This page contains some exposition about the classification of regular-faced polyhedra. See also the list of convex regular-faced polyhedra with conditional edges.

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

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 [1] and proven complete by
Victor Zalgaller in 1969 [2].
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 A_{4},
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.

M_{1} |
Tetrahedron |

M_{2} |
Square pyramid |

M_{3} |
Pentagonal pyramid |

M_{4} |
Triangular cupola |

M_{5} |
Square cupola |

M_{6} |
Pentagonal cupola |

M_{7} |
Tridiminished icosahedron |

M_{8} |
Bilunabirotunda |

M_{9} |
Pentagonal rotunda |

M_{10} |
Trucated tetrahedron |

M_{11} |
Truncated cube |

M_{12} |
Truncated dodecahedron |

M_{13} |
Tridiminished rhombicosidodecahedron |

M_{14} |
Parabidiminished rhombicosidodecahedron |

M_{15} |
Dodecahedron |

M_{16} |
Truncated octahedron |

M_{17} |
Great rhombicuboctahedron |

M_{18} |
Great rhombicosadodecahedron |

M_{19} |
Truncated icosahedron |

M_{20} |
Triangular hebesphenorotunda |

M_{21} |
Hebesphenomegacorona |

M_{22} |
Sphenocorona |

M_{23} |
Sphenomegacorona |

M_{24} |
Disphenocingulum |

M_{25} |
Snub disphenoid |

M_{26} |
Snub cube |

M_{27} |
Snub dodecahedron |

M_{28} |
Snub square antiprism |

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.)

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 [3] and Yu. A. Pryakhin [4] established by 1973 that there are 6 noncomposite
convex regular-faced polyhedra with conditional edges. These are the polyhedra labelled
Q_{1} through Q_{6} 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 [5] and by A. V. Timofeenko [6]. However, both classifications
originally omitted the polyhedra P_{4,30} and P_{4,31} 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 [7].

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.

See the list of convex regular-faced polyhedra with conditional edges.

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 [8] 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 [9]. Some examples of such polyhedra have been produced by Roger Kaufman and others.

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* [10], in particular the
Stewart toroids.

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

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

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

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

[5] 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.

[6] 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.

[7] A. V. Timofeenko, article in preparation in early 2011.

[8] 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.

[9] 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.

[10] Bonnie M. Stewart, *Adventures Among the Toroids*, 1970; 2nd ed. 1980.

Robert R Tupelo-Schneck / <schneck@gmail.com> / 2011-03-31