.png Schlegel half-solid runcinated 8-cell.png Schlegel half-solid bitruncated 8-cell.png Schlegel half-solid cantitruncated 8-cell.png Schlegel half-solid runcitruncated 8-cell.png Schlegel half-solid omnitruncated 8-cell.png
B4 4-cube t0.svg 4-cube t1.svg 4-cube t01.svg 4-cube t02.svg 4-cube t03.svg 4-cube t12.svg 4-cube t012.svg 4-cube t013.svg 4-cube t0123.svg
 
Name 16-cell rectified
16-cell
truncated
16-cell
cantellated
16-cell
runcinated
16-cell
bitruncated
16-cell
cantitruncated
16-cell
runcitruncated
16-cell
omnitruncated
16-cell
Coxeter
diagram
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schläfli
symbol
{3,3,4} t1{3,3,4}
r{3,3,4}
t0,1{3,3,4}
t{3,3,4}
t0,2{3,3,4}
rr{3,3,4}
t0,3{3,3,4} t1,2{3,3,4}
2t{3,3,4}
t0,1,2{3,3,4}
tr{3,3,4}
t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
Schlegel wireframe 16-cell.png Schlegel half-solid rectified 16-cell.png Schlegel half-solid truncated 16-cell.png Schlegel half-solid cantellated 16-cell.png Schlegel half-solid runcinated 16-cell.png Schlegel half-solid bitruncated 16-cell.png Schlegel half-solid cantitruncated 16-cell.png Schlegel half-solid runcitruncated 16-cell.png Schlegel half-solid omnitruncated 16-cell.png
B4 4-cube t3.svg 4-cube t2.svg 4-cube t23.svg 4-cube t13.svg 4-cube t03.svg 4-cube t12.svg 4-cube t123.svg 4-cube t023.svg 4-cube t0123.svg

It in a sequence of regular polychora and honeycombs with tetrahedral vertex figures.

{p,3,3}
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ... {∞,3,3}
Image Stereographic polytope 5cell.png Stereographic polytope 8cell.png Stereographic polytope 120cell faces.png H3 633 FC boundary.png
Coxeter diagrams
subgroups
1 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4 CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
6 CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
12 CDel nodes 11.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png CDel branch 11.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.pngCDel 8.pngCDel node.png CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.pngCDel infin.pngCDel node.png
24 CDel nodes 11.pngCDel 2.pngCDel nodes 11.png CDel branch 11.pngCDel splitcross.pngCDel branch 11.png Cdel tet4 1111.png Cdel tetinfin 1111.png
Cells
{p,3}
CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.png
Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexahedron.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel nodes 11.pngCDel 2.pngCDel node 1.png
Dodecahedron.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.png
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel branch 11.pngCDel split2.pngCDel node 1.png
H2 tiling 237-1.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 238-1.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
H2 tiling 23i-1.png
{∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.png

It in a sequence of regular polychora and honeycombs with cubic cells.

{4,3,p}
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
{4,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{4,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png
{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{4,3,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel ultra.pngCDel node.pngCDel split1.pngCDel branch.pngCDel uaub.pngCDel nodes 11.png
{4,3,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
{4,3,8}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
... {4,3,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Image Stereographic polytope 8cell.png Cubic honeycomb.png H3 435 CC center.png H3 436 CC center.png
Vertex
figure
CDel node 1.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
8-cell verf.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cubic honeycomb verf.png
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.png
Order-5 cubic honeycomb verf.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 63-t2.png
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.png
H2 tiling 237-4.png
{3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 238-4.png
{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.png
H2 tiling 23i-4.png
{3,∞}
CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel labelinfin.png

See also

Notes

References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • H.S.M. Coxeter:
    • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • 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 [1]
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)

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