Euclidean honeycombs and higher dimensional space filling like tetracombs etc. often can be seen as an infinte repetition of some finite layer sequence.

This deserves an own notational concept.

Well we have already lace towers as a notational aid to denote stacked vertex layers in finite polytopes. Thus e.g. ofx-&#xt denotes a regular pentagon, xux-&#xt denotes a regular hexagon, xfoo-5-oofx-&#xt denotes a regular dodecahedron, xxxx-4-oxxo-&#xt denotes a small rhombicuboctahedron, etc. (Dashes here only were inserted for the ease of reading, those are not a part of the notation itself.) Thus we have already some notational aid wrt. the multilayer concept, esp. when cosidering true facets (at the antipodal ends of that axial direction) and about pseudo facets as the other vertex layer sections, as well as wrt. to the lacing elements.

But when it comes to some kind of repetinioning, we are still without a clou on how to denote that. - For that reason, I'd like to take some rescue to the notation in musics. There too sometimes some passages are to be repeated. Notationally those usually are denoted that way:

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`||: .... | .... | .... | .... :||`

So what about enclosing repeating layers in our context as well within ':' characters? And, in order to provide a further additional clue on how often the repetition would have to take place, we could add at the end of that axial notation part ('&#xt') the corresponding repetition number, or, in case of our infinite structures, simply an 'i' (for 'infinitely often').

Thus chon = x4o3o4o would become chon = :x:4:o:4:o:&#xti or alternatively (in a different orientation) as chon = :qoo:3:oqo:3:ooq:3*a&#xti. And rich = o4x3o4o could be denoted as rich = :oq:4:xo:4:oo:&#xti = :xxo:3:xox:3:oxx:3*a&#xti. I think, that these few examples already make the concept quite clear, and that this concept does allow for lots of applications.

(For sure, whenever we have just a single repetition layer, as in the first provided example, this is just a different description of a cross-product with aze, i.e. :x:4:o:4:o:&#xti = x-infin-o x4o4o.)

Wendy, as I suppose, would make lots of use from this new notational concept, as she already had considered lots of high-dimensional lattice geometries, esp. those A_n, D_n and E_n geometries, and has deduced from such repeating layerwise considerations most of her layerwise understandings of the corresponding cell polytopes. So it would be quite easy to her, to put her already derived applications into that new notational representation.

Esp. wrt. her 'laminates', which she so far only denotes by some L (laminate), some letter (representing type) and optionally some number (representing subtype) - e.g. LA1. That L-notation might serve for her purposes quite well, as she knows what these character-number-suffixes are chosen for, but for all the other willing readers that notation is a bluddy horror. I suppose, that this - much more Dynkin style - notation finally could help out of that mess too.

The most important thing here, for sure, is that this new notation then allows for a corresponding concise description of all the respective subelements as well, which no longer have to take refuge to totally different notational concepts any more.

--- rk