*Note:* I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow may have a better knowledge of the literature here.

A *monohedral* polyhedron is one whose faces are all congruent.

We have a classification of all convex isohedral (face-transitive) 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, the Johnson solids offer 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 - but see the next two bullet points). (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 math.SE 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 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.