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Convex solids can have all sorts of symmetries:

  • the platonic solids are vertex and face-transitive, meaning there is a subgroup of the rotations of 3-dimensional space which can bring any vertex onto another one (and the same for faces). The list there is limited to the 5 platonic solids.

  • face transitive (or isohedral) solids include the Catalan solids, the (infinite family) of dipyramids and the (infinite family) of trapezohedra. Note that without further restricitions these solids can come in infinite families: the rhombic dodecahedron has an infinite number of deltoidal cousins (see here ); it also fits in a one-parameter family of dodecahedra called pyritohedra (see here ); the dodecahedron and the triakis tetrahedron fit in the one-parameter family called tetartoid (see here ); dipyramids and trapezohedron also admit alls sorts of deformations beside the number of faces.

  • there is a much weaker symmetry one can ask for. Let's call it pseudo-Catalan (for lack of a better name). Fix a "centre" $C$. The convex solid is pseudo-Catalan, if each face can be sent to another face by a rotation with centre $C$ or a reflection (whose plane goes through $C$). Note that there is no requirement that this rotation (+ reflection) preserve the whole solid. An example of such a solid which is not a Catalan solid is the gyrate deltoidal icositetrahedron.

Question: is there a list of pseudo-Catalan solids?

  • note that there would be a last category, where the solid is convex and all the faces are congruent (a convex monohedral solid). The difference with the previous category is that translations are now allowed. In particular, to check that a solid belongs to the previous category, the choice of $C$ (and the fact that all rotations and reflections are constrained by this point) is important. Examples of such solids are the the triaugmented triangular prism and the gyroelongated square dipyramid.

[Edits: clarified definitions as to reflect the content of the comments below)]

ARG
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  • What do you mean by translation within your addendum? As you are considering convex solids, coplanar translated faces would give rise for a much larger combined face in contrast to your search. Non-coplanar translated faces (at one side of the solid) would imply that the deeper face completely gets lost by convex hull argument. If those would be more or less diametrally antipodes, i.e. the solids happens to lie somehow inbetween, then by virtue of your search assumption there will be a rotation-reflection to match those. - I don't get the difference. – Dr. Richard Klitzing Nov 23 '19 at 11:41
  • Solids which fit in my addendum (but not in the other items) include the triaugmented triangular prism and the gyroelongated square dipyramid. Basically, I only require that all faces of the solid be congruent. – ARG Nov 24 '19 at 05:10
  • I included this addendum only to make sure that the category considered in the question was clearly defined. There, I require that there is a point, so that all faces can be sent to each other using only rotations [& reflections], but do not require these rotations to preserve the whole solid. The difference is apparent if you look at the symmetry group of the gyrate deltoidal icositetrahedron (4-fold Antiprismatic a.k.a. D4v) whereas the deltoidal icositetrahedron (a Catalan solid) has full Octahedral symmetry (a.k.a Oh) – ARG Nov 24 '19 at 05:20
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    And yes, I expect the number of solids in the addendum to be fairly large (it should contain many infinite families). But to be honest, I could not find many descriptions of such solids (which do not fit in the previous families). – ARG Nov 24 '19 at 05:32
  • Thus you seem to require that your needed rotations or rotation-reflections, even so not necessarily being a symmetry of the whole solid, still have to fix the solid center. Am I right? Else the 2 examples of your first comment surely could pass too. – Dr. Richard Klitzing Nov 24 '19 at 12:02
  • indeed, two rotations with different centre can generate a translation. So the rotations are all with respect to the same centre. – ARG Nov 24 '19 at 14:35
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    This question has now been cross-posted to MO (and there are some examples of solids there which fit the required conditions). – ARG Dec 02 '20 at 10:56
  • Concerning your first point, a disphenoid (a tetrahedron with opposite edges equal) is vertex- and face-transitive (by $180^\circ$ rotations) but not regular. – mr_e_man Mar 15 '22 at 04:06
  • @mr_e_man Thanks for the correction. I should have said "vertex and face transitive solids include the platonic solids" as I did in my second point. – ARG Mar 15 '22 at 06:53

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