by **Klitzing** » Sun Jul 26, 2015 6:57 am

This research, in fact, was the first idea on how to extrapolate the Johnson solids idea onto higher dimensions: Considering all convex polytopes in N dimensions, the (N-1) dimensional facets of which would be regular ones in turn. And that research already has been completed way back around 1979. The couple Roswitha and Gerd Blind then published the complete enumeration - outside the combinatorically huge amount of possible diminishings of the 600-cell. But in the meantime even that number has been computer-counted. - So nothing new in here.

In 4D we have (beyond the well-known archimedean ones):

- the dipyramid of the tetrahedron

- the dipyramid of the icosahedron

- the pyramid of the octahedron

- the pyramid of the icosahedron

- the pyramid of the octahedron attached onto a rectified pentachoron

- and all those numerous diminishings of the hexacosachoron

in all spaces beyond:

- the dipyramid of the simplex

- the pyramid of the cross-polytope

You then just need to select the ones which have tetrahedra solely from that above shortlist:

- the 2 mentioned dipyramids

- plus the 3 known regular ones

That's it.

Btw., a still on-going research is a different extrapolation of Johnson's idea: Considering all convex polytopes in N dimensions, the 2-dimensional faces of which would be regular ones. This is what is called CRF (convex regular faced). Lots of individual findings with respect to N=4 already can be found in this forum. In fact, even for N=4 no complete enumeration has been achieved so far.

--- rk