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The definition of platonic solids is the following (see Wikipedia):

In Euclidean geometry, a Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex.

So, I am curious, that what is the name of the class of solids obtained by omitting the regularity and the equality of the degree of vertexes in the definition of platonic solids?

For example the triangular and pentagonal bipyramid are in this class, but they aren't platonic solids.

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    $\begingroup$ Apart from the cube and the dodecahedron, these would be the deltahedra I guess. $\endgroup$
    – user856
    Commented Apr 19, 2015 at 8:37
  • $\begingroup$ 'Regularity' of platonic solids refers to the fact that all the faces are congruent regular polygons (& also all the vertices lie on a spherical surface). 'Equality of degree of vertices' of platonic solids refers to the fact that all the vertices are identical i.e. identical faces meet at each vertex of the platonic solid. $\endgroup$ Commented May 19, 2015 at 20:00

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These would be the monohedral Johnson solids.

The Johnson solids are the 92 convex polyhedra whose faces are all regular polygons, other than the 5 Platonic solids, 13 Archimedean solids, prisms, and antiprisms.

"Monohedral" means that all the faces are congruent; this is a weaker condition than isohedral, which means that the symmetry group of the polyhedron can carry any face to any other.

It turns out that the only monohedral Johnson solids are deltahedra, i.e. all the faces are equilateral triangles, and there are five. Besides the two bipyramids you mentioned, there are the gyroelongated square bipyramid, triaugmented triangular prism, and snub disphenoid.

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