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?


There is some information on your first extension to 4D in Anton Sherwood's Johnson solids webpages. He calls the class CRFs: convex and regular-faced. A subset of your second extension to 4D was apparently studied by Roswitha Blind in 1979: convex, with facets regular polyhedra, but not necessarily uniform. Although there is a wealth of information in these webpages, I am not finding it easy to extract a clear status summary.

          A "segmentochoron."
Added after the OP's comment:

Blind, Roswitha. "Konvexe Polytope mit regulären Facetten im $\mathbb{R}^n$, ($n \ge 4$)." In Contributions to Geometry, pp. 248-254. Birkhäuser, Basel, 1979.


  • 1
    $\begingroup$ Thanks. It seems that Blind polytopes are exactly what my second definition suggests (excluding the uniform polychora of course) and that they have all been enumarated. I will have a look at the papers by Blind, but they are in German, so it may take a while. $\endgroup$ – FusRoDah Mar 29 '18 at 8:28
  1. 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.

  2. 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.

  3. 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.)

  4. 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


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