Tesseract

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 < x_i < 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.

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