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 use of the term polychoron is not entirely standard. Its use has been advocated by Norman Johnson and George Olshevsky. See the Uniform Polychora Project.)
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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.
Classification
Polychora may be classified based on properties such as convexity and symmetry.
- A polychoron is convex if its cells do not intersect each other, and non-convex otherwise. Non-convex 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 regular if its cells are all alike.
- 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.
Categories
The following lists the various categories of polychora classified according to the criteria above:
- 3 convex semiregular polychora
- 47 non-prismatic convex uniform polychora (includes the 6 convex regular polychora and the 3 convex semiregular polychora)
- Non-convex uniform polychora (last known count is somewhere around 1,800)
- Prismatic uniform polychora:
- Polyhedral prisms (infinite family)
- Duoprisms (infinite family)
- Tessellations of 3-space:
- 28 Andreini tessellations: uniform convex polyhedral tessellations (including the regular cubic tessellation)
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
- 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 made up of 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
External links
- Polychoron on Mathworld
- Four dimensional figures page
- Multidimensional glossary – compiled by George Olshevskyes:Polícoro