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Is there a name for the class of polyhedra that are (a) convex, and (b) have regular polygons for faces? I don't want to invent a name if a name already exists.

Call the class P. P would include,

  1. the 'cap' of an icosahedron, i.e. the 5 triangles round a vertex plus their pentagonal base,

  2. the rest of the icosahedron after removal of the cap,

  3. the so called "siamese" dodecahedron,

  4. a polyhedron of 3 pentagons and 5 (equilateral) trianges. It has one triangle connected to three pentagons, so the vertices there are (triangle-pentagon-pentagon), and there is then room for four triangles to fill the space left at the other end.

  5. plus of course all the Platonic, Archimedean and prism/antiprism polyhedra.

A very simple idea, but I can't find on the web a word in use for the set P.

Martin
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3 Answers3

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I do not think that there is a universally used and understood term for this class of polyhedra. That is, this particular class of polyhera does not have a name which is generally used in the literature. In my own writing, I would probably spell it out—this kind of object is a convex polyhedron with regular faces (CPRF, for short).

However, I believe that "semiregular polyhedron" is close, as long as a definition or some explanation is given. From the above linked Wikipedia page, a semiregular polyhedron is

a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron,

which is not quite what the question asks for (as this definition imposes an additional condition on the configuration of vertices), but this seems reasonably close, and the term "uniform polyhedron" is more commonly used for vertex transitive polyhedra with regular faces.

For what it is worth, there is an interesting discussion of these kinds of polyhedra in

Johnson, N. W., Convex polyhedra with regular faces, Can. J. Math. 18, 169-200 (1966). ZBL0132.14603.

Indeed, as pointed out in a comment, some authors refer to such objects as "Johnson solids", likely due to the above-cited paper.

Xander Henderson
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  • Chris, Dan, Xander, thank you so much for your prompt and useful answers. I have found the full pdf text of Johnson's paper here: https://www.researchgate.net/profile/Norman-Johnson-2/publication/268494379_Convex_Polyhedra_with_Regular_Faces/links/54e0ed250cf29666379121c7/Convex-Polyhedra-with-Regular-Faces.pdf – Martin Jul 12 '23 at 07:21
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These are called Johnson solids.

Dan Asimov
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As the other answers already refer to, your search is quite close to the term of "Johnson solids", which indeed refers to the 92 solids listed within that already mentioned paper of him. However, that term then is meant to be exclusive instead of your inclusive quest. I.e. that count of 92 is beyond the Archimedeans and the 2 infinite series of prisms and antiprisms.

As a further addition onto the other answers I would point to the German couple Roswitha & Gerd Blind, who within the 1980s had elaborated the similar quest for any higher dimension as well, i.e. the search for all convex polytopes with regular facets. They even provide a complete short-list of those, i.e. an explicite enumeration of all beyond the various diminishings of the 120-cell. Again, their set is meant as an exclusive one only.

https://polytope.miraheze.org/wiki/Blind_polytope

On the other hand there still is a different extrapolation from the 3D quest as well: the search for convex polytopes with regular faces. Those meanwhile became known as "CRF"s accordingly. Those clearly encompass both the Blind polytopes as well as the Jonson solids and all the convex uniforms too. But they allow for much more as well, for instance all the prisms of the Johnson solids. That research is still on-going. Lots of those can be found here:

https://bendwavy.org/klitzing/explain/johnson.htm#crf

--- rk

Dr. Richard Klitzing
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