## Number of CRF polychora

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

### Re: Number of CRF polychora

Nice puzzle. Not sure I know the answer... but there's something about these constructions where you have an initial polytope P, layered on top of a phi-scaled P, and then some other vertex layers to close things up. In that sense, they are all generalizations of teddi (as triangle || phi*triangle || dual triangle).

Am I close?
quickfur
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### Re: Number of CRF polychora

quickfur wrote:Nice puzzle. Not sure I know the answer... but there's something about these constructions where you have an initial polytope P, layered on top of a phi-scaled P, and then some other vertex layers to close things up. In that sense, they are all generalizations of teddi (as triangle || phi*triangle || dual triangle).

Am I close?

Well teddi is obviously contained both in the 3||teddi and the tetrahedral ursachoron. But it is mibdi, which is contained in the 3-mibdi-laced-wedge. Sure, mibdi as such is just an ike diminishing as teddi is. So they surely are related somehow by the pentagon logic xfo&#x. - But beyond?

In fact those 3 have a much closer relationship than you assumed so far. ...

--- rk
Klitzing
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### Re: Number of CRF polychora

As there were no further trials
here finally the answer comes :

Code: Select all
o3o           o3o                   o3o           o3o

x3o                         +     x3o                   x3o     =     x3o                   x3o

f3o                               f3o                                 f3o
o3x           o3x                               o3x                   o3x           o3x

{3} || teddi              +       tetrahedral ursachoron      =       3-mibdi-laced-wedge

Nice find, ain't it?

--- rk
Klitzing
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### Re: Number of CRF polychora

So you're saying that 3-mibdi-laced-wedge is just a stacking of tetrahedral ursachoron with {3} || teddi? Haha... and I was under the (wrong) impression that it was elementary! (I.e. can't be decomposed into smaller CRFs).

Now, I wonder if we would get something interesting if we started with an x5o3x ursachoron (don't remember what's the final name we agreed on for this), and stacked it with {6} || x5o3x? Would it be CRF? Can it be made CRF? Will it have nice symmetries?
quickfur
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### Re: Number of CRF polychora

If you would give the tetra-ursachoron four of those augments (3||teddi) what would it look like? Would it have pentagonal diprisms? I assume it would be non-convex.
ndl
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### Re: Number of CRF polychora

I'm actually not sure - neither tet-ursachoron nor 3||teddi seems to have dichoral angles listed on Klitzing's site as of yet. Anyone up for deriving the relevant angles?
Trionian

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### Re: Number of CRF polychora

The dichoral angle between two adjacent teddies in the octahedral ursachoron is 112.4555°.

I can't seem to find my 3||teddi model...
quickfur
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### Re: Number of CRF polychora

I've got all the vertices here:

Code: Select all
1.00000000000000000  0.70710678118654752  0.00000000000000000  0.00000000000000000
-1.00000000000000000  0.70710678118654752  0.00000000000000000  0.00000000000000000
0.00000000000000000 -0.70710678118654752  1.00000000000000000  0.00000000000000000
0.00000000000000000 -0.70710678118654752 -1.00000000000000000  0.00000000000000000
0.00000000000000000  1.85122958682191612  0.61803398874989485  1.14412280563536860
0.00000000000000000  1.85122958682191612 -0.61803398874989485  1.14412280563536860
1.00000000000000000  0.43701602444882107  1.61803398874989485  1.14412280563536860
-1.00000000000000000  0.43701602444882107  1.61803398874989485  1.14412280563536860
1.00000000000000000  0.43701602444882107 -1.61803398874989485  1.14412280563536860
-1.00000000000000000  0.43701602444882107 -1.61803398874989485  1.14412280563536860
1.61803398874989485 -0.43701602444882107  1.00000000000000000  1.14412280563536860
1.61803398874989485 -0.43701602444882107 -1.00000000000000000  1.14412280563536860
-1.61803398874989485 -0.43701602444882107  1.00000000000000000  1.14412280563536860
-1.61803398874989485 -0.43701602444882107 -1.00000000000000000  1.14412280563536860
0.61803398874989485 -1.85122958682191612  0.00000000000000000  1.14412280563536860
-0.61803398874989485 -1.85122958682191612  0.00000000000000000  1.14412280563536860
1.61803398874989485  1.14412280563536860  0.00000000000000000  1.85122958682191612
-1.61803398874989485  1.14412280563536860  0.00000000000000000  1.85122958682191612
0.00000000000000000 -1.14412280563536860  1.61803398874989485  1.85122958682191612
0.00000000000000000 -1.14412280563536860 -1.61803398874989485  1.85122958682191612
0.00000000000000000  1.41421356237309505  0.00000000000000000  2.99535239245728471
1.00000000000000000  0.00000000000000000  1.00000000000000000  2.99535239245728471
1.00000000000000000  0.00000000000000000 -1.00000000000000000  2.99535239245728471
-1.00000000000000000  0.00000000000000000  1.00000000000000000  2.99535239245728471
-1.00000000000000000  0.00000000000000000 -1.00000000000000000  2.99535239245728471
0.00000000000000000 -1.41421356237309505  0.00000000000000000  2.99535239245728471

Convex hull in Stella gives me some offending tets rising from each edge of lowest tet in rosette form with inner edge being only 1/phi. Those are not going to be pentagonal dipyramids after all just regular peppies with no way to fit anything in those holes to make it convex.

aug ursa.JPG (61.65 KiB) Viewed 94 times
ndl
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### Re: Number of CRF polychora

If you're looking for a CRF that contains pentagonal dipyramids, there's the various augmented 5,10-duoprisms. The 5,10-duoprism admits augmentations with 5-prism pyramids, and since the dichoral angle between the 5-prism and the pentagonal pyramid is exactly 18°, placing two augments on adjacent 5-prisms will result in the pentagonal pyramids of the respective augments lying on the same hyperplane, so they will merge into pentagonal dipyramids. You can have all 10 pentagonal prisms augmented, which produces a CRF with 10 pentagonal dipyramid cells. (The ring of decagonal prisms is also augmentable with 10-prism||5-gon at the same time, so you can have an omniaugmented 5,10-duoprism that contains pentagonal dipyramids and pentagonal cupolae.)
quickfur
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