Tesseract

Geometry

In a square, each vertex has two perpendicular edges incident to it, while a cube has three. A tesseract has four. Canonical coordinates for the vertices of a tesseract centered at the origin are (±1, ±1, ±1, ±1), while the interior of the same consists of all points (x0, x1, x2, x3) with -1 

Related Topics:
Vertex - Perpendicular

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The first illustration shows how 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. The second picture accounts for the fact that each edge of a tesseract is of the same length. This picture also enables the human brain to find a multitude of cubes that are nicely interconnected. The third diagram finally orders 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 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.

Related Topics:
Network topology - Parallel computing

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A tesseract is bound by eight hyperplanes. 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. The vertex figure of the tesseract is a regular tetrahedron. Thus the tesseract is given Schläfli symbol {4,3,3}. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4}.

Related Topics:
Hyperplane - Vertex figure - Schläfli symbol - Dual polytope - Hexadecachoron

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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.

Related Topics:
Projection - Cubical - Envelope

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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 parallelopipeds, giving a total of 8 possible parallelopipeds. The images of the tesseract's cells under this projection are precisely these 8 parallelopipeds. This projection is also the one with maximal volume.

Related Topics:
Rhombic dodecahedral - Parallelopipeds

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The square, cube, and tesseract are all examples of measure polytopes in their respective dimensions.

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Tesseracts are also bipartite graphs, just as a path, rectangle, cube and tree are.

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The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4x4 square.

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