## Obtaining uniform polytopes by removing degenerate elements

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

### Obtaining uniform polytopes by removing degenerate elements

This was inspired by posts on the other thread.

In here, I will determine the representations of many of the degenerate Wythoffian polychora (those that have {6/2} faces, degenerate cells etc). This first post will cover the linear o3o3o3o and related. The 3-D ones are all lised on Klitzing's website.

NOTE: to avoid redundancies, if a polychoron is identical to one listed, such as x3x3o3o and x3x3/2o3o, it will only be written once:

o3o3o3o:
x3o3o3o - pen
o3x3o3o - rap
x3x3o3o - tip
x3o3x3o - srip
x3o3o3x - spid
o3x3x3o - deca
x3x3x3o -grip
x3x3o3x - prip
x3x3x3x - gippid

o3o3o3/2o:
o3o3x3/2x - 4-covered pen
o3x3o3/2x - pinnip + 5 2thah
reduced o3x3o3/2x by 2thah - pinnip (in rap regiment, cells octs and trips)
x3o3o3/2x - 2-covered firp
hemi x3o3o3/2x - firp (in rap regiment, cells tets and trips)
o3x3x3/2x - sirdop + 20{6/2}
reduced o3x3x3/2x by {6/2} - sirdop (in srip regiment, cells choes, tuts, trips)
x3o3x3/2x - piphid + 10 2trip
reduced x3o3x3/2x by 2trip - piphid (in spid regiment, cells tets, trips, coes)
x3x3o3/2x - pirpop + 5 2thah
reduced x3x3o3/2x by 2thah - pirpop (in srip regiment, cells tuts, trips, hips)
x3x3x3/2x - ripdip + 10 2trip
reduced x3x3x3/2x by 2trip - ripdip (in prip regiment, cells choes, toes, hips)

o3o3/2o3o:
o3x3/2x3o - 6-covered pen
x3x3/2x3o - nipdip + 10{6/2}
reduced x3x3/2x3o by {6/2} - nipdip (in spid regiment, cells tets, trips, choes)
x3x3/2x3x - 2garpop + 20{6/2}
reduced x3x3/2x3x by {6/2} - garpop (in srip regiment, cells choes and hips)

o3o3/2o3/2o:
o3x3/2x3/2x - 2rap+firp (?) (cells "2oct+6{4}", 3tet, trip)
x3o3/2x3/2x - 3firp+5 2thah
x3x3/2x3/2x - garpop + ? (cells "2oct+6{4}", "cho+4{6/2}", hip, 2trip)

o3/2o3o3/2o:
x3/2x3x3/x - ? (cells are "cho+4{6/2}"s and 2trips)

o3/2o3/2o3/2o:
x3/2x3/2x3/2x - 4pinnip (?) (cells are "2oct+6{4}"s and 2trips)
Dionian

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### Re: Obtaining uniform polytopes by removing degenerate eleme

Here is something to do that I find interesting. Take any existing verf and cut anything from it that "looks like" a vertex figure, by taking a subset of its vertices. Now, as long as the resulting figure is a lace simplex between Whytoffians in the same family, it should be possible to reverse-engineer this "fake" verf into a Dynkin diagram, which -- while not necessarily describing a valid polytope -- should at least describe a valid polytope family.

To see this in practice, imagine taking the verf of sishi -- a v-sized dodecahedron -- and cut from it the segmentotope vx5oo#v, a pentagonal podium. The 5/2-verf lacing between the two pentagons can be thought of as x5/4x, leading to the Dynkin diagram o5o3x5/4x5/2*b, which is indeed a Grunbaumian member of a real polytope family!

Interestingly, this means that there should be a way to find every polytope family in n dimensions, just by taking every set of n vertices from a Whytoffian vertex figure, making them into a simplex, and finding the family of the would-be omnitruncate with that as its verf.
polychoronlover
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### Re: Obtaining uniform polytopes by removing degenerate eleme

Of course, the o5x5/3x5o, x5o5/3o5x and x5x5/3x5x are also double-covers, surtopes coinside by pairs.
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the dream we dream together is reality.

wendy
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### Re: Obtaining uniform polytopes by removing degenerate eleme

polychoronlover wrote:Here is something to do that I find interesting. Take any existing verf and cut anything from it that "looks like" a vertex figure, by taking a subset of its vertices. Now, as long as the resulting figure is a lace simplex between Whytoffians in the same family, it should be possible to reverse-engineer this "fake" verf into a Dynkin diagram, which -- while not necessarily describing a valid polytope -- should at least describe a valid polytope family.

To see this in practice, imagine taking the verf of sishi -- a v-sized dodecahedron -- and cut from it the segmentotope vx5oo#v, a pentagonal podium. The 5/2-verf lacing between the two pentagons can be thought of as x5/4x, leading to the Dynkin diagram o5o3x5/4x5/2*b, which is indeed a Grunbaumian member of a real polytope family!

Interestingly, this means that there should be a way to find every polytope family in n dimensions, just by taking every set of n vertices from a Whytoffian vertex figure, making them into a simplex, and finding the family of the would-be omnitruncate with that as its verf.

Nice. I did notice once that some of these have verfs that looks like pieces cut out from a larger polytope (like x3/2x3o4o, whose verf is a unit-edged square pyramid I'm fairly sure - but x3/2x3o4o is what could best be described as "3hex+8oct"). I might actually start this sometime, by looking into more of these.
Dionian

Posts: 56
Joined: Sat Mar 18, 2017 1:42 pm

### Re: Obtaining uniform polytopes by removing degenerate eleme

wendy wrote:Of course, the o5x5/3x5o, x5o5/3o5x and x5x5/3x5x are also double-covers, surtopes coinside by pairs.

Yes, I did notice that those (and their conjugates from o5/3o5o5/3o) are degenerate polychora (check Klitzing's site).