http://bendwavy.org/klitzing/explain/dynkin.htm In 2014 Mrs. Krieger extended her zoo of lace prisms, towers, and simplexes a bit. She introduced a further qualifier "z" (zero). If this qualifier precedes the lacing edge lengths ("...&#zx.."), then the respective segmental heights happen to become zero. (In that this qualifier adds just some optional further information to the reader.) But there is a bit more to that: For lace prisms the bases usually are part of the figure. And for lace towers the extremal layers similarily. But not the intermediate ones. Those happen to be just pseudo facets, sectioning the tower into segments. But for degenerate lace towers there might occur situations, where this splitting of the vertex set into subsets (those co-hyperrealmic "layers") would result in pseudo facets solely. This then is a case, where this preceding "z" qualifier surely would be required! – A simple example here would be the ike, when being represented in briquet symmetry: fxo ofx xof&#zx. (None of those golden rectangles belongs to the elements of ike.)
She then also allowed for a suffixing "z" too. This one would build up the structure as devised without that additional "z" first, and thereafter scales all lacing edges down, such that the segmental heights all become zero. – This probably would be rather seldomly used explicitly. But implicitly it provides a different pictoral representation for the segmentochora, in fact the "telescope view" from infinity. (The pictures provided within this website however usually provide views from nearby, thus relatively scaling the two bases additionally in a perspectivic way.)
A further concept closely related to the former zoo is that of lace cities. Here the name derives from the picturesque description of a city: being a set of towers. So re-consider the just described lace towers. If each of those layers on its own can be described as a tower, if further all those towers of one dimension less are describable within the same symmetry group, then there is an arrangement of the higher tower in the manner of a street, while the lower towers are given as sets of layers at the specific grounds. This yields a 2 dimensional graphical arrangement of dynkin symbols, each of 2 dimensions less. Take for example the octahedron (oct). It can be given as the lace tower (in fact only a lace prism) xo3ox&#x, i.e. x3o || o3x. Now x3o = x o&#x and o3x = o x&#x. Thus we have for lace city
z has only one meaning, the construction-node that joins to the first node. That's it. The flat lace-tower is a programming feature.
The whole notation that i use, is specifically designed to be 'fairly easy' for a computer to parse, and someone like quickfur to turn a polytope. But even though it's all written before lace-towers etc, the thing is reasonably robust.
So let's parse what's going on.
When the symbol is linearised, the 'penciled in' line that describes the symmetry is the 'trace'. So o---o---o-5-o is a picture which has no direction, but if i write it as o3o3o5o, it's a trace. Normally we follow the regular line, if one is there, to minimalise the trace. The trace-count is 1--2--3-F-4 (which is why i have alternate letters for the structural branches!). The main reason the trace reset exists, is that z points to the first node after a reset, and zz the second. So if you write, eg o3o3o3z&o3o3o3z, the second z points to trace-node 5, and would be *e in Klitzing-style, while the first is *a. If you write a3o3o3z2o3o3o3z, both z's refer to the same node a, the group consists of a triangle-loop, with a tail of o3o3o3o.
& resets the trace-count for the z and zz node. 2 (or R), does not.
# creates a mirrorless mirror node. That means, that if you put an x after a 2, it creates a prism in both cases, so x3o2x and x3o&x are trips or triangle-prisms. But if we want to kill the mirror, we need to put something like # (now i have your attention, gosub hashnode...) So putting x3o2# is actually valid, but it's defined as separate to the base-figure, so we use x3o&#.
A z, creates a loop. It's different in function to the * marker, since the * identifies the current trace-node with a node marked 'a', but this structure does not have a reset-trace scheme, so *a when it comes to be the second part of a product, could become *g. So, eg the z points to a in both a3o3o2o3o3o3z and o3o3o&a3o3o3z , you see here we used the & to reset the trace count, so we go 1,2,3;1,2,3,4=1. This would be eg o3o3o o3o3o3*a vs o3o3o o3o3o3*c resp.
Putting z at the beginning of an trace then creates one of two effects. By itself, it is self-referent, so z = *a (by itself), which loops and removes the z node, and hence the whole symmetry. It's cunning really, but the CD diagram has no way of removing symmetry, or describing a point, since a single o by itself, is a line, not a point.
So putting z after &# makes &#z, which does in order
& new symmetry, with reset trace counters (ie no references back to the previous symmetries)
# create an axis with no mirror
z subsume this node into the first node of the current trace (ie it copies node 1)
The effect is to create an axis, and then to immediately destroy it. But it allows lacings that are in the same node to operate lace operations.
The order of precidence is that 'structural devices' build the kaleidoscope, are always interpreted before 'decorative devices' which create the thing the kaleidoscope shows.
You see, I keep a close eye on all this stuff, because the key thing is that one needs to program it with minimum effort.
The j node is a decorative node, which tells you to look for irregular(symmetry), eg j5j2j5j = j(5,2,5). This is a simple lookup table that returns the irregular figure as requested. But since j5j2j5j has a symmetry of 5,2,5, we still allow the base structure to shine through.
The idea then with the mirror-less mirror, is that you could create a compound before the &#, which extends only as far as the previous &, and then 'lace the bits together, in the order given'. A lace-tower, is like those chinese-lantens or the flat-pack christmas decorations that create a solid device when you pull it out. Something like oxoo5ooxo is simply four figures written in the same symmetry. When you add &#, it adds the third dimension. When you put 'x', it connects the vertices over every symmetry, with a length x (decorative). When you put 't', (structural), it means pick the mess up from the first row and let it dangle. So the order of precidence is tht # and t are structural, and x is decorative, so the structural takes precidence over the decorative.
The role of z then is to zap the current node to the first trace-count. It does not reset the trace-count itself. It's a structural node, which means that 333z is different to 333 by symmetry (the first is the triangle-tilings, the second is the fivechoron, so z affects how the kaleidoscope is made, and is thus structural). When you impose decorations on it, it's like moving one or more points in say 333, and then creating a depth perpendicular to it &#, and then pulling the stuff around so the lacing is tight.
Something like qoo3oqo3oooAooq&#xz uses the group 3,3,A of order 192, and creates inside it, three different 16chora, of edge q=sqrt 2. The vertical into 5d is created by &# and destroyed by z, and the x node is then some kind of lacing to these figures. The z for t just highlights the fact it is actually flat relative to the &# axis.