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A tesseract, or four-dimensional hypercube, is a regular four-dimensional polytope, or polychoron, with eight cubical cells.

It can be thought of as the four-dimensional analogue of the cube: roughly speaking, the tesseract is to the cube as the cube is to the square.

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 (x0x1x2x3) 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.

Hypercubes in fiction

Robert Heinlein mentioned hypercubes in at least two of his science-fiction stories. And He Built a Crooked House (1940) described a house built as a net (i.e. an unfolding of the cells into three-dimensional space) of a tesseract. It collapsed, becoming a real hyperdimensional tesseract. Glory Road (1963) included the foldbox, a hyperdimensional packing case that was bigger inside than outside.

A hypercube is also used as the main deus ex machina of Robert J. Sawyer[?]'s book Factoring Humanity[?].

The tesseract is mentioned in the children's fantasy novel A Wrinkle In Time[?], by Madeleine L'Engle[?], as a way of introducing the concept of higher dimensions, but the treatment is extremely vague.

See also: 3-sphere hypersphere

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