# Regular-faced Polyhedra

A regular-faced polyhedron is a polyhedron whose faces are regular polygons.

## Strictly convex regular-faced polyhedra

It is well-known that the strictly convex regular-faced polyhedra comprise two infinite families (the prisms and antiprisms), the 5 Platonic solids, the 13 Archimedean solids, and the 92 Johnson solids. (Thus there are 110 strictly convex regular-faced polyhedra in addition to the prisms and antiprisms; alternately one might count 108, since the cube is a prism and the octahedron is an antiprism.)

The complete list was given by Norman Johnson in 1966  and proven complete by Victor Zalgaller in 1969 . The proof proceeds by an exhaustive case-analysis over all ways to attach regular faces together which preserve convexity as well as the property of being noncomposite. A regular-faced polyhedron is noncomposite if there is no plane which divides the polyhedron into two regular-faced polyhedra. Zalgaller established that the noncomposite strictly convex regular-faced polyhedra were the prisms and antiprisms (excepting A4, the octahedron, which is composite) and a further 28 polyhedra. Zalgaller concluded by stating the theorem that all junctions of those polyhedra along entire faces which are strictly convex are in the list of 110, establishing the classification.

### The 28 non-composite strictly convex regular-faced polyhedra (besides prisms and antiprisms)

 M1 Tetrahedron M2 Square pyramid M3 Pentagonal pyramid M4 Triangular cupola M5 Square cupola M6 Pentagonal cupola M7 Tridiminished icosahedron M8 Bilunabirotunda M9 Pentagonal rotunda M10 Trucated tetrahedron M11 Truncated cube M12 Truncated dodecahedron M13 Tridiminished rhombicosidodecahedron M14 Parabidiminished rhombicosidodecahedron

 M15 Dodecahedron M16 Truncated octahedron M17 Great rhombicuboctahedron M18 Great rhombicosadodecahedron M19 Truncated icosahedron M20 Triangular hebesphenorotunda M21 Hebesphenomegacorona M22 Sphenocorona M23 Sphenomegacorona M24 Disphenocingulum M25 Snub disphenoid M26 Snub cube M27 Snub dodecahedron M28 Snub square antiprism

## Relaxing strict convexity

If the requirement of strict convexity is relaxed, allowing adjacent faces of the polyhedron to be co-planar, further convex regular-faced polyhedra arise. In effect we allow a wider variety of faces, not only regular polygons but also parquet faces which are composed of regular polygons. We call the "edges" between components of a single parquet face conditional edges; the "vertices" of the components, when the vertices are in the interior of a parquet face or its edges, are conditional vertices. Note that conditional edges are edges of dihedral angle π, and conditional vertices are vertices where the planar angles of the faces incident on a vertex sum to 2π. (In a strictly convex polyhedron, dihedral angles are strictly less than π and the planar angles of the faces incident on a vertex have a sum strictly less than 2π.)

### Convex regular-faced polyhedra with conditional edges

Allowing conditional edges, but not conditional vertices, there are five varieties of parquet faces possible: the rhombus "3+3" composed of two equilateral triangles; the square and triangle "3+4"; the pentagon and triangle "3+5"; the square and two opposite triangles "3+4+3"; and the pentagon and two opposite triangles "3+5+3".

These polyhedra admit a classification in the same manner as the strictly convex ones; there are 78. B. A. Ivanov  and Yu. A. Pryakhin  established by 1973 that there are 6 noncomposite convex regular-faced polyhedra with conditional edges. These are the polyhedra labelled Q1 through Q6 in the list. An initial list of 70 composite convex regular-faced polyhedra with conditional edges was completed in 2008 in parallel by A. M. Gurin and Victor Zalgaller  and by A. V. Timofeenko . However, both classifications originally omitted the polyhedra P4,30 and P4,31 from the list; they alone have the property that they are composite, but there is no plane which divides one of them into two components one of which is noncomposite; instead, each is divisible into two composite components. Timofeenko had realized this omission by 2010 .

Without any knowledge of that work done in Russia, Bonnie Stewart considered polyhedra with regular 3+3 rhombus faces in his Adventures among the Toroids by the 1970s. The 71 such polyhedra were listed by Alex Doskey, Roger Kaufman, and Steve Waterman in 2006, though they did not know if their list was complete.

### Convex regular-faced polyhedra with conditional vertices

Allowing conditional vertices means that we must consider many infinite families of polyhedra: consider as an example the family of rectangular solids composed of a rod of cubes extended to arbitrary lengths. However, Yu. A. Pryakhin  realized that this class of polyhedra still admits a finitary classification into "types" of polyhedra sharing a common network of faces where at corresponding faces the corresponding angles are equal. For example, all rectangular solids are of a single "type". In fact, Pryahkin stated this result for the larger class of convex polyhedra whose faces are equiangular, not just regular. Furthermore Pryahkin stated the result that there are exactly 23 types of convex parquet polygons, defined by the interior angles at the vertices, which are decomposable into equiangular polygons.

Completing the classification of convex regular-faced polyhedra with conditional vertices is ongoing work for A. V. Timofeenko and his students . Some examples of such polyhedra have been produced by Roger Kaufman and others.

## Non-convex regular-faced polyhedra

For completeness I mention the non-convex regular-faced polyhedra. Self-intersecting such polyhedra turn up in the study of uniform star polyhedra. Non-self-intersecting non-convex regular-faced polyhedra were studied by Bonnie Stewart in his masterful Adventures among the Toroids , in particular the Stewart toroids.

## References

   Norman W. Johnson, Convex polyhedra with regular faces, Canadian Journal of Mathematics 18 (1966), 169-200.

   V. A. Zalgaller, Convex polyhedra with regular faces, Zap. Nauchn. Semin. LOMI 2 (1967), 5-221; English translation: Consultants Bureau, New York, 1969.

   B. A. Ivanov, Polyhedra with faces composed of regular polygons, Ukr. Geom. Sb. 10 (1971), 20-34.

   Yu. A. Pryakhin, On convex polyhedra with regular faces, Ukr. Geom. Sb. 14 (1973), 83-88.

   A. M. Gurin and V. A. Zalgaller, On the history of the study of convex polyhedra with regular faces and faces composed of regular ones, Trudy S.-Peterburg Mat. Obshch. 14 (2008), 215-294; English translation: Proceedings of the St. Petersburg Mathematical Society, Volume XIV, American Mathematical Society Translations, Series 2, Volume 228 (2009), 169-229.

   A. V. Timofeenko, Junction of noncomposite polygons, Algebra i Analiz 21 (2009), no. 3, 165-209; English translation: St. Petersburg Math. J. 21 (2010), no. 3, 483-512. doi:10.1090/S1061-0022-10-01105-2.

   A. V. Timofeenko, Corrections to "Junction of noncomposite polyhedra", Algebra i Analiz 23 (2011), no. 4; English translation: St. Petersburg Math. J. 23 (2012), no. 4, 779-780. https://doi.org/10.1090/S1061-0022-2012-01217-3.

   Yu. A. Pryahkin , Convex polyhedra whose faces are equiangular or composed of such, Zap. Nauchn. Semin. LOMI 45 (1974), 111-112; English translation: Journal of Mathematical Sciences 10 (1978), no. 3, 486-487. doi:10.1007/BF01476855.

   A.V. Timofeenko, Convex polyhedra with parquet faces, Doklady Akademii Nauk 428 (2009), no. 4, 454-457; English translation: Doklady Mathematics 80 (2009), no. 2, 720-723. doi:10.1134/S1064562409050238.

   Bonnie M. Stewart, Adventures Among the Toroids, 1970; 2nd ed. 1980.

Robert R Tupelo-Schneck / <schneck@gmail.com> / 2011-03-31, reference  updated and CC0 added 2020-05-25 To the extent possible under law, Robert R Tupelo-Schneck has waived all copyright and related or neighboring rights to http://tupelo-schneck.org/polyhedra. This work is published from: United States.