A few monostratic polychora

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

A few monostratic polychora

Postby username5243 » Sun Apr 16, 2017 12:57 pm

Recently, I was thinking about the 4-d partial Stott expansion series from hex to sidpith. In particular, the lace tower descriptions of those. I figured out that:

hex = pt || oct || pt
pex hex = pt || oct || oct || pt, or line || esquidpy || line
quawros = line || esquidpy || esquidpy || line, or square || squobcu || square
pacsid pith = square || squobcu || squobcu || square, or cube || sirco || cube
sidpith = cube || sirco || sirco || cube


In particular, this should mean that there should be a series from the oct pyramid to cube || sirco.

Specifically:

Start with octpy (oct pyramid). this has as cells 1 oct (octahedron) + 8 tets (tetrahedra).

The next one is line || esquidpy. This can also be thought of as a square pyramidal prism with two square pyramidal pyramids on the bases. Its cells are 1 esquidpy (elongated square dipyramid) + 8 tets + 4 trips (triangular prisms).

The next one is square || squobcu. It is formed by joining 2 square || squacu together at a cupola. Its cells are 1 squobcu (square orthobicupola) + 2 cubes + 8 tets + 8 trips.

And finally there is cube || sirco, with cells 1 cube (top) + 6 cubes (sides) + 8 tets + 12 trips + 1 sirco (small rhombicuboctahedron).

Is this a valid partial expansion series, and are there any others like it?
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Re: A few monostratic polychora

Postby student91 » Sun Apr 16, 2017 4:55 pm

username5243 wrote:[...]
Is this a valid partial expansion series, and are there any others like it?
Certainly it is valid!
For these kind of expansions, it is best to understand the Coxeter-Dynkin notation used around here. It starts out with the Coxeter-Dynkin diagram which you can Google about, it is not that easy to comprehend, but we can help with the aspects you don't understand. This particular expansion uses [2,2]-symmetry.

In the notation of Wendy's used here, certain lengths are given their own symbol, e.g.
o=0
x=1
q=sqrt(2)
f=(1+sqrt(5))/2=golden ratio
w=x+q=1+sqrt(2)
F=x+f=(3+sqrt(5))/2

Now your expansion starts with an oct in [2,2]-symmetry. This is q2o2o+o2q2o+o2o2q, which is written as qoo2oqo2ooq&#zx.
Then we add a point, so we get qoo2oqo2ooq&#zx || o2o2o.
Then the expansions consist of adding a value of x to one of the nodes.
The first expansion wxx2oqo2ooq&#zx || x2o2o
The second expansion wxx2xwx2ooq&#zx || x2x2o
The last expansion wxx2xwx2xxw&#zx || x2x2x.
That's about it. Practically every Stott-expansion can be given in this way, and when it is given in this way it certainly is a Stott-expansion. I am not fully aware of similar expansions, I assume there are many, but Klitzing probably knows these better.
I hope I have informed you well, feel free to ask any questions.
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Re: A few monostratic polychora

Postby Klitzing » Sun Apr 16, 2017 10:17 pm

username5243 wrote:Start with octpy (oct pyramid). this has as cells 1 oct (octahedron) + 8 tets (tetrahedra).

The next one is line || esquidpy. This can also be thought of as a square pyramidal prism with two square pyramidal pyramids on the bases. Its cells are 1 esquidpy (elongated square dipyramid) + 8 tets + 4 trips (triangular prisms).

The next one is square || squobcu. It is formed by joining 2 square || squacu together at a cupola. Its cells are 1 squobcu (square orthobicupola) + 2 cubes + 8 tets + 8 trips.

And finally there is cube || sirco, with cells 1 cube (top) + 6 cubes (sides) + 8 tets + 12 trips + 1 sirco (small rhombicuboctahedron).

Right you are. It is the partial Stott expansion when considering the pyramidal full octahedral symmetry, starting with
ox3oo4oo&#x = octpy
<-> esquippidpy
<-> squicuf
<-> ox3oo4xx&#x = cubasirco
In fact, the to be applied expansion would require a breakdown of the full starting (resp. ending) symmetry into some true subsymmetry. Here that one clearly is the axially directed (pseudo-pyramidal heights) of o2o2o2o, i.e. some sort of o2o2o||o2o2o.
username5243 wrote:Is this a valid partial expansion series, and are there any others like it?

It is valid; both as partial Stott expansion and student91 already showed you a way to implement it by mere normal Stott expansion (within that lesser symmetry). And yes, there are others too.

E.g. you could use any lower dimensional sequence of partial Stott expansions and lift that into the next dimension by mere prisms or (pseudo)pyramids. So you could expand from tet to tricu, from squippy to squacu, from peppy to pecu. - This already shows that you are not reduced to cartesian-axially expansions only. There are trigonal and pentagonal ones too (the square happens to be cartesian again).

Or you could go dimensionally up. Then you would have other subsymmetries. So you could expand only tetrahedrally (4 radial directions) the cubically symmetrical figures. Or just hexadecachorically (8 radial directions) the tesseractically symmetric ones.

Virtually you could use any possible subsymmetry and expand according patches of bounding elements into the according directions. It is only that most of these partial Sott expansions in general would produce new to be filled in elements, which no longer are regular polygons only / Johnson solids only / CRFs only. Rather some rhombs, shields or other stuff might turn up too. But the more interesting cases for sure are those, where we remain fully within the CRF regime.

--- rk
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