I'm back with (what I hope is) a new CRF

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

I'm back with (what I hope is) a new CRF

Postby New Kid on the 4D analog of a Block » Tue Aug 06, 2019 6:32 am

There's most likely a flaw in this proposed CRF, as to be expected with my "work," but here goes anyway.

A loop of six haps (xo6ox tower) goes through the polychoron, manifesting as columns of three haps in each of the two identical "halves." They make contact with the two hexagons of the "middle boundary." Six peppys connect to each of the "top and bottom" haps with one triangular face, while their pentagons connect to the pentagons of the middle boundary. 36 tets fill in all gaps.
"Central" hap of one half highlighted:
Screen Shot 2019-08-05 at 10.57.04 PM.png
(50.57 KiB) Not downloaded yet

"Top and bottom" haps of one half highlighted:
Screen Shot 2019-08-05 at 10.57.31 PM.png
(43.81 KiB) Not downloaded yet


Like D4.8.x and other known CRFs, it has a near-miss Johnson solid as a middle boundary. This middle boundary is a dodecahedron with two antipodal pentagons replaced with hexagons, and two new pentagons added. (D4.4's middle boundary is like a gyrated, rectified version of this, in case that helps.)
"Top" view:
Screen Shot 2019-08-05 at 10.56.31 PM.png
Screen Shot 2019-08-05 at 10.56.31 PM.png (10 KiB) Viewed 131 times

Facing one of the pentagons:
Screen Shot 2019-08-05 at 10.56.36 PM.png
Screen Shot 2019-08-05 at 10.56.36 PM.png (9.15 KiB) Viewed 131 times


I checked and double-checked all the dihedral angles, and didn't see any in excess of 360°. (There were some in the mid-350s, though...) I checked CRF-list.xls, and didn't see any polychora with 6 haps among their cells.
So, what's wrong with this one?
New Kid on the 4D analog of a Block
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Re: I'm back with (what I hope is) a new CRF

Postby Klitzing » Tue Aug 06, 2019 7:41 pm

At least the pentagonal variant of one of those halves does indeed exist. That one then is nothing but papadoe = pentagonal antiprism atop dodecahedron. Its existance thereby is being seen as a bidiminishing of ikadoe = icosahedron atop dodecahedron, as 2 antipodal tips of the icosahedron get removed at the top level base, while the cutting plane just rasps tangential at the opposing pentagon faces of the bottom level base. And that ikadoe moreover is nothing but a vertex-first subsegment of the 600-cell.

Within lace city display ikadoe is being represented as
Code: Select all
  o5o  o5x      x5o  o5o 
                          
x5o      f5o  o5f      o5x

Here you can see the icosahedron at the top level, showing up its subdimensional layers point atop pentagon atop dual pentagon atop point. Below you see the accordingly oriented dodecahedron level, again with its subdimensional layers pentagon atop f=(1+rt(5))/2 scaled pentagon atop dual f-pentagon atop dual x=unit scaled pentagon.

Accordingly, chopping off those opposite pair of icosahedral tips, you'd get the lace city display of papadoe as
Code: Select all
       o5x      x5o       
                          
x5o      f5o  o5f      o5x

and therefore the pentagonal variant of your idea thus would look like
Code: Select all
       o5x      x5o       
                          
x5o      f5o  o5f      o5x
                          
       o5x      x5o       

i.e. the lace tower pap || doe || pap.

The problem with your hexagonal variant now is that, in order to maintain all faces planar and all unit-edged within that medial layer polyhedron, then those lateral pentagons thereof would no longer remain equiangular, and therefore not regular. But then for your potential CRF you need to errect pyramids on those pentagons. But it is obvious that the lacing edges of such a non-regular pentagonal based pyramid cannot be all of the same size. - Thus your polychoron does exist and is a nice Variation of the above with a different, hexagonal rotational symmetry. But it cannot be made all unit-edged. And therefore fails to be a CRF.

--- rk
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Re: I'm back with (what I hope is) a new CRF

Postby New Kid on the 4D analog of a Block » Wed Aug 07, 2019 1:28 am

Thank you for the clarification.

So I don't make the same mistake again, does the medial layer polyhedron of a CRF (if it has one) need to be unit-edged, regular-faced, and planar-faced? I don't think all these criteria are necessary; D4.8.x has a skew boundary...

And if that doesn't give us an easy way to find potential polyhedra to use as medial layers, is there an easy way to determine whether some lace diagram (for example, the invalid one of the hexagonal variant)
Code: Select all
       o6x      x6o       
                         
x6o      N6o  o6N      o6x
                         
       o6x      x6o

or some polyhedron can be a valid medial layer for a CRF?

Or... is it pointless to search for such small CRFs when one of you surely would have discovered them already?
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Re: I'm back with (what I hope is) a new CRF

Postby Klitzing » Wed Aug 07, 2019 5:32 am

Just consider what CRF means. It is the abreviation of convex regular-faced (polychoron). Thus all polygonal faces have to be planar, unit-edged, and equi-angular (i.e. regular-faced). Moreover all edges throughout the polychoron have to be the same size. Within an intermedial layer however you also might find pseudo faces, i.e. faces which only occur in that section, but else are contained within some through-going polyhedral cell. Those clearly neither have to be equiangular nor need to be completely planar (then not even being pseudo faces, just edge circuits). Even so, those all have to be unit-edged too.

--- rk
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