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 square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes. The tesseract is also called an 8-cell, C8, (regular) octachoron, or cubic prism. It is the four-dimensional measure polytope, taken as a unit for hypervolume. Harold Scott MacDonald Coxeter labels it the γ4 polytope. The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope.
Construction The construction of a tesseract can be visualized through the analogy of dimensions in the following steps:
One can take out two points with a certain length that form a line segment. If another identical line segment is its length in a perpendicular direction from itself, it sweeps out and forms a square (2-cube). The results have four points and four line segments, which are called vertices and edges, respectively. Moving the square with the same length in the direction perpendicular to the plane it lies on generates a cube (3-cube). The results have eight vertices, twelve edges, and six squares. The squares are called the faces. Moving the cube with the same length again into the fourth-dimensional space generates a tesseract (4-cube). A tesseract is bounded by eight cubes (its cells). Each cube shares each of its faces with another cube. Three cubes and three squares meet at each edge. Four cubes, six squares, and four edges meet at every vertex. Collectively, the tesseract consists of eight cubes, twenty-four squares, thirty-two edges, and sixteen vertices. The tesseract, like both the square and the cube, is a member of the hypercube's family.
An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract, each of which can tile 3-space. 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). One of these unfoldings is the Dali cross, named after Spanish surrealist artist Salvador Dalí, whose 1954 painting Corpus Hypercubus depicted it. It consists of eight cubes, four cubes stacked vertically and four more attached to the second-from-top of the first four.
Word origin
The Oxford English Dictionary traces the word tesseract to Charles Howard Hinton's 1888 book A New Era of Thought. Hinton originally spelled the word as tessaract, changing it to tesseract in his 1904 book The Fourth Dimension. The term derives from the Ancient Greek téssara (τέσσαρα "four") and aktís (ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices. The word "tesseract" has been adopted for numerous other uses in popular culture, including as a plot device in works of science fiction, often with little or no connection to the four-dimensional hypercube.
Properties The eight cells of a tesseract may be regarded in three different ways as two interlocked rings of four cubes. 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 4-4 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 the 16-cell with Schläfli symbol {3,3,4}. The tesseract, with 16 vertices, is the convex hull of a compound of two 16-cells, with 8 vertices each, in an exact dimensional analogy to the cube, with 8 vertices, which is the convex hull of a compound of two regular tetrahedra, with 4 vertices each. Each edge of a regular tesseract is of the same length. This 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. The tesseract can be decomposed into smaller 4-polytopes. It is the convex hull of the compound of two demitesseracts (16-cells). It can also be triangulated into 4-dimensional simplices (irregular 5-cells) that share their vertices with the tesseract. It is known that there are 92487256 such triangulations and that the fewest 4-dimensional simplices in any of them is 16. The dissection of the tesseract into instances of its characteristic simplex (a particular orthoscheme with Coxeter diagram ) is the most basic direct construction of the tesseract possible. The characteristic 5-cell of the 4-cube is a fundamental region of the tesseract's defining symmetry group, the group which generates the B4 polytopes. The tesseract's characteristic simplex directly generates the tesseract through the actions of the group, by reflecting itself in its own bounding facets (its mirror walls).
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