| Tesseract 8-cell (4-cube) | |
|---|---|
 | |
| Type | Convex regular 4-polytope |
| Schläfli symbol | {4,3,3} t0,3{4,3,2} or {4,3}×{ } t0,2{4,2,4} or {4}×{4} t0,2,3{4,2,2} or {4}×{ }×{ } t0,1,2,3{2,2,2} or { }×{ }×{ }×{ } |
| Coxeter diagram |      |
| Cells | 8 {4,3}  |
| Faces | 24 {4} |
| Edges | 32 |
| Vertices | 16 |
| Vertex figure |  Tetrahedron |
| Petrie polygon | octagon |
| Coxeter group | B4, [3,3,4] |
| Dual | 16-cell |
| Properties | convex, isogonal, isotoxal, isohedral, Hanner polytope |
| Uniform index | 10 |
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube.[1] Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes.
The tesseract is also called an 8-cell, C8, (regular) octachoron, or cubic prism. It is the four-dimensional measure polytope, taken as a unit for hypervolume.[2] Coxeter labels it the γ4 polytope.[3] The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope.
The Oxford English Dictionary traces the word tesseract to Charles Howard Hinton's 1888 book A New Era of Thought. The term derives from the Greek téssara (τέσσαρα 'four') and aktís (ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as tessaract.[4]
- ^ "The Tesseract - a 4-dimensional cube". www.cut-the-knot.org. Retrieved 2020-11-09.
- ^ Elte, E. L. (1912). The Semiregular Polytopes of the Hyperspaces. Groningen: University of Groningen. ISBN 1-4181-7968-X.
- ^ Coxeter 1973, pp. 122–123, §7.2. illustration Fig 7.2C.
- ^ "tesseract". Oxford English Dictionary (Online ed.). Oxford University Press. 199669. (Subscription or participating institution membership required.)