## 5D castellations?

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

### 5D castellations?

Just a quick thought on 5D CRFs, the castellated rhodoperihedral prism is xofox5ooooo3xFoFx&#xt (reversed from Klitzing's tower since I like to have the higher number first).

Do either of the following exist in 5D, then?

xofox5ooooo3xFoFx3ooooo&#xt
xofox5ooooo3ooooo3xFoFx&#xt

Or maybe something along those lines?

Since I'd love to some rhodochoric-axial symmetries in 5D CRFs.

Also a more relevant reason: if they do exist, surely they would contain some 4D analog of the J91 as tera!

Edit: Just realised, either of my constructions above would surely lead to dodecahedral pyramids, which cannot be made CRF. So instead, how about:

ooooo5xofox3ooooo3xFoFx&#xt

which would convert them to icosahedral pyramids, which are CRF?

Keiji

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### Re: Johnsonian Polytopes

Keiji wrote:Just a quick thought on 5D CRFs, the castellated rhodoperihedral prism is xofox5ooooo3xFoFx&#xt (reversed from Klitzing's tower since I like to have the higher number first).

Me too. I always write o5x3o instead of o3x5o, for example. But it's just a matter of convention, since it doesn't matter what order you write it in, it still denotes the same thing.

[...]Edit: Just realised, either of my constructions above would surely lead to dodecahedral pyramids, which cannot be made CRF. So instead, how about:

ooooo5xofox3ooooo3xFoFx&#xt

which would convert them to icosahedral pyramids, which are CRF?

This is an interesting thought indeed! Problem is, my 5D visualization is nowhere near as facile as my 4D visualization, so I have trouble seeing what kind of shape would be produced in the lateral cells, or even whether this lace tower can be made CRF. I've to admit my CD diagram skillz are mainly driven by visualization, and I'm pretty helpless if I can't "see" what's going on with the symbols.

P.S. On a related note, if we go the other direction to 3D analogues of the castellated prism, the most direct analogue I can think of is a prism with pentagons lacing the top and bottom faces, with alternating orientations (i.e., alternately sharing an edge or a vertex), with the gaps filled in (possibly) equilateral triangles and isosceles triangles with phi-edges. However, unlike in 4D, the pentagons don't merge into a single face, and I've no idea what order of polygons would be needed for the top/bottom faces. My wild guess is phi-scaled decagons, but a more careful calculation would be needed to confirm that.

Also on a related note, I note that many of the 4D CRFs we've been finding actually have 3D analogues, albeit non-CRFs, but can be included if we relaxed our requirements to permit triangles with phi-edges or √2-edges. For example, the square ursahedron would be admitted, having √2-edged triangles around the bottom, and everything else remains regular polygons. Similarly, if we permitted bisected hexagons, then many polyhedra would be admitted which have direct analogues with 4D CRFs that have triangular cupolae. While this goes outside the scope of the present thread, studying what additional polyhedra are admitted under these relaxed requirements may suggest 4D CRF possibilities, due to 4D having a greater inventory of cell shapes to choose from, many of which have non-CRF 3D analogues.
quickfur
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### Re: Johnsonian Polytopes

Well, my lace tower would be - take your pick! - (Hc4 || φ2 H4 || φ Cr4) or (srix || φ2 ex || φ rahi) or (cant600cell || φ2 600cell || φ rect120cell) - plus the reverse of the first two layers of course.

Would the center position then be ooooo5xofox3oFoFo (i.e. id || φ2 ike || φ id + reverse)? Could this be CRF, or am I missing something?

Keiji

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### Re: Johnsonian Polytopes

In the polygloss, the distinction is made between placing the large number first ("dodecahedrally") or small number first ("icosahedrally"). The various short notations permit both, so the dodecahedral form for the tesseract is x4o3o3o, while the icosahedral form is o3o3o4x.

I mainly use the icosahedral form, because this is the way the stott matrix is usually implemented. But a few changes in the rules could easily lead to the icosahedral form. For some project like the johnson CRH project, it is useful to have a standard form so that the same thing does not appear twice.
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the dream we dream together is reality.

wendy
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### Re: Johnsonian Polytopes

Keiji wrote:Well, my lace tower would be - take your pick! - (Hc4 || φ2 H4 || φ Cr4) or (srix || φ2 ex || φ rahi) or (cant600cell || φ2 600cell || φ rect120cell) - plus the reverse of the first two layers of course.

Well, I know that, but how those layers fit in 5D -- I only know in theory, I can't "see" it in my mind's eye, so I'm not confident I'll get it right.

Would the center position then be ooooo5xofox3oFoFo (i.e. id || φ2 ike || φ id + reverse)? Could this be CRF, or am I missing something?

I'll have to sit down and work this through...
quickfur
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### Re: Johnsonian Polytopes

Keiji wrote:[...]
ooooo5xofox3ooooo3xFoFx&#xt
[...]

Hold on a sec, are you sure you didn't have a typo here? If I unpacked it correctly, the layers should be: o5x3o3x || o5o3o3F || o5f3o3o || .... The first symbol describes the cantellated 600-cell, which contains o5x3o (icosidodecahedral) cells; as far as I know, the icosidodecahedral pyramid is non-CRF (the circumradius is phi*edge_length). But in your post you alluded to the icosahedral pyramid, so perhaps you meant to have o5o3x3o in the first layer instead?
quickfur
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### Re: Johnsonian Polytopes

No, I definitely meant icosahedral pyramids - I imagined they would be produced by the phi-squared 600-cell ... except wait, the 600-cell is made of tetrahedra

xofox5ooooo3xFoFx3ooooo and xofox5ooooo3ooooo3xFoFx&#xt would surely have dodecahedral pyramids, though, so that is no use.

Are you sure ooooo5xofox3ooooo3xFoFx&#xt would have icosidodecahedral pyramids? Wouldn't the icosidodecahedra appear at positions not being made into pyramids or am I just spouting rubbish?

Keiji

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### D4.10 temp

Let's unpack ooooo5xofox3ooooo3xFoFx&#xt:
Code: Select all
`o   x   o   xo   o   o   Fo 5 f 3 o 3 oo   o   o   Fo   x   o   x`

So the first two layers are o5x3o3x || o5o3o3F. If you look at the .5.3. positions (i.e. the first 3 columns), you have o5x3o || o5o3o, that is, icosidodecahedron || point, IOW, icosidodecahedral pyramid.

In the original castellated prism, we have:
Code: Select all
`x   o   xo   o   Ff 5 o 3 oo   o   Fx   o   x`

Here, the .5. position of the first two layers yield: x5o || o5o, i.e., pentagon || point, or pentagonal pyramid, which is CRF. Now, the interesting part about the castellated prism, is that there are no lacing edges from the .3. position (last 2 columns) between the first and second layers; instead, o3x laces directly to the 3rd layer's o3o, giving us o3x || o3o, or dual_triangle||point, that is, triangular pyramid, or tetrahedron.

In your proposed lace tower, however, assuming that we have a similar lacing from the 1st layer directly to the 3rd layer (I'm not 100% sure about this, because this effect depends on the relative circumradii of the layers in that region, and like I said, I can't visualize 5D well enough to be able to tell at a glance whether the conditions are right for this to happen), then we would have x3o3x lacing directly to f3o3o, which gives us cuboctahedron || phi-scaled tetrahedron, which is non-CRF.

Now suppose the .3. position doesn't skip the 2nd layer, then we would have x3o3x || o3o3F, or cuboctahedron || phi2-scaled dual tetrahedron, which is again non-CRF.

quickfur
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### Re: Johnsonian Polytopes

Hrrmm...

So going back to my first idea of xofox5ooooo3xFoFx3ooooo...

Code: Select all
`x   o   x   oo   o   F   of 5 o 3 o 3 oo   o   F   ox   o   x   o`

...not sure how I came to the conclusion that this would create dodecahedral pyramids, actually. Would this work after all?

Keiji

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### Re: Johnsonian Polytopes

Wouldn't this have either x5o3x || o5o3F or x5o3x || f5o3o in the .5. positions? So they would be non-CRF.

OTOH, if what we're trying to do is to find 4D bilbiro analogues, then perhaps we're looking at the wrong elements: we should rather be looking at the lateral cells of these towers; who cares if the lacing pyramids are non-CRF, as long as the analogue to the bilbiros' positions are CRF? I've to admit, however, that my CD-diagram-fu is not powerful enough to figure out how to read a lace tower "sideways", as it were.

Another approach is to go up another level of abstraction: instead of worrying about the fiddly details of the exact CD symbols, what if we looked at it from a "constructive" angle? From this POV, basically the castellated prism consists of a top and bottom facet, which can be any shape that serves our purpose, really, with pentagonal pyramids and tetrahedra serving as the lacing cells, and the gaps in between filled up by bilbiro's. Since we're trying to find the analogue of the bilbiros, we can leave that part undecided for the time being. This leaves the pentagonal pyramids and tetrahedra. The thing of interest here is that the pentagonal pyramids from the top/bottom are bridged via their apices with a unit edge, whereas the tetrahedra's apices actually coincide.

This suggests that we could construct some manner of 5D castellated prism by using icosahedral pyramids, bridged by an edge between their apices, and some kind of analogue of the tetrahedra, perhaps 5-cells sharing a vertex. So this tells us that we want some 4D CRF that has icosahedral cells, which narrows down the possibilities to only two: o5o3x3x, and o5o3x3o. Unfortunately, this rules out having 5-cells in our prism, because neither o5o3x3x nor o5o3x3o have tetrahedra as cells. o5o3x3x has truncated tetrahedra, but since their circumradius is greater than their edge length, we can't make pyramids out of them. So this leaves only o5o3x3o, which has octahedra. Now luckily, octahedra do have CRF pyramids. So this tells us that our prism will have icosahedral pyramids and octahedral pyramids, and now we know that the top/bottom facets will be o5o3x3o. Furthermore, the icosahedral pyramids will be separated by an edge, whereas the octahedral pyramids will share vertices, so this suggests a 5-layer construction where the icosahedral pyramids lace layers 1&2 and 4&5, and the octahedral pyramids lace layers 1&3&1. This then gives us this lace tower:
Code: Select all
`o5o3x3oo5o3o3AB5o3o3oo5o3o3Ao5o3x3o`

for some as-yet unknown values of A and B. So then the challenge is to find some A and B, such that the result will be CRF. Unfortunately, this is now out of my depth. Anybody want to jump in to help?
quickfur
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### Re: Johnsonian Polytopes

quickfur wrote:Wouldn't this have either x5o3x || o5o3F or x5o3x || f5o3o in the .5. positions? So they would be non-CRF.

By the logic in your previous posts it was lacing directly between layers 1 & 3 and 2 & 4.

xf5oo3xo should exist, as xx5oo3xo is dodecahedral (peri)cupola. Though it is close, since xo5oo3xx does not exist

Keiji

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### Re: Johnsonian Polytopes

Keiji wrote:
quickfur wrote:Wouldn't this have either x5o3x || o5o3F or x5o3x || f5o3o in the .5. positions? So they would be non-CRF.

By the logic in your previous posts it was lacing directly between layers 1 & 3 and 2 & 4.

Well, I don't know that for sure, that's why I said it could be either x5o3x || o5o3F (if it laces from 1 to 2) or x5o3x || f5o3o (if it laces from 1 to 3). Whether lacing edges skip an intermediate layer depends on whether the circumradius of the edge's endpoints is greater than the intermediate layer's inradius (i.e., the vertices overhang the intermediate layer), or whichever radius corresponds to that particular position. So you'd have to calculate the radii for the various layers in order to be sure which layer laces to which.

xf5oo3xo should exist, as xx5oo3xo is dodecahedral (peri)cupola. Though it is close, since xo5oo3xx does not exist

xf5oo3xo = x5o3x || f5o3o, so at the .5. position you have x5o || f5o, that is, pentagon || phi-scaled pentagon. I'm not sure I understand where you're getting xx5oo3xo from?
quickfur
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