# What are the known convex polyhedra with congruent faces?

A monohedral polyhedron is one whose faces are all congruent. Note that this is a weaker condition than being isohedral (face-transitive).

We have a classification of all convex isohedral polyhedra, consisting of 30 classes of assorted finite polyhedra and infinite families. See Wolfram Mathworld for a list.

In the process of writing this answer, I was trying to find instances of convex polyhedra which are monohedral but not isohedral, and struggled to find a classification of such shapes or even a list of known instances with an outline of which cases remain open.

The instances I know of beyond the isohedral polyhedra given above:

• When the faces are regular polygons, we can search among the Johnson solids to find three non-isohedral examples: the snub disphenoid, the triaugmented triangular prism, and the gyroelongated square bipyramid.

• The pseudo-deltoidal icositetrahedron, the dual of the pseudorhombicuboctahedron. In footnote 46 on page 185 of Advances in Discrete and Computational Geometry: Proceedings of the 1996 AMS-IMS-SIAM Joint Summer Research Conference, Discrete and Computational Geometry, it is remarked that no other convex non-isohedral monohedra are known with non-triangular faces (or at least, none were known in 1996). (Here is a Google Books link to the relevant section of the previous source.)

• However, in a result apparently unknown to the above source, Ed Pegg provides what seems to be a counterexample in this mathSE question; manually cutting out and folding the net, it does indeed seem to fold into a convex polyhedron.

• In the article The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra (Grünbaum, B., 2010), the author includes the rhombic icosahedron and the Belinski dodecahedron, and cites Belinski as proving that these are the only convex monohedra with centrally symmetric faces (such polyhedra termed isozonohedra) not already listed among the isohedra.

• In David Eppstein's paper On Polyhedral Realization with Isosceles Triangles (link to arXiv abstract), three infinite families of convex monohedra not given above are listed, with isosceles triangle faces: one consisting of a heightened antiprism with pyramids glued to each large opposite face (called a "gyroelongated bipyramid"), a variant of the previous shape in which two halves are rotated about a skew hexagonal cross-section (called a "twisted gyroelongated bipyramid"), and a shape called the "biarc hull" which I can't describe very well but is related to a sphericon. They cite the paper Some New Tilings of the Sphere with Congruent Triangles by Robert Dawson, most of which seem not to translate to monohedral polyhedra but which I haven't checked fully.

• The answers to this math.SE question, which provide monohedral polyhedra combinatorially equivalent to the icosahedron but with non-equilateral faces. (The scalene case is not always convex, but can be made so with small distortions to the angles of the triangles involved.) The book Advances in Discrete and Computational Geometry mentioned above goes into more detail about the possibility of such polyhedra.

Given the apparent lack of a classification of more restricted classes of polyhedra, I expect that no complete classification exists. However, I'd like to better understand the scope of which examples and impossibility results are known, as I haven't been able to find much in the way of a definitive source tackling this question. I'm hoping this question and its answers can at least serve as a better overview of known convex monohedra than the scattered state of information that seems to exist on the problem at present.

• Interesting question. I am not aware of a classification, but I am also interested in the higher-dimensional analogue of this question. I think it is unclear whether the problem of finding monohedral polytopes becomes easier or harder in $d\ge 4$ (I speak of identical facets). In 3D I would probably attempt a first classification by the gonality of the faces: monohedral polyhedra with pentagon faces are probably easiest to classify (are there any others than the regular dodecahedron?). Next up the ones with 4-gonal faces. And triangular faces are probably the hardest. – M. Winter Nov 2 '20 at 11:31
• @M. Winter: For pentagonal faces, I know of the tetartoid and the pyritohedron, both listed under en.wikipedia.org/wiki/Dodecahedron#Other_pentagonal_dodecahedra. These are combinatorially equivalent to the regular dodecahedron in their incidence structure, but have less symmetry. – RavenclawPrefect Nov 9 '20 at 2:32
• Update: I have cross-posted this question to MathOverflow at mathoverflow.net/questions/376071/…. – RavenclawPrefect Nov 10 '20 at 1:07