Tesseract

Tesseract
8-cell
4-cube

Schlegel diagram
Type Convex regular 4-polytope
Schläfli symbol {4,3,3}
{4,3}x{}
{4}x{4}
{4}x{}x{}
{}x{}x{}x{}
Coxeter-Dynkin diagram



Cells 8 (4.4.4)
Faces 24 {4}
Edges 32
Vertices 16
Vertex figure
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". 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. Some people have called the same figure a tetracube, and also simply a hypercube (although the term hypercube is also used with dimensions greater than 4).

Contents

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}. Constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol {4,3} × { }. As a duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol {4}×{4}.

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

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

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 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

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.[1] 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).
A stereoscopic 3D projection of a tesseract.

Perspective projections


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
A 3D projection of an 8-cell performing a double rotation about two orthogonal planes
Perspective with hidden volume elimination. The red corner is the nearest in 4D and has 4 cubical cells meeting around it.

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 projection

(Edges are projected onto the 3-sphere)

2D orthographic projections

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph
Dihedral symmetry [12/3] [4]
This oriention shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:4:6:4:1. There's only one overlapping vertex pair, at the center.

Related uniform polytopes

Name tesseract rectified
tesseract		 truncated
tesseract		 cantellated
tesseract		 runcinated
tesseract		 bitruncated
tesseract		 cantitruncated
tesseract		 runcitruncated
tesseract		 omnitruncated
tesseract
Coxeter-Dynkin
diagram
Schläfli
symbol		 {4,3,3} t1{4,3,3} t0,1{4,3,3} t0,2{4,3,3} t0,3{4,3,3} t1,2{4,3,3} t0,1,2{4,3,3} t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
B4 Coxeter plane graph
 
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-Dynkin
diagram
Schläfli
symbol		 {3,3,4} t1{3,3,4} t0,1{3,3,4} t0,2{3,3,4} t0,3{3,3,4} t1,2{3,3,4} t0,1,2{3,3,4} t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
B4 Coxeter plane graph

See also

Notes

  1. ^ "Unfolding an 8-cell". http://unfolding.apperceptual.com/. 

References

External links

5-cell 8-cell 16-cell 24-cell 120-cell 600-cell

{3,3,3}
pentachoron

{4,3,3}
tesseract

{3,3,4}
hexadecachoron

{3,4,3}
icositetrachoron

{5,3,3}
hecatonicosachoron

{3,3,5}
hexacosichoron