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# Tesseract

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.

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.

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