## Inconsistency in defining regularity

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### Inconsistency in defining regularity

Why is it that non-compound polytopes must be flag-transitive to be considered regular, but compound polytopes do not? For example, of the 5 so-called “regular” compound polyhedra, 4 of them (those with icosahedral symmetries) are not transitive on their flags.
Mecejide
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### Re: Inconsistency in defining regularity

a regular compound surely has to be regular
cf. "A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds." on https://en.wikipedia.org/wiki/List_of_regular_polytopes_and_compounds#Three_dimensional_compounds
whereas
a compound of regulars usually does not
e.g. the compound of a cube and an octahedron definitely cannot map a square onto a triangle.

btw.
regular polytopes are fully transitive
whereas
fully transitive polytopes needn't be regular
cf. https://math.stackexchange.com/questions/2350100/totally-transitive-polytopes-which-are-not-regular

--- rk
Klitzing
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### Re: Inconsistency in defining regularity

I count among the regulars these compounds.

1. Stella Octangula = 3+3
2. Stella Tegmata = 3+3,4
3. Stella Prismata = 4,3+3
4, The mete star. {3,3,3}_120 The mirrors of [3,3,5] include those of [3,3,3] but pairs (other than opposites) do not share more than one mirror.

Among the star-compounds, 4+4, {6/2,6}, {3*6} {6/2*6}, 3+3,4,3 and 4,3+3,4. and their duals.

In all of these cases, except (4), the symmetry is that transitive on the flag, with internal dividing mirrors. The '+' sign is a division between the two nodes on the other side of the polygon, eg 3+3 = {3,3,4/2:}.

The mete star, and the bulk of the polygrams, are 'lattice-stars', in that there are mirrors that do not pass through the elements of the symmetry of the lesser group.

So coxeter describes 'compounds of regulars in a regular' as for 'regular compounds'. They are not 'compounds that are flag-transitive'.
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wendy
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### Re: Inconsistency in defining regularity

wendy wrote:I count among the regulars these compounds.

1. Stella Octangula = 3+3
2. Stella Tegmata = 3+3,4
3. Stella Prismata = 4,3+3
4, The mete star. {3,3,3}_120 The mirrors of [3,3,5] include those of [3,3,3] but pairs (other than opposites) do not share more than one mirror.

1. Compound of two pentachora {(3, 3, 3)}
2. Comopund of two icositetrachora {(3, 4, 3)}
3. Compound of six tesseracts {(4, (3, 3))}
4. Compound of six hexadecachora {((3, 3), 4)}

Here I used the parentheses indicate compounds of two polytopes in opposite orientations—the Stella Octangula, Stella Tegmata, and Stella Prismata would be {(3, 3)}, {(3, 3), 4}, and {4, (3, 3)}, respectively.
Mecejide
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### Re: Inconsistency in defining regularity

Is there a list anywhere of regular 4D compounds? I only found the partial list of uniform compounds on Klitzings website which doesn't specify which ones are regular. I always thought the best way of finding regular compounds was to take a regular polytope and imagine it being inserted into another larger symmetry type and then repeating it as many times as needed to complete all the points of symmetry.
ndl
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### Re: Inconsistency in defining regularity

ndl wrote:Is there a list anywhere of regular 4D compounds? I only found the partial list of uniform compounds on Klitzings website which doesn't specify which ones are regular. I always thought the best way of finding regular compounds was to take a regular polytope and imagine it being inserted into another larger symmetry type and then repeating it as many times as needed to complete all the points of symmetry.

By my definition, there are 6—the stellated decachoron, the stellated icositetrachoron, the great icositetrachoron, the stellated tetracontaoctachoron, the great stellated tetracontaoctachoron, and the grand tetracontaoctachoron.
Mecejide
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### Re: Inconsistency in defining regularity

I'm sure i scribbled a list of the regular polytopes out somewhere.

There are 184 regular compounds, of which 46 are listed in Coxeter's "Regular Polytopes". Being too cheap to use the 7c for a photocopy, i copied the list by hand, and ended up with 53. On the facing page, it went to 54. But they just kept on coming.

These are only those that have vertex in vertex or face in face, as per Coxeter's definition. He missed a couple of them.
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wendy
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