In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells.
The term polychoron (plural polychora), from the Greek roots poly ("many") and choros ("room" or "space") and has been advocated by Norman Johnson and George Olshevsky, but it is little known in general polytope theory. Other names for polychoron include: polyhedroid and polycell.
Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space.
Polychora are closed four-dimensional figures. The most familiar example of a polychoron is the tesseract or hypercube, the 4D analogue of the cube. A tesseract has vertices, edges, faces, and cells. A vertex is a point where four or more edges meet. An edge is a line segment where three or more faces meet, and a face is a polygon where two cells meet. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Furthermore, the following requirements must be met:
- Each face must join exactly two cells.
- Adjacent cells are not in the same three-dimensional hyperplane.
- The figure is not a compound of other figures which meet the requirements.
Euler characteristic 
For example, the convex regular 4-polytopes:
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- A polychoron is convex if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the polychoron is contained in the polychoron or its interior; otherwise, it is non-convex. Self-intersecting polychora are also known as star polychora, from analogy with the star-like shapes of the non-convex Kepler–Poinsot polyhedra.
- A polychoron is uniform if it has a symmetry group under which all vertices are equivalent, and its cells are uniform polyhedra. The faces of a uniform polychoron must be regular.
- A uniform polychoron is semi-regular if its cells are regular polyhedra. The cells may be of two or more kinds, provided that they have the same kind of face.
- A semi-regular polychoron is said to be regular if its cells are all of the same kind of regular polyhedron; see regular polyhedron for examples. There are only 3 cases identified by Thorold Gosset in 1900: the rectified 5-cell, rectified 600-cell, and snub 24-cell.
- A regular polychoron which is also a convex polychoron is said to be a convex regular polychoron.
- A polychoron is prismatic if it is the Cartesian product of two lower-dimensional polytopes. A prismatic polychoron is uniform if its factors are uniform. The hypercube is prismatic (product of two squares, or of a cube and line segment), but is considered separately because it has symmetries other than those inherited from its factors.
- A 3-space tessellation is the division of three-dimensional Euclidean space into a regular grid of polyhedral cells. Strictly speaking, tessellations are not polychora as they do not bound a "4D" volume, but we include them here for the sake of completeness because they are similar in many ways to polychora. A uniform 3-space tessellation is one whose vertices are related by a space group and whose cells are uniform polyhedra.
The following lists the various categories of polychora classified according to the criteria above:
- Convex uniform polychora (64, plus two infinite families)
- Non-convex uniform polychora (10 + unknown)
- Infinite uniform polychora of Euclidean 3-space (uniform tessellations of convex uniform cells)
- Infinite uniform polychora of hyperbolic 3-space (uniform tessellations of convex uniform cells)
- 41 unique dual convex uniform 4-polytopes
- 17 unique dual convex uniform polyhedral prisms
- infinite family of dual convex uniform duoprisms (irregular tetrahedral cells)
- 27 unique convex dual uniform honeycombs, including:
- Weaire-Phelan structure periodic space-filling honeycomb with irregular cells
These categories include only the polychora that exhibit a high degree of symmetry. Many other polychora are possible, but they have not been studied as extensively as the ones included in these categories.
See also 
- Convex regular 4-polytope
- The 3-sphere (or glome) is another commonly discussed figure that resides in 4-dimensional space. This is not a polychoron, since it is not bounded by polyhedral cells.
- The duocylinder is a figure in 4-dimensional space related to the duoprisms. It is also not a polychoron because its bounding volumes are not polyhedral.
- Vialar, T. (2009). Complex and Chaotic Nonlinear Dynamics: Advances in Economics and Finance. Springer. p. 674. ISBN 978-3-540-85977-2.
- Capecchi, V.; Contucci, P., Buscema, M., D'Amore, B. (2010). Applications of Mathematics in Models, Artificial Neural Networks and Arts. Springer. p. 598. doi:10.1007/978-90-481-8581-8. ISBN 978-90-481-8580-1.
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
- A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
- H.S.M. Coxeter:
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
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- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation 
- Weisstein, Eric W., "Polychoron", MathWorld.
- Weisstein, Eric W., "Polyhedral formula", MathWorld.
- Weisstein, Eric W., "Regular polychoron Euler characteristics", MathWorld.
- Four dimensional figures page, George Olshevsky.
- Olshevsky, George, Polychoron at Glossary for Hyperspace.
- Uniform Polychora, Jonathan Bowers
- Uniform polychoron Viewer - Java3D Applet with sources
- Dr. R. Klitzing, polychora
|Fundamental convex regular and uniform polytopes in dimensions 2–10|
|Family||An||BCn||Dn||E6 / E7 / E8 / F4 / G2||Hn|
|Uniform polyhedron||Tetrahedron||Octahedron • Cube||Demicube||Dodecahedron • Icosahedron|
|Uniform polychoron||5-cell||16-cell • Tesseract||Demitesseract||24-cell||120-cell • 600-cell|
|Uniform 5-polytope||5-simplex||5-orthoplex • 5-cube||5-demicube|
|Uniform 6-polytope||6-simplex||6-orthoplex • 6-cube||6-demicube||122 • 221|
|Uniform 7-polytope||7-simplex||7-orthoplex • 7-cube||7-demicube||132 • 231 • 321|
|Uniform 8-polytope||8-simplex||8-orthoplex • 8-cube||8-demicube||142 • 241 • 421|
|Uniform 9-polytope||9-simplex||9-orthoplex • 9-cube||9-demicube|
|Uniform 10-polytope||10-simplex||10-orthoplex • 10-cube||10-demicube|
|Uniform n-polytope||n-simplex||n-orthoplex • n-cube||n-demicube||1k2 • 2k1 • k21||n-pentagonal polytope|
|Topics: Polytope families • Regular polytope • List of regular polytopes|