TITLE: The "Johnson polychora"
QUESTION [9 upvotes]: Firstly, a definition:  

A convex polyhedron, whose faces are regular polygons (2D polytopes).  

This includes the 92 Johnson solids, 13 Archimedean solids, 5 Platonic solids and two infinite familes - prisms and antiprisms.
However, I see two ways of extending this definition to 4D:

1) A convex polychoron (4-polytope), whose 2D faces are regular polygons, ie. its 3D faces satisfy the first definiton.  
2) A convex polychoron, whose 3D faces are regular polyhedra (3D polytopes).  

The list I gave above is known to be complete. But is there (at least partial) progress on the 4D case? The second definition is much more restrictive, so maybe a complete classification of polychora, which satisfy 2) exists. Can you point me to some reference, please?

REPLY [4 votes]: Even so the above cited webpages indeed are being hosted by Anton Sherwood, everything what belongs under https://bendwavy.org/klitzing/home.htm is my own content, he just provides webspace to me.
The set of convex polytopes, where their facets are bound to be regular, indeed are the polytopes already investigated by Gerd and esp. his wife Roswitha Blind, cf. https://bendwavy.org/klitzing/explain/johnson.htm#blind. - What they proved could be enlisted nearly fully as this $complete$ listing

the bipyramid of the tetrahedron, and similarily the bipyramid on every higher dimensional regular simplex
the pyramid on the octahedron, and similarily the pyramid on every higher dimensional regular crosspolytope
the pyramid on the icosahedron, and the bipyramid on the icosahedron
the adjoin of an octahedral pyramid and the rectified 5-cell (aka apiculated rectified 5-cell)
millions of more or less symmetrical edge-facetings of the 600-cell (using tetrahedra and icosahedra for cells), e.g. the snub 24-cell is one of those.

The set of convex polytopes, where only its polygonal faces are bound to be regular, are known as CRF (convex, regular faced polytopes). This set is much vaster and by no means fully classified. Even so under the restriction to 4D cases only, a large list of examples is being provided on my wbsite, cf. https://bendwavy.org/klitzing/explain/johnson.htm#crf. (Directly above the table even a downloadable spreadsheet with the full so far known listing is available.)
The picture displayed above not only is my proprietary, it moreover displays the first example historically being known what later became known as the set of scalifomrs, cf. https://bendwavy.org/klitzing/explain/scaliform.htm (symmetry acts transitively on the vertex set, edges are all of the same size, polygonal faces are regular). The displayed picture e.g. has for facets a total of 2 truncated tetrahedra, 8 trigonal cupola, and 6 tetrahedra (the latters acting as digonal antiprisms).

--- rk