Regular polychoron

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A regular polychoron is a polychoron whose flags are identical under its symmetry group. In 3D Euclidean space, there are 16 finite regular polychora and 1 honeycomb that are not skew, and an unknown number of additional regular polychora that are skew.

Regular convex polychora[edit | edit source]

The regular convex polychora are the 4-dimensional counterparts to the Platonic solids and the regular convex polygons. There are 6 regular convex polychora. Apart from the infinite set of regular convex polygons, there are more regular convex polytopes of rank 4 than of any other rank.

The regular convex polychora
Solid Number of facets Schläfli symbol Image
5-cell 5 {3,3,3}
8-cell 8 {4,3,3}
16-cell 16 {3,3,4}
24-cell 24 {3,4,3}
120-cell 120 {3,3,5}
600-cell 600 {5,3,3}

Proof of completeness[edit | edit source]

Recursive geometric proof[edit | edit source]

The following proof shows that the above 6 polychora are the only convex regular polychora, using the completeness of the Platonic solids. For a proof that the Platonic solids are the only regular convex polyhedra, see Regular polyhedron § Geometric proof.

Proof —

The facet of a regular convex polychoron must be itself a regular convex polyhedron. More subtly, the vertex figure of a regular convex polychoron must also be a regular convex polyhedron. We can see this because the dual of a regular convex polychoron must be itself a regular convex polychoron, and so its vertex figure is dual to the facet of another regular convex polychoron.

Since there are only 5 regular convex polyhedra:

there are only a finite number of ways to combine them with one as the facet and one as the vertex figure. Two polytopes can be combined if the vertex figure of the facet matches the face of the vertex figure. So we can make a polychoron with {5,3} as the facet and {3,4} as the verf because the 3s match, but {3,5} does not match with itself.

There are in total 11 combinations to be checked:

Five of them are not in the above list of 6. And while they are arguably convex they are all apeirochora of differing types and thus not valid regular convex polychora:

Schläfli-Hess polychora[edit | edit source]

There are also 10 finite self-intersecting regular polychora, known as the Schläfli-Hess polychora:

Proof of completeness[edit | edit source]

Recursive proof[edit | edit source]

The proof of the completeness of the regular convex polychora can be extended into a wider recursive proof that the 10 Schläfli-Hess polychora above, along with the 6 regular convex polychora, are the only finite planar polychora.

Proof —

As before the facet and vertex figure of a regular polychoron is a regular polyhedron. The possible components are then:

The valid combinations, excluding those enumerated in the previous proof, are then:

Four of these are not among the Schläfli-Hess polychora. We can apply Van Oss's criterion to show that all of these must be infinite dense polychora[1] and are thus discarded:

  • {3,5/2,3}
  • {4,3,5/2}
  • {5/2,3,4}
  • {5/2,3,5/2}

Regular compounds[edit | edit source]

Regular tesselations[edit | edit source]

Euclidean[edit | edit source]

There is also a single regular honeycomb of 3-dimensional Euclidean space:

Hyperbolic[edit | edit source]

There are infinitely many regular honeycombs in 3-dimensional space. These can be classified as compact, paracompact or hypercompact depending on their symmetry group. There are 4 compact and 11 paracompact regular hyperbolic honeycombs, all the remaining honeycombs are hypercompact.

Compact regular honeycombs in 3 
Image
Schläfli symbol {5,3,4} {4,3,5} {3,5,3} {5,3,5}
Paracompact regular honeycombs in 3 
Image
Schläfli symbol {6,3,3} {4,4,3} {6,3,4} {6,3,5}
Image
Schläfli symbol {4,3,6} {5,3,6} {3,4,4} {4,4,4}
Image
Schläfli symbol {3,3,6} {3,6,3} {6,3,6}

Regular skew polychora[edit | edit source]

Finite regular skew polychora[edit | edit source]

There are 18 finite regular skew polychora. The Petrial tesseract can be obtained as the Petrie dual of the tesseract, and the other 17 regular skew polychora can be obtained by applying κ  to the Petrial tesseract or any of the 16 planar polychora above.

Regular skew honeycombs[edit | edit source]

A section of the apeir tetrahedron with a facet highlighted in yellow.

In addition to the single regular planar honeycomb there are 7 regular skew honeycombs in 3-dimensional Euclidean space:

  • - Mucubic honeycomb
  • - Apeir tetrahedron
  • - Petrial apeir tetrahedron
  • - Apeir octahedron
  • - Petrial apeir octahedron
  • - Apeir cube
  • - Petrial apeir cube

The mucubic honeycomb is the Petrie dual of the cubic honeycomb.

Skew apeirochora in 4-dimensions[edit | edit source]

The regular polyhedra have several regular apeirohedra spanning 3 dimensions. e.g. the mucube and the blended triangular tiling. One could speculate potentially exists analogous regular apeirochora spanning 4-dimensions, and indeed these do exist.

Blends[edit | edit source]

Regular skew apeirochora in 4 dimensions can be made by blending regular honeycombs with a 2-dimensional polytope. This category gives 16 blends made from the eight regular honeycombs with either the dyad or the apeirogon.

Blended apeirochora
Honeycomb Blended by ...
Digon Apeirogon
Cubic honeycomb Blended cubic honeycomb Helical cubic honeycomb
Apeir tetrahedron Blended apeir tetrahedron Helical apeir tetrahedron
Apeir octahedron Blended apeir octahedron Helical apeir octahedron
Apeir cube Blended apeir cube Helical apeir cube
Mucubic honeycomb Blended mucubic honeycomb Helical mucubic honeycomb
Petrial apeir tetrahedron Blended Petrial apeir tetrahedron Helical Petrial apeir tetrahedron
Petrial apeir octahedron Blended Petrial apeir octahedron Helical Petrial apeir octahedron
Petrial apeir cube Blended Petrial apeir cube Helical Petrial apeir cube

These are analogous to the 12 blended apeirohedra in 3 dimensions.

Pure apeirochora[edit | edit source]

External links[edit | edit source]

References[edit | edit source]

  1. Coxeter (1973:275)

Bibliography[edit | edit source]

  • Coxeter, Donald (1973). Regular polytopes (3 ed.). Dover. ISBN 0-486-61480-8. OCLC 798003.
  • McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF). Discrete Computational Geometry.