198.144.199.xxx (talk) Little tigher explanation |
Josh Grosse (talk | contribs) Hopefully cleaned it up a bit, added some notes |
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A '''tesseract''', or hypercube, is the four dimensional object. In a square, each vertex has two perpendicular edges incident to it, while a cube has three. A hypercube has four. So, 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 (x<sub>0</sub>, x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) with -1 < x<sub>i</sub> < 1. |
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A tesseract is bound by eight hyperplanes, each of which intersects it to form a cube. Two cubes and so three squares intersect at each edge. There are three cubes meeting at every vertex, the vertex polyhedron of which is a regular tetrahedron. Thus the tesseract is given Schläfi notation {4,3,3}. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices. The square, cube, and tesseracts are all examples of ''measure polytopes'' in their respective dimensions. |
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In a cube, every edge is shared by 2 squares. In a tesseract, 3 squares meet at every edge. A tesseract has 16 vertices, 32 edges, 24 squares, and 8 cubes. |
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A tesseract is defined by the set of points: |
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<b>{(''x'', ''y'', ''z'', ''h''): 0 <u><</u> ''x'' <u><</u> 1, 0 <u><</u> ''y'' <u><</u> 1, 0 <u><</u> ''z'' <u><</u> 1, 0 <u><</u> ''h'' <u><</u> 1}</b> |
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Revision as of 05:16, 18 November 2001
A tesseract, or hypercube, is the four dimensional object. In a square, each vertex has two perpendicular edges incident to it, while a cube has three. A hypercube has four. So, 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 < xi < 1.
A tesseract is bound by eight hyperplanes, each of which intersects it to form a cube. Two cubes and so three squares intersect at each edge. There are three cubes meeting at every vertex, the vertex polyhedron of which is a regular tetrahedron. Thus the tesseract is given Schläfi notation {4,3,3}. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices. The square, cube, and tesseracts are all examples of measure polytopes in their respective dimensions.
See also [1] for an illustration.