Tesseract

A net of a tesseract.

In geometry, the tesseract 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.

The tesseract is also called an 8-cell, C8, (regular) octachoron, octahedroid,cubic prism, and tetracube (although this last term can also mean a polycube made of four cubes). It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of hypercubes or "measure polytopes".

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 (téssereis aktines or "four rays"), referring to the four lines from each vertex to other vertices. In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract".

    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:

    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:

    • 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
    A diagram showing how to create a tesseract from a point
    An animation of the shifting in dimensions as shown above

    It is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space.

    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.

    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

    The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space. An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract. 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).

    Stereoscopic 3D projection of a tesseract (parallel view)

    Alternative projections


    A 3D projection of a tesseract 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 Other F4 A3
    Graph
    Dihedral symmetry [2] [12/3] [4]

    Tessellation

    The tesseract, along with all hypercubes, tessellates Euclidean space. The self-dual tesseractic honeycomb consisting of 4 tesseracts around each face has Schläfli symbol {4,3,3,4}. Hence, the tesseract has a dihedral angle of 90°.

    Related uniform polytopes

    Convex p-gonal prismatic prisms
    Name {3}×{}×{} {4}×{}×{} {5}×{}×{} {6}×{}×{} {7}×{}×{} {8}×{}×{} {p}×{}×{}
    Coxeter
    diagrams



    Image




    Cells 3 {4}×{}
    4 {3}×{}
    4 {4}×{}
    4 {4}×{}
    5 {4}×{}
    4 {5}×{}
    6 {4}×{}
    4 {6}×{}
    7 {4}×{}
    4 {7}×{}
    8 {4}×{}
    4 {8}×{}
    p {4}×{}
    4 {p}×{}
    Net
    Name tesseract rectified
    tesseract
    truncated
    tesseract
    cantellated
    tesseract
    runcinated
    tesseract
    bitruncated
    tesseract
    cantitruncated
    tesseract
    runcitruncated
    tesseract
    omnitruncated
    tesseract
    Coxeter
    diagram

    =

    =
    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
    B4
     
    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

    =

    =

    =

    =

    =

    =
    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
    B4

    It is in a sequence of regular 4-polytopes 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
    Cells
    {p,3}

    {3,3}

    {4,3}

    {5,3}

    {6,3}

    {7,3}

    {8,3}

    {∞,3}

    It is in a sequence of regular 4-polytope and honeycombs with cubic cells.

    {4,3,p}
    Space S3 E3 H3
    Form Finite Affine Compact Paracompact Noncompact
    Name
    {4,3,3}
    {4,3,4}


    {4,3,5}
    {4,3,6}


    {4,3,7}
    {4,3,8}

    ... {4,3,∞}

    Image
    Vertex
    figure

    {3,3}

    {3,4}


    {3,5}

    {3,6}


    {3,7}

    {3,8}


    {3,∞}

    In popular culture

    Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and fiction. Notable examples include:

    • Crucifixion (Corpus Hypercubus)—oil painting by Salvador Dalí featuring a four-dimensional hypercube unfolded into a three-dimensional Latin cross
    • The Grande Arche, a monument and building near Paris, France said to resemble the projection of a hypercube
    • "And He Built a Crooked House"—a science fiction story featuring a building in the form of a four-dimensional hypercube written by Robert Heinlein (1940).

    Notes

    References

    • 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)
    • T. Gosset (1900) On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan.
    • T. Proctor Hall (1893) "The projection of fourfold figures on a three-flat", American Journal of Mathematics 15:179–89.
    • Norman Johnson Uniform Polytopes, Manuscript (1991)
      • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
    • Victor Schlegel (1886) Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper, Waren.

    External links

    • Weisstein, Eric W., "Tesseract", MathWorld.
    • Olshevsky, George, Tesseract at Glossary for Hyperspace.
      • 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 10, George Olshevsky.
    • Richard Klitzing, 4D uniform polytopes (polychora), x4o3o3o - tes
    • The Tesseract Ray traced images with hidden surface elimination. This site provides a good description of methods of visualizing 4D solids.
    • Der 8-Zeller (8-cell) Marco Möller's Regular polytopes in R4 (German)
    • WikiChoron: Tesseract
    • HyperSolids is an open source program for the Apple Macintosh (Mac OS X and higher) which generates the five regular solids of three-dimensional space and the six regular hypersolids of four-dimensional space.
    • Hypercube 98 A Windows program that displays animated hypercubes, by Rudy Rucker
    • ken perlin's home page A way to visualize hypercubes, by Ken Perlin
    • Some Notes on the Fourth Dimension includes very good animated tutorials on several different aspects of the tesseract, by Davide P. Cervone
    • Tesseract animation with hidden volume elimination
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