Polychoron

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.)

Definition

Polychora are closed four-dimensional figures. We can describe them further only through analogy with such three dimensional polyhedron counterparts as spheres, pyramids, cylinders and cubes.

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.

Classification

Polychora may be classified based on properties like "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 polyhedra.

*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 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.

Categories

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

Uniform polychora
* Convex uniform polychora (64, plus two infinite families)
** 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.

* 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)
** 33 uniform convex polyhedral tessellations including:
*** 4 regular tessellation of hyperbolic space
** Unknown others

* Abstract regular polychora
** There are two special abstract regular polychora: 11-cell, and 57-cell.

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.

ee 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.
*The Klein bottle is the generalization of the Möbius strip to 4D. It is also not a polychoron because it is 2-manifold (a 2-dimensional surface, not bounded by 3-dimensional volumes).

References

* 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:
** H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: "Uniform Polyhedra", Philosophical Transactions of the Royal Society of London, Londne, 1954
** H.S.M. Coxeter, "Regular Polytopes", 3rd Edition, Dover New York, 1973
* 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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 22) H.S.M. Coxeter, "Regular and Semi Regular Polytopes I", [Math. Zeit. 46 (1940) 380-407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, "Regular and Semi-Regular Polytopes II", [Math. Zeit. 188 (1985) 559-591]
** (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
* M. Möller: "Definitions and computations to the Platonic and Archimedean polyhedrons", thesis (diploma), University of Hamburg, 2001

External links

*Mathworld | urlname=Polychoron | title=Polychoron
* [http://members.aol.com/Polycell/uniform.html Four dimensional figures page]
*GlossaryForHyperspace | anchor=Polychoron | title=Polychoron
* [http://www.polytope.net/hedrondude/polychora.html Jonathan Bowers: Uniform polychora]
* [http://www.pentatope.de/applets/polyc.html Uniform polychoron Viewer] Dead link|date=April 2008 - applet with sources (requires Java and Java3d)