Completeness of isohedral figure classification

The table below was simplified from the table at this Wikipedia article.

Here is a table of isohedral figures from Wikipedia. Is it proven that all isohedral figures must be one of these things – a Platonic solid, a Catalan solid, or one of these pyramid-stacking “polar” figures?

Faces	Face config.	Class	Name	Symmetry	Order
4	V33	Platonic	tetrahedron	Td, [3,3], (*332)	24
4	V33	Platonic	tetragonal disphenoid	D2d, [2+,2], (2*)	4
4	V33	Platonic	rhombic disphenoid	D2, [2,2]+, (222)	4
6	V34	Platonic	cube	Oh, [4,3], (*432)	48
6	V34	Platonic	cube	D3d, [2+,6]	12
6	V34	Platonic	cube	(2*3)	12
6	V34	Platonic	trigonal trapezohedron	D3	6
6	V34	Platonic	asymmetric trigonal trapezohedron	[2,3]+, (223)	6
8	V43	Platonic	octahedron	Oh, [4,3], (*432)	48
8	V43	Platonic	octahedron	D4h,[2,4],(*224)	16
8	V43	Platonic	octahedron	D2h,[2,2],(*222)	8
8	V43	Platonic	square bipyramid	D2d,[2+,4],(2*2)	8
8	V43	Platonic	rhombic bipyramid	D2d,[2+,4],(2*2)	8
8	V43	Platonic	square scalenohedron	D2d,[2+,4],(2*2)	8
12	V35	Platonic	regular dodecahedron	Ih, [5,3], (*532)	120
12	V35	Platonic	regular dodecahedron	Th, [3+,4], (3*2)	24
12	V35	Platonic	regular dodecahedron	T, [3,3]+, (*332)	12
12	V35	Platonic	pyritohedron	T, [3,3]+, (*332)	12
12	V35	Platonic	tetartoid	T, [3,3]+, (*332)	12
20	V53	Platonic	regular icosahedron	Ih, [5,3], (*532)	120
12	V3.62	Catalan	triakis tetrahedron	Td, [3,3], (*332)	24
12	V(3.4)2	Catalan	rhombic dodecahedron	Oh, [4,3], (*432)	48
12	V(3.4)2	Catalan	rhombic dodecahedron	Td, [3,3], (*332)	24
12	V(3.4)2	Catalan	deltoidal dodecahedron	Td, [3,3], (*332)	24
24	V3.82	Catalan	triakis octahedron	Oh, [4,3], (*432)	48
24	V4.62	Catalan	tetrakis hexahedron	Oh, [4,3], (*432)	48
24	V3.43	Catalan	deltoidal icositetrahedron	Oh, [4,3], (*432)	48
48	V4.6.8	Catalan	disdyakis dodecahedron	Oh, [4,3], (*432)	48
24	V34.4	Catalan	pentagonal icositetrahedron	O, [4,3]+, (432)	24
30	V(3.5)2	Catalan	rhombic triacontahedron	Ih, [5,3], (*532)	120
60	V3.102	Catalan	triakis icosahedron	Ih, [5,3], (*532)	120
60	V5.62	Catalan	pentakis dodecahedron	Ih, [5,3], (*532)	120
60	V3.4.5.4	Catalan	deltoidal hexecontahedron	Ih, [5,3], (*532)	120
120	V4.6.10	Catalan	disdyakis triacontahedron	Ih, [5,3], (*532)	120
60	V34.5	Catalan	pentagonal hexecontahedron	I, [5,3]+, (532)	60
2n	V33.n	Polar	trapezohedron	Dnd, [2+,2n], (2*n)	4n
2n	V33.n	Polar	trapezohedron	Dn, [2,n]+, (22n)	2n
2n	V33.n	Polar	asymmetric trapezohedron	Dn, [2,n]+, (22n)	2n
2n	V42.n	Polar	regular n-bipyramid	Dnh, [2,n], (*22n)	4n
2n	V42.2n	Polar	isotoxal 2n-bipyramid	Dnh, [2,n], (*22n)	4n
2n	V42.2n	Polar	2n-scalenohedron	Dnd, [2+,2n], (2*n)	4n
4n	V42.2n	Polar	2n-scalenohedron	Dnd, [2+,2n], (2*n)	4n

Short answer

Yes – for convex polyhedra the classification is finished. Branko Grünbaum and Geoffrey C. Shephard completed it in their study of isohedral (face-transitive) tilings of the sphere, and the result is exactly what your table shows:

• 5 Platonic solids

• 13 Catalan solids (duals of the Archimedean solids)

• Three infinite polar families

  • the (general) scalenohedra / isosceles tetrahedra
  • the $n$-bipyramids (duals of prisms)
  • the $n$-trapezohedra (duals of antiprisms)

Together these make up 30 “classes’’: 28 individual solids and 2 one-parameter families whose face-count grows with $n$. MathWorld’s Isohedron page quotes the same total and traces it to the Grünbaum–Shephard work (MathWorld), while Wikipedia summarises the duality-based reasoning – Platonic ↔ Platonic, Archimedean → Catalan, prism → bipyramid, antiprism → trapezohedron (Wikipedia).

What the proof looks like (in outline)

1. Work on the sphere. Any convex polyhedron gives a tiling of the unit sphere by projecting its faces outward.

2. Classify all face-transitive spherical tilings. Grünbaum & Shephard (1981) listed every way a single tile can repeat around the sphere under a finite symmetry group.

3. Lift the tilings back to space. Each tiling corresponds to either

   • a Platonic solid,
   • the dual of a uniform spherical tiling (hence a Catalan solid), or
   • a 2-cap tiling whose dual is a bipyramid or trapezohedron. No other convex embedding exists – that’s the heart of the proof.

Because the argument happens on the sphere, it automatically rules out any stray convex isohedra that look different but would project to one of the known patterns.

Limits of the result

• If you drop convexity (allow self-intersection or star faces) there are many more isohedra, starting with the four Kepler–Poinsot polyhedra, and no complete list is known.
• If you go up one dimension (4-polytopes) or down to tilings of the plane, the problem is again open.
• Even in 3-D, the weaker “all faces merely congruent’’ condition (monohedral) is not classified; several sporadic convex examples are known that are not isohedral (Mathematics Stack Exchange).

So, within the usual confines of convex solids the answer is yes – every isohedron is either Platonic, Catalan, or one of the polar stack-of-pyramids families on your list. Outside that safe zone, the zoo is still very much alive and un-catalogued.

Do you have the citation for the study by Branko Grünbaum and Geoffrey C. Shephard?

Below is the full bibliographic information that most authors cite when they refer to the Grünbaum–Shephard classification of isohedral (face-transitive) spherical tilings—which, in turn, yields the list of convex isohedral polyhedra you asked about.

Ref.	Standard citation (APA-style)	Notes
[GS 1981a]	Grünbaum, B., & Shephard, G. C. (1981). Spherical tilings with transitivity properties. In C. Davis, B. Grünbaum & F. A. Sherk (Eds.), The Geometric Vein: The Coxeter Festschrift (pp. 65-98). Springer-Verlag. https://doi.org/10.1007/978-1-4612-5648-9_4	The definitive, worked-out classification of all isohedral, isotoxal and isogonal tilings of the 2-sphere. Convex isohedral polyhedra correspond to these tilings via radial projection and duality. (SpringerLink)
[GS 1981b]	Grünbaum, B., & Shephard, G. C. (1981). Patterns on the 2-sphere. Mathematika, 28(1), 1-35. https://doi.org/10.1112/S0025579300015321	A shorter companion journal article that previews some of the same results in a more compressed form. (londmathsoc.onlinelibrary.wiley.com)

Either reference is usually accepted; the book chapter [GS 1981a] is the most frequently quoted source when people assert that “every convex isohedron is Platonic, Catalan, or one of the three polar families.”