Hello everyone
I've had my fair share of staring at the disphenocingulum as well now, and an idea popped up, which might give some (AFAIK new) CRF polytope
quickfur wrote:Yet another idea: this was inspired by staring at the disphenocingulum (J90). It appears to be a belt of triangles capped by two spheno complexes (pairs of lune complexes, i.e. 2 triangles + 2 squares). If we remove the spheno complexes, we get a distorted concave polyhedron that can be deformed into a hexagonal prism. So this made me think of a possible 4D analogue, where you start with some kind of duoprism, then deform it so that one ring of cells no longer lies on a great circle, causing the cells on the other ring to become prisms of non-uniform polygons. Then cover up these non-uniform prisms with some kind of cell complex such that it becomes CRF again.
I thought of that ring as an antiprism, If you'd have a prism there, It'd become the square orthobicupola. For this construction, you start with a hexagonal prism, and remove the hexagonal faces. You are now left with a ring of 6 squares, on which you want to place the spheno complex.
Since we are looking at the hexagonal prism, which has orthogonality all over the place, all 6 top points will stay in the same plane. Therefore, we deform the spheno complex until all outer edges lie on the same plane, and paste it to the irregular hexagonal prism on both sides, to obtain the square orthobicupola.
If we'd start with the hexagonal
antiprism, a deformation
would result in a shift from the plane on which all points were. If we now stick the spheno complex to one side, the antiprism is deformed, such that it nicely fits a 90 degree rotated spheno complex on the other side.
If such a CRF exists, it would imply:
1) The existence of some kind of cell complex that ends with a base in the shape of a prism of a non-uniform polygon (specifically, it has to be a prism with unit edge lengths and square lacing faces, but not necessarily regular top/bottom polygons); and:
2) The dichoral angles at the base of this cell complex is small enough that you can at least attach 3 of them in a loop to form regular polygons in the orthogonal ring. So that means the dichoral angle must be ≤ 60°, so that at least it would work with an orthogonal ring of triangular prisms. If it's ≤ 45°, you could form an orthogonal ring of cubes; if it's ≤ 36°, you could form an orthogonal ring of pentagonal prisms. But I'm not holding out much hope for higher-order orthogonal rings; what about at least the ≤ 60° case?
Does anyone know of a 4D cell complex that might potentially have these properties?
Since I only saw an antiprism in the bispenocingulum, I did not get this part. I also had some difficulty finding a satisfactory 4D analogue of the antiprism. This is also where the likely new CRF will pop up
. The first way to generalize the concept of an antiprism, is to consider the top polygon as the dual of the bottom polygon. This would result in 4D in segmentochora such as tet||inverted tet (16-cell), cube||oct and ike||doe. However, Having square or pentagonal faces greatly diminishes our freedom of squishing, which we do not want if we want to create some disphenocingulum-like CRF.
we also do not want a regular prism, because they are too orthogonal, and will not do stuff that is interesting enough. (the vertices remain in the same hyperplane upon squishing)
What I then considered, is placing a 90 degrees
rotated ike on top of another ike (resp oct, tet). In lace-city-ness:
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prism: (ike, oct, tet)
x20 x20 x20 x20 x20 x20
02f 02f 02q 02q 02x 02x
f2x f2x x20 x20
02f 02f
x20 x20
'antiprism':
x20 02x x20 02x x20 02x
02f f20 02q q20 02x x20
f2x x2f x20 02x
02f f20
x20 02x
the nice thing here is that, because of the symetry of ike, the 90-degree rotation will result in quite a symetric constellation
. It is easiest to visualize with the 3-interlocking-golden-rectangles way of viewing ike. now, if we rotate ike, and look at the two wide rectangles, we see that the horizontal and vertical rectangles will find each other, meaning that we could also have rotated around their long axes
For the tet, since it's dual is its 90-degree-rotated self, we have that this antiprism-thing is indeed CRF.
With the oct-constellation, an additional 4 octs and 6 tets will fill the gaps, for a total of 6 octs and 6 tets. I do not know how to verify whether this is CRF, or know of any polytope of which this is a snippet?
The ike gaps are then filled with 6 tets (on the x-edges of the rectangles) and AFAI could see, 20 octs :O
edit: 30 tets and 8 octs (you could also see this constellation as snub oct || dual snub oct, on the basis of which I start to doubt the CRF-ness
)
if, and that's a big if, these things (especially ike-like) happen to be CRF, we could start with the process which leads to the bisphenocingulum in 3D (finally). We only need to find some complex which has a distorted version of ike as a convex hull. For this, I looked at the crown jewels in 3D a lot, and recalled my post in the snub disphenoid based CRF thread, which showed:
oct -> pentagonal bipyramid (adding an edge)
ike -> sphenocorona (deleting an edge and squishing until we have two squares)
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o o | o
f | B o
x (x)| x
f | B o
o o | o
(o) (o) |
f | x
x x | x x
f | D
o o | o o
ike -> sphenomegacorona (stretching a point to become a line)
ike -> hebesphenomegacorona (stretching two points to become lines)
Now, especially the sphenocorona is a diminishing of ike, and therefore could be equipped with two squippies and five tets to make the ike-hull. The constellation would then be:
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ike -> ike-hull || other ike->hull
02x | 02x || x20
f20 | x20 || 02D
x2f | x2B || B2x
f20 | D20 || 02x
02x | 02x || x20
Here I chose for opposing-oriented sphenocoronas, hoping that they will exactly cancel each other's curvature in the ring-of-20-octs-and-6-tets
I know this is also a (very) long shot, and might be proven rong from like step 1, but I considered the symetry-lining-up thingy in ike too beatiful to not share