Polychoron

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In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora), from the Greek root poly, meaning "many", and choros meaning "room" or "space". It is also called a 4-polytope or polyhedroid. The two-dimensional analogue of a polychoron is a polygon, and the three-dimensional analogue is a polyhedron.

(Note that the term polychoron is a recent invention and has limited usage at present. It has been advocated by Norman Johnson and George Olshevsky—see the Uniform Polychora Project—but it is little known in general polytope theory.)

1 Definition
2 Classification
3 Categories
4 See also
5 External links

[edit] Definition

A polychoron is a closed four-dimensional figure with 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.

[edit] Classification

Polychora may be classified based on properties such as convexity and symmetry.

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 solids.

A polychoron is uniform if it has a symmetry group under which all vertices are equivalent, and its cells are uniform polyhedra. The edges of a uniform polychoron must be equal in length.

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.

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 3-dimensional Euclidean space into a regular grid of polyhedral cells. Strictly speaking, tessellations are not polychora (because 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.

[edit] Categories

The following lists the various categories of polychora classified according to the criteria above:

Uniform polychora

Convex uniform polychora (64 plus set of duoprims)
- 47 non-prismatic convex uniform polychora including:
  - 6 convex regular polychora
- Prismatic uniform polychora:
  - {} x {p,q} : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercube)
  - hyperprisms built on antiprisms (infinite family)
  - {p} x {q} : Duoprisms (infinite family)
Non-convex uniform polychora (10 + unknown)
- 10 Schläfli-Hess polychora
- 57 hyperprisms built on nonconvex uniform polyhedra
- Unknown total number of nonconvex uniform polychora: The Uniform Polychora Project now counts 1849 known cases. [1]

Infinite uniform polychora of Euclidean 3-space (uniform tessellations of convex uniform cells)
- 28 convex uniform honeycombs: uniform convex polyhedral tessellations, including:
  - 1 regular tessellation: (cubic honeycomb)

Infinite uniform polychora of hyperbolic 3-space (uniform tessellations of convex uniform cells)
- at least 33 uniform convex polyhedral tessellations including:
  - 4 regular tessellation of hyperbolic space
- Unknown others

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.

[edit] See also

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.
Polytope