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Concept
Tesseract
Summary
In geometry, a tesseract is the four-dimensional analogue 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 six square faces, the hypersurface of the tesseract consists of eight 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. It is the four-dimensional hypercube, or 4-cube as a member of the dimensional family of hypercubes or measure polytopes. Coxeter labels it the polytope. The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope.
The Oxford English Dictionary traces the word tesseract to Charles Howard Hinton's 1888 book A New Era of Thought. The term derives from the Greek téssara (τέσσαρα 'four') and from aktís (ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as tessaract.
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}, with which it can be combined to form the compound of tesseract and 16-cell.
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.
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