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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the tesseract, also called an 8-cell or regular octachoron, is the four-dimensional
Fourth dimension

In physics and mathematics, a vector of n real number can be understood as a Coordinate system in an n-dimensional Euclidean space. When n = 4, the set of all such locations is called 4-dimensional Euclidean space....
 analog of the cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
. The tesseract is to the cube as the cube is to the square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
. Just as the surface of the cube consists of 6 square faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
, the hypersurface of the tesseract consists of 8 cubical cells
Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
. The tesseract is one of the six convex regular 4-polytope
Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
s.

A generalization of the cube to dimensions greater than three is called a “hypercube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
”, “n-cube” or “measure polytope”.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the tesseract, also called an 8-cell or regular octachoron, is the four-dimensional
Fourth dimension

In physics and mathematics, a vector of n real number can be understood as a Coordinate system in an n-dimensional Euclidean space. When n = 4, the set of all such locations is called 4-dimensional Euclidean space....
 analog of the cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
. The tesseract is to the cube as the cube is to the square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
. Just as the surface of the cube consists of 6 square faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
, the hypersurface of the tesseract consists of 8 cubical cells
Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
. The tesseract is one of the six convex regular 4-polytope
Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
s.

A generalization of the cube to dimensions greater than three is called a “hypercube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
”, “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
Charles Howard Hinton

Charles Howard Hinton was a United Kingdom mathematics and writer of science fiction works titled Scientific Romances. He was interested in higher dimensions, particularly the fourth dimension, and is known for coining the word tesseract and for his work on methods of visualising the geometry of higher dimensions....
 in his book A New Era of Thought
A New Era of Thought

A New Era of Thought is a non-fiction work written by Charles Howard Hinton, was published in 1888 and reprinted in 1900 by Swan Sonnenschein & Co....
, from the Greek
Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
 “” (“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 a hypercube can be of any dimension).

Geometry


The tesseract can be constructed in a number of different ways. As a regular polytope
Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
 with three cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
s folded together around every edge, it has Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 . Constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol x. As a duoprism
Duoprism

In geometry, a duoprism is a polytope resulting from the Cartesian product of two polytopes of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an -polytope, where n and m are 2 or higher....
, a Cartesian product
Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
 of two squares
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
, it can be named by a composite Schläfli symbol x.

Since each vertex of a tesseract is adjacent to four edges, the vertex figure
Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
 of the tesseract is a regular tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol .

The standard tesseract in Euclidean 4-space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 is given as the convex hull
Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
 of the points (±1, ±1, ±1, ±1). That is, it consists of the points:

A tesseract is bounded by eight hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
s (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 AB.
  • 2-dimensional: Two parallel lines 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.


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. A multitude of cubes that are nicely interconnected. The vertices of the tesseract with respect to the distance along the edges, with respect to the bottom point. This view is of interest when using tesseracts as the basis for a network topology
Network topology

Network topology is the study of the arrangement or mapping of the elements of a Computer networking, especially the physical and logical interconnections between nodes....
 to link multiple processors in parallel computing
Parallel computing

Parallel computing is a form of computing in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved Concurrency ....
: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.

Tesseracts are also bipartite graph
Bipartite graph

In the mathematics field of graph theory, a bipartite graph is a graph whose vertex can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets....
s, just as a path, square, cube and tree are.

Projections to 3 dimensions

The cell-first parallel projection
Graphical projection

Graphical projection is a protocol by which an image of an imaginary Three-dimensional space object is projected onto a planar surface without the aid of mathematical calculation....
 of the tesseract into 3-dimensional space has a cubical
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
 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 cuboid
Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
al 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
Hexagonal prism

In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce...
. 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
Rhombic dodecahedron

The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
 envelope. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
s, 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.

Unfolding the tesseract


The tesseract can be unfolded into eight cubes, just as the cube can be unfolded into six squares (view animation). An unfolding of a polytope is called a net
Net (polyhedron)

In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron....
. 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
Tree (graph theory)

In mathematics, more specifically graph theory, a tree is a graph in which any two Vertex are connected by exactly one path . Alternatively, any connectedness graph with no Cycle is a tree....
 together with a perfect matching
Matching

In the mathematical discipline of graph theory a matching or edge-independent set in a graph is a set of edges without common vertex . It may also be an entire graph consisting of edges without common vertices....
 in its complement
Complement graph

In graph theory the complement or inverse of a graph is a graph on the same vertices such that two vertices of are adjacent if and only if they are not adjacent in ....
).

Image gallery


Perspective projections
Stereographic Polytope 8cell

Stereographic projection
Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point — the projection point....

(Edges are projected onto the 3-sphere
3-sphere

In mathematics, a '3-sphere' is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space....
)
Schlegel diagram
Schlegel diagram

In geometry, a Schlegel diagram is a projection of a polytope from into through a point beyond one of its facets. The resulting entity is a polytopal subdivision of the facet in that is combinatorially equivalent to the original polytope....

Cell-centered
Schlegel diagram
Schlegel diagram

In geometry, a Schlegel diagram is a projection of a polytope from into through a point beyond one of its facets. The resulting entity is a polytopal subdivision of the facet in that is combinatorially equivalent to the original polytope....

Vertex-centered
A 3D projection of an 8-cell performing a double rotation
SO(4)

In mathematics, SO is the four-dimensional rotation group; that is, the group of rotations about a fixed point in four-dimensional Euclidean space....
 about two orthogonal planes.
Orthogonal projections


Projection inside
Petrie polygon
Petrie polygon

In geometry, a Petrie polygon is a skew polygon such that every two consecutive Edge belong to a Face of a regular polyhedron.This definition extends to higher regular polytopes....

A net
Net (polyhedron)

In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron....
 of a tesseract.
3d Stereographic Projection Tesseract

A stereoscopic
Stereogram

A stereogram is an optical illusion of depth created from flat, two-dimensional image or images. Originally, stereogram referred to a pair of stereo images which could be viewed using stereoscope....
 3D projection of a tesseract.		 Perspective with hidden volume elimination. See Fourth dimension
Fourth dimension

In physics and mathematics, a vector of n real number can be understood as a Coordinate system in an n-dimensional Euclidean space. When n = 4, the set of all such locations is called 4-dimensional Euclidean space....
. The red corner is the nearest in 4D and has 4 cubical cells meeting round it.


See also

  • Fourth dimension
    Fourth dimension

    In physics and mathematics, a vector of n real number can be understood as a Coordinate system in an n-dimensional Euclidean space. When n = 4, the set of all such locations is called 4-dimensional Euclidean space....
  • Hypercube
    Hypercube

    In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
     - The dimensional family of polytopes:
    • Square
      Square (geometry)

      In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
       -
    • Cube
      Cube

      A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
       -
    • Tesseract -
    • Penteract
      Penteract

      In Fifth dimension geometry, a penteract is a name for a Fifth dimension hypercube with 32 Vertex , 80 Edge s, 80 square Face , 40 cubic Cell , and 10 tesseract hypercells....
       -
    • Hexeract
      Hexeract

      A hexeract is a name for a six-dimensional hypercube with 64 Vertex , 192 Edge s, 240 square Face , 160 cubic Cell , 60 tesseract hypercell, and 12 penteract 5-faces....
       -
    • Hepteract
      Hepteract

      A hepteract is a seven-dimensional hypercube with 128 Vertex , 448 Edge s, 672 square Face , 560 cubic Cell , 280 tesseract hypercell, 84 penteract 5-faces, and 14 hexeract 6-faces....
       -
    • Octeract
      Octeract

      An octeract is an eight-dimensional hypercube with 256 Vertex , 1024 Edge s, 1792 square Face , 1792 cubic Cell , 1120 tesseract hypercell, 448 penteract 5-faces, 112 hexeract 6-faces, and 16 hepteract 7-faces....
       -
    • Enneract
      Enneract

      An enneract is a nine-dimensional hypercube with 512 Vertex , 2304 Edge s, 4608 square Face , 5376 cubic Cell , 4032 tesseract hypercell, 2016 penteract 5-faces, 672 hexeract 6-faces, 144 hepteract 7-faces, and 18 octeract 8-faces....
       -
    • 10-cube or dekeract
      10-cube

      A 10-cube or dekeract is a ten-dimensional hypercube with 1024 Vertex , 5120 Edge s, 11520 square Face , 15360 cubic Cell , 13440 tesseract hypercell, 8064 penteract 5-faces, 3360 hexeract 6-faces, 960 hepteract 7-faces, 180 octeract 8-faces, and 20 enneract 9-faces....
       -
  • Uniform polychoron#Tesseract family
    Uniform polychoron

    In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
     - uniform polychora based on tesseract
  • Other regular polytopes
    • Simplex
      Simplex

      In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
       - simplest polytope family
    • Cross-polytope
      Cross-polytope

      In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
       - The dual family to hypercubes
    • Demihypercube - a family of alternated hypercubes
    • List of regular polytopes
      List of regular polytopes

      This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
  • 3-sphere
    3-sphere

    In mathematics, a '3-sphere' is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space....
  • Metatron's Cube
  • Logical connective
    Logical connective

    In logic, two sentences may be joined by means of a logical connective to form a compound sentence. The truth-value of the compound is uniquely determined by the truth-values of the simpler sentences....
    s
  • Ludwig Schläfli
    Ludwig Schläfli

    Ludwig Schl?fli was a Switzerland geometry and complex analysis who was one of the key figures in developing the notion of higher dimensional spaces....
     - Polytopes
    Ludwig Schläfli

    Ludwig Schl?fli was a Switzerland geometry and complex analysis who was one of the key figures in developing the notion of higher dimensional spaces....
  • Hasse diagram
    Hasse diagram

    In the mathematics discipline known as order theory, a Hasse diagram is a simple picture of a finite partially ordered set, forming a Graph drawing of the transitive reduction of the partial order....


External links

  • Marco Möller's Regular polytopes in R4 (German)
  • 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.
  • A Windows
    Microsoft Windows

    Microsoft Windows is a series of software operating systems and graphical user interfaces produced by Microsoft. Microsoft first introduced an operating environment named Windows in November 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces ....
     program that displays animated hypercubes, by Rudy Rucker
    Rudy Rucker

    Rudolf von Bitter Rucker is an American mathematician, computer scientist and science fiction author, and is one of the founders of the cyberpunk literary movement....
  • A way to visualize hypercubes, by Ken Perlin
    Ken Perlin

    Ken Perlin is a professor in the Department of Computer Science at New York University. His research interests include graphics, animation, multimedia, and science education....
  • includes very good animated tutorials on several different aspects of the tesseract, by
  • by Bill Price