Inconsistency in defining regularity

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

Inconsistency in defining regularity

Postby Mecejide » Fri Apr 05, 2019 2:14 pm

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.
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Re: Inconsistency in defining regularity

Postby Klitzing » Sat Apr 06, 2019 9:47 am

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
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Re: Inconsistency in defining regularity

Postby wendy » Sat Apr 06, 2019 12:29 pm

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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Re: Inconsistency in defining regularity

Postby Mecejide » Mon Apr 08, 2019 4:14 pm

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.

What about these four:
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.
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