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
`o5o3x3o`

o5o3o3A

B5o3o3o

o5o3o3A

o5o3x3o

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?