4-polytope

Criteria

Like all polytopes, 4-polytopes may be classified based on properties like "convexity" and "symmetry".

• A 4-polytope 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 4-polytope is contained in the 4-polytope or its interior; otherwise, it is non-convex. Self-intersecting 4-polytopes are also known as star 4-polytopes, from analogy with the star-like shapes of the non-convex star polygons and Kepler–Poinsot polyhedra.
• A 4-polytope is regular if it is transitive on its flags. This means that its cells are all congruent regular polyhedra, and similarly its vertex figures are congruent and of another kind of regular polyhedron.
• A convex 4-polytope is semi-regular if it has a symmetry group under which all vertices are equivalent (vertex-transitive) and its cells are regular polyhedra. The cells may be of two or more kinds, provided that they have the same kind of face. There are only 3 cases identified by Thorold Gosset in 1900: the rectified 5-cell, rectified 600-cell, and snub 24-cell.
• A 4-polytope 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 4-polytope must be regular.
• A 4-polytope is scaliform if it is vertex-transitive, and has all equal length edges. This allows cells which are not uniform, such as the regular-faced convex Johnson solids.
• A regular 4-polytope which is also convex is said to be a convex regular 4-polytope.
• A 4-polytope is prismatic if it is the Cartesian product of two or more lower-dimensional polytopes. A prismatic 4-polytope 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 tiling or honeycomb of 3-space is the division of three-dimensional Euclidean space into a repetitive grid of polyhedral cells. Such tilings or tessellations are infinite and do not bound a "4D" volume, and are examples of infinite 4-polytopes. A uniform tiling of 3-space is one whose vertices are congruent and related by a space group and whose cells are uniform polyhedra.

Classes

The following lists the various categories of 4-polytopes classified according to the criteria above:

The truncated 120-cell is one of 47 convex non-prismatic uniform 4-polytopes

Uniform 4-polytope (vertex-transitive):

• Convex uniform 4-polytopes (64, plus two infinite families)
  • 47 non-prismatic convex uniform 4-polytope including:
    • 6 Convex regular 4-polytope
  • Prismatic uniform 4-polytopes:
    • {} × {p,q} : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercube)
    • Prisms built on antiprisms (infinite family)
    • {p} × {q} : duoprisms (infinite family)
• Non-convex uniform 4-polytopes (10 + unknown)
  

  The great grand stellated 120-cell is the largest of 10 regular star 4-polytopes, having 600 vertices.
  • 10 (regular) Schläfli-Hess polytopes
  • 57 hyperprisms built on nonconvex uniform polyhedra
  • Unknown total number of nonconvex uniform 4-polytopes: Norman Johnson and other collaborators have identified 1849 known cases (convex and star), all constructed by vertex figures by Stella4D software.^[5]

Other convex 4-polytopes:

• Polyhedral pyramid
• Polyhedral prism

The regular cubic honeycomb is the only infinite regular 4-polytope in Euclidean 3-dimensional space.

Infinite uniform 4-polytopes of Euclidean 3-space (uniform tessellations of convex uniform cells)

• 28 convex uniform honeycombs: uniform convex polyhedral tessellations, including:
  • 1 regular tessellation, cubic honeycomb: {4,3,4}

Infinite uniform 4-polytopes of hyperbolic 3-space (uniform tessellations of convex uniform cells)

• 76 Wythoffian convex uniform honeycombs in hyperbolic space, including:
  • 4 regular tessellation of compact hyperbolic 3-space: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}

Dual uniform 4-polytope (cell-transitive):

• 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:
  • Rhombic dodecahedral honeycomb
  • Disphenoid tetrahedral honeycomb

Others:

• Weaire–Phelan structure periodic space-filling honeycomb with irregular cells

The 11-cell is an abstract regular 4-polytope, existing in the real projective plane, it can be seen by presenting its 11 hemi-icosahedral vertices and cells by index and color.

Abstract regular 4-polytopes:

• 11-cell
• 57-cell

These categories include only the 4-polytopes that exhibit a high degree of symmetry. Many other 4-polytopes are possible, but they have not been studied as extensively as the ones included in these categories.

Notes

Bibliography

• H.S.M. Coxeter:
  • H. S. M. Coxeter, M. S. Longuet-Higgins and 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, edited 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]
  • (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
• Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation