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In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.[1] Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

The tesseract is also called an 8-cell, C8, (regular) octachoron, octahedroid,[2] cubic prism, and tetracube.[3] It is the four-dimensional hypercube, or 4-cube as a member of the dimensional family of hypercubes or measure polytopes.[4] Coxeter labels it the {\displaystyle \gamma _{4}} polytope.[5] The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope.

According to the Oxford English Dictionary, the word tesseract was first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek téssara (τέσσαρα 'four') and aktís (ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices. In this publication, as well as some of Hinton's later work, the word was occasionally spelled tessaract.[6]

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