New lace term?

[student91]

In the D4.11 & D4.12 topic Klitzing and I were discussing whether or not we would introduce a new lace term U.
The Idea for this term came when I tried to represent D4.11 in it's demitesseractic symmetry. D4.11 can indeed be represented as such. It would then be the "union" of f3x3o *b3o, x3o3f *b3x and o3o3x *b3F. This "union" is derived as the convex hull of these three things when placed around the same origin.
I at first thought this was a degenerate lace simplex, but Klitzing pointed out it was not. He said it was a degenerate lace tower, which is indeed much more accurate. Nevertheless I think this "union" might need a new lace term, because it's degenerate.

a lace tower can be seen as several polytopes made of the same dynkin graph, with their origins distributed over one dimension. The coordinates can be given as ([dynkin polytope], height)

a lace city can be seen as several polytopes with the same dynkin graph, with their origins distributed over two dimensions. Hence, their coordinates can be given as ([dynkin polytope], [x-position], [y-poisition])

Now the new "union" operator would take several polytopes with the same dynkin graph, and distribute their origins over zero dimensions, i.e. don't distribute it at all. In fact, it is pretty similar to the compound operator, with the additional step that you take the convex hull of the resulting vertices. EDIT: definition [1]
When we leave it's definition as such, it would be only usefull for convex polytopes. We could also define it as [take all the vertices of the dynkin polytopes with the same origin and the same orientation] and then [connect all vertices with an x-edge, if possible].
The first definition would give only convex polytopes, thus is very suited for the CRF project, but not for much else.
The second definition can give any polytope, but you have to be really sure that you have everything right. EDIT: definition [2]
A notation we could introduce can be &U, for union according to definition 1, or &U(x), for union according to definition two with x-edges. I like the U, because it looks a bit similar to the Union sign of set theory.
If you want it to be defined a bit differently, or have a suggestion for a different notation, or just read the topic, please let me know what you think.

[Klitzing]

In the D4.11 & D4.12 topic Klitzing and I were discussing whether or not we would introduce a new lace term U.
The Idea for this term came when I tried to represent D4.11 in it's demitesseractic symmetry. D4.11 can indeed be represented as such. It would then be the "union" of f3x3o *b3o, x3o3f *b3x and o3o3x *b3F. This "union" is derived as the convex hull of these three things when placed around the same origin.
I at first thought this was a degenerate lace simplex, but Klitzing pointed out it was not. He said it was a degenerate lace tower, which is indeed much more accurate.

Yep. that very case solves that way.

Nevertheless I think this "union" might need a new lace term, because it's degenerate.

Well, just for incidently having zero height, I don't believe we really need for something new.

a lace tower can be seen as several polytopes made of the same dynkin graph, with their origins distributed over one dimension. The coordinates can be given as ([dynkin polytope], height)

a lace city can be seen as several polytopes with the same dynkin graph, with their origins distributed over two dimensions. Hence, their coordinates can be given as ([dynkin polytope], [x-position], [y-poisition])

Well, in fact this is just 2 different instances of the same thing:   parallel-space coordinates   x   perpendicular-space coordinates.
It is just that in lace towers the para-space is 1-dimensional, and in lace cities it happens to be 2-dimensional. But this concept is way more General.

Now the new "union" operator would take several polytopes with the same dynkin graph, and distribute their origins over zero dimensions, i.e. don't distribute it at all. In fact, it is pretty similar to the compound operator, with the additional step that you take the convex hull of the resulting vertices.
When we leave it's definition as such, it would be only usefull for convex polytopes. We could also define it as [take all the vertices of the dynkin polytopes with the same origin and the same orientation] and then [connect all vertices with an x-edge, if possible].
The first definition would give only convex polytopes, thus is very suited for the CRF project, but not for much else.
The second definition can give any polytope, but you have to be really sure that you have everything right.
A notation we could introduce can be &U, for union according to definition 1, or &U(x), for union according to definition two with x-edges. I like the U, because it looks a bit similar to the Union sign of set theory.
If you want it to be defined a bit differently, or have a suggestion for a different notation, or just read the topic, please let me know what you think.

We already encountered the need for compounds within perp-space at special positions of para-space. - Within lace cities this can be resolved as usual: just display the compound at that position (e.g. that of rico:

Code: Select all

      o3x   x3x   x3o     
                          
                          
   x3x   u3x   x3u   x3x   
                          
                          
x3o   x3u  oH3Ho  u3x   o3x
                          
                          
   x3x   u3x   x3u   x3x   
                          
                          
      o3x   x3x   x3o     

, where u=2x and H=3x).
Within lace towers however it would conflict with the linear sequencing of stacks. But when allowing zero heights within towers, it would not conflict any longer. Note that generally not all vertex layers are connected by lacing edges to the very neighbours. It happens every now and then, that lacing edges surpass one or more layers inbetween. Same here as well: the 2 layers of those compound components will generally not being laced mutually, rather they are laced to other layers independingly.

But you might wish to point out such zero height layers. This too already has been done in the past by means of according parantheses. (E.g. the lace tower of the icosidodecahedron in vertex first orientation would be
oxFf(oX)fFxo 2 ofxF(Xo)Fxfo &#xt,
where F=ff and X=2f.)

In that sense then also that degenerate tower with full hexic symmetry of pretasto could be given more visibly as
(f[x)o] 3 (x[o)o] 3 (o[f)x] *b3 (o[x)F] &#xt.

--- rk

[wendy]

I'm not really overfussed with the idea of 'unions', since the figure is probably not a union (a union is more like a compound than a lacing), and the whole point of the lace-notation is that the edge length already gives enough information to construct everything.

A compound is several polytopes tied to the same symmetry, eg xo3ox or oq3oo4xo, being first the hexagram as two triangles, and second, the compound of a dual cube and octahedron, where the edges bisect each other.

A lace prism is something you do to a compound: you lace its vertices together with extra edges, A very famous 3d flat lace prism is "ofx2fxo2xof&#x". This is three golden rectangles, laced together to form an icosahedron. Similar kinds of constructs exist to make the 2_21, the 27-vertex-polypeton.

flat lace-prisms are supprisingly common. Often one pulls these things out of lattices, so one has the compound formed by, eg

/6/ + 3/3, a polytope having 126 vertices, being 70+56, according to the decimal count. This is a lace compound, that might be written as xo3oo3oo3ox3oo3oo3xo&#x. It is, of course, no other than the 2_31, or 5/B.

Many lace cities can be wrapped into flat lace prisms. For example, one has x5x2o5o + fo5of2ox5xo + xo5ox2fo5of + o5o2f5f, which is a presentation of x3o3o5o, against an equatorial girth. You see much more when this is rolled out into a lace-city, since you can see the various sections of the rings it is made of.

The thing is that even something like oo3ox4oo&xt, while it be flat in 3d, comes out as a proper pyramid in hyperbolic 4space, so it's probably no different to ox3oo5oo&x or oo3oo5ox&x, which are also valid pyramids.

[Klitzing]

Other, quite elementary flat segmentochora are provided at my new flat complexes page of my incmats website. (Scroll down to section "Flat Segmentochora".)

Much more interesting, because of stacking snubs to Wythoffians, are the ones provided at the same page at section "Flat Segmentotera".

--- rk

[Klitzing]

Similar kinds of constructs exist to make the 2_21, the 27-vertex-polypeton.

would you mind to elaborate on that one?
I.e. on a degenerate (= 0 height) lace tower or lace simplex description of "jak"?

--- rk

[student91]

I'm not really overfussed with the idea of 'unions', since the figure is probably not a union (a union is more like a compound than a lacing), and the whole point of the lace-notation is that the edge length already gives enough information to construct everything.

Indeed the word union does not accurately describe the process I described. I chose this word because of my shallow vocabulary, so if you have a better word for this process, I will eagerly use it. (of course, if we're going to introduce it.)

[...]flat lace-prisms are supprisingly common. Often one pulls these things out of lattices, so one has the compound formed by, eg

When they're so common, how do you describe them? do you describe them all as either lace-simplices or lace-towers, or are there some exceptions to this? if so, how are these exceptios described?

/6/ + 3/3, a polytope having 126 vertices, being 70+56, according to the decimal count. This is a lace compound, that might be written as xo3oo3oo3ox3oo3oo3xo&#x. It is, of course, no other than the 2_31, or 5/B.

Many lace cities can be wrapped into flat lace prisms. For example, one has x5x2o5o + fo5of2ox5xo + xo5ox2fo5of + o5o2f5f, which is a presentation of x3o3o5o, against an equatorial girth. You see much more when this is rolled out into a lace-city, since you can see the various sections of the rings it is made of.

The thing is that even something like oo3ox4oo&xt, while it be flat in 3d, comes out as a proper pyramid in hyperbolic 4space, so it's probably no different to ox3oo5oo&x or oo3oo5ox&x, which are also valid pyramids.

I'm very unfamiliar with hyperbolic geometry, so I don't know if my new lace term would ruin those properties of flat lace-prisms.
Anyhow, I still think the lace-tower representation of bilbiro is, although a lot more accurate, still not completely accurate. In the 2-2-2 representation of bilbiro, f2o2o + x2x2f + o2F2x, there indeed is no x-lacing between f2o2o and o2F2x. However, there is a f-lacing between these two. This means, it's "structure" is a x-f-x-triangle, with all edges scaled down to zero. This certainly is not a lace-simplex, because it's "structure" is not a simplex with all edges x. (this in contrary to wendy's ike, that has a x-x-x-structure, and thus is a valid simplex). You could indeed argue that it is a lace-tower, but that would assume the angle between the x-edges (in the x-f-x triangle) to be 180. This angle is rather undefined, so such an assumption is a bit like assuming 0/0=1. (I'm sorry for that bad analogy, couldn't make up a better one)
Furthermore I think this "union" differs from the degenerate lace-towers on your site in the part that this union has all it's vertices at the exterior of the polytope. This makes it possible to represent a CRF in a subsymmetry of it. furthermore this makes all vertices "see" each other, whereas in the things on your site, the "bottom" of the tower doesn't "see" the top, and thus doesn't try to connect with it.
A last argument I had is that the order of the "dynkin polytopes" doesn't matter in this representation. When you treat the bilbiro-"union" as a lace-tower, you have to find out what parts have an x-lacing to what, in order to find out whether it's a degenerate tower, or rather a degenerate simplex. I think this discussion is superfluous, as all heights have to be 0 anyways (here I'm implying another argument: my "union" would describe things of the same dimension as the "base-symmetry", whereas degenerate lace-towers resp. simplices treat it as something that might be a higher-dimensional thing, but it occasionally doesn't have). This means, you can see what parts do have a lacing, using the radius-spreadsheed, and setting the lacing-height to zero, instead of having to find this out beforehand, in order to fit in the current system of operands.

So basically what I'm trying to introduce is an operator that throws all vertices of the "dynkin polytopes" together (like a compound), and then additionally connects all vertices with each other that need to be connected. What vertices need to be connected is decided by what definition you would take. Definition [1] would connect all vertices if their connection doesn't go through the polytope, and definition [2] would connect all vertices that have a connection with a set length (e.g. length x). As far as I know, such an operator doesn't exist yet. Maybe you don't like the "throwing everything on a pile" though. I think this operator may prove usefull for the partial stott-expansions that led to D4.11 etc.

[wendy]

I have decided that there is room to create student91's request, although at a slightly different point. Don't worry, i did this to r.klitzing's notation too, because there are 'global' interests to protect.

U is already in use as the horogon. It is the 1d tiling, which can be made into a limiting polygon. When you use it as eg &U, you end up with a long stick, divided into prisms. (Prism comes from greek 'prisma' off-cut, like what you cut off a long stick.) In any case, you have F=f*f, H=h+h, so i suppose, U=u*u.

'z' is the appropriate symmetry for a point, since something like 'o' implies that it's a zero-length line (but lives in a 1d space). z is a node that here loops onto itself, and destroys any symmetry. In any case &#z can be read as 'zero-height/dimension' (added here).

Thatching

Ever since Hedrondude rolled out the regiments, i have been trying to find a term to make a non-convex hull. For example, the regiment (figures sharing vertices + edges) that contains {5/2,5} and {3,5/2}, the first of these is the 'most convex' of these. But you can't call it convex, and it's hardly a hull, so i decided instead that it needs a new word. That word is 'thatching', here 'edge-thatching', since the edges are kept proud. Vertex-thatching is the default meaning of thatch, so it does make a convex hull.

For the present case, you can use something like &#z as a suffix, since the edges will come from the stott-differences. If you want to further imply an edge, you can add that after this, ie &#zx.

Lace Union

I suppose we could keep this term for a thatched compound, especially if it is expressed as a laced layer. I'm all ears, though.

Lace projections

Sometimes you want to project a lace-tower onto the base, for the purpose of modeling it. In order to do this, one would still construct the figure as a lace tower, and then zero the height. Lacing edges would then be shortened by projection, but differently for different shells.

The proposal here is to write this by replacing 't' by 'z', eg 'oo3oo4ox&#xz' is the shape formed by the atomium at http://www.atomium.be/ , which is evidently a spherated edgeform of this.

Lace towers

It is still appropriate to designate a series of layers from a tiling, as a lace tower, since we have ample tools now to determine the height. These are to be read as 0=o, 1=x, against o3o3o5o. So the inner shell (marked *), is o3x3o5o. From this to the x3o3x5x, there is a flat lace segmentochoron in &#xt. So the whole 'cage' comes up as a dozen polychora, laced together by unit lacings, but it's still flat. If you replaced it bt &#ft, it would stand upright as a kind of pyramid.

The letters 'a' to 'd' here are differences between layers, is used to verify that thing is flat.

Code: Select all

  1 0 0 0    10.472136
  0 1 0 0    37.885438  *
  0 0 0 1    54.832815  a   0 1 0 -1
  0 0 1 0    82.249223  b   0 0 1 -1
  1 1 0 0    86.249223  c   1 1 -1 0
  1 0 0 1   109.665631  a
  1 0 1 0   147.554175  b
  0 1 0 1   181.442719  d   1 -1 1 -1
  0 1 1 0   229.803398  b
  0 0 1 1   270.164078  a
  1 1 0 1   274.164078  c
  1 1 1 0   332.996894  b
  1 0 1 1   379.829710  a
  0 1 1 1   506.439634
  1 1 1 1   653.993788


Thank you for your help.

Wendy

[wendy]

Here are answers to questions from student91 and from r.klitzing.

When they're so common, how do you describe them?

Normally, the layer is expressed as a compound. You can see an example in Richard's lace city at the centre, where he writes "oH3Ho". This is how they normally deal with sections which are best described as a compound of two polyhedra.

I'm very unfamiliar with hyperbolic geometry, so I don't know if my new lace term would ruin those properties of flat lace-prisms.

It's not your worry to see if this ruins hyperbolic geometry. This is a concern i have to look at when I look at these things. If the thing falls in the same radius, then it does not effect hyperbolic geometry.

Anyhow, I still think the lace-tower representation of bilbiro is, although a lot more accurate,

I was a bit worried about what you were doing with ball-point pens (biro) to small furry creatures (bilbies sing. bilby), until i found that a bilbiro is actually J91.

One of the real worries, is that quickfur is actually progressing to reading these lace towers in etc, so this is also a concern.

would you mind to elaborate on that one?
I.e. on a degenerate (= 0 height) lace tower or lace simplex description of "jak"?

You can make this figure from, eg xoo3oxo2oxo3oox2oox3xoo&#xt. It's kind of like the cuboctahedron, if you look at it in three hedrices. For example, if x3o is 'w' then -w would be o3x, and the jak comes at (w,-w,0) EP3C (where 3C rotates the triangles by 120 degrees in the plane, a complex version of 'change sign'.

a lace city can be seen as several polytopes with the same dynkin graph, with their origins distributed over two dimensions. Hence, their coordinates can be given as ([dynkin polytope], [x-position], [y-poisition])

Instead of 'x, y', you can divide these by symmetric rays, called x3o, and o3x, and plot the missing coordinate out as a lace-compound with the last coordinate as a o3o. You would be supprised how many interesting figures come from doing this. Some figures, like x3o3o5o, look the same in either axis of o5o2o5o, while x3o4o3o, is presented differently in the o3o2o3o, since these stand in the ratio of 1:q.

[Klitzing]

@student91:

You surely are free to start from existing concepts and build thereon your owns. We generally cannot hinder that. But you should be careful in doing so. Because we don't want any 2nd Babylonian confusion. You should always verify whether your ideas are unique and indeed new, and that those cannot be represented by the so far existing system. To that end you are surely supposed to understand the existing system correctly, esp. right into its corresponding specific details. - And here it seems to me that you might have some missconceptions.

A lace tower canbe considered in its easiest cases just as a stack of lace prisms. This already make clear, that lace prism bases clearly are external facets. Within such a stack (as a tower), some such intermediate "bases" will become internal, and therefore then represent pseudo facets only.

Just as the stacking of lace prisms already tells, the lacing edges of either segment then can have any arbitrary angle between them. They definitely are not bound to align. It is just the axis, which is assumed to be continued. In fact any lace prism has 2 parallel bases, each being perpendicular to the axis. Accordingly the bases and those intermediate pseudo facets of a tower too will be all perpendicular to the common axis, or, taken differently, are mutually parallel. - This latter wording also then will hold for degenerate towers too. If the segment heights would calculate as zero you clearly have no true axis. But still a base (or sectioning pseudo facet) normal. And therefore, in that special case, parallelity of bases (or sectioning pseudo facets) just reads as coplanarity (of corresponding dimension for sure).

But there is also a different feature in real lace towers. It is that not all neighbouring pairs of bases and/or pseudo facets ought to be conected directly by lacings of the assigned length. If the respective segment heights become small enough, there might be edges connecting one vertex layer e.g. to the one but next. And such cases might even intertwine, resulting in cases where neighbouring layers are not being laced at all, each just being connected to other ones. - Such an behaviour for sure becomes more frequent, the lower the respective heights would be. So it is not at all surprising, when you consider degenerate towers, i.e. zero heights.

Lace simplices on the other hand imply (just by the definition of "simplex") that all layers shall be laced pairwise.

That is, bilbiro in fact is a degenerate lace tower, when considering the "layers" f2o2o, x2x2f, and o2F2x. This is because layers f2o2o and x2x2f are laced by unit edges and then calculate to a zero height of that partial lace prism. Same holds for the layers x2x2f and o2F2x. But the layers f2o2o and o2F2x shall not be laced within bilbiro. (In fact those could so only by means f scaled edges.) Thus this results therefore in just fxo2oxF2ofx&#xt. (But cf. also below!)

But then you are right, it truely differs a little bit from other degenerate lace towers or lace prisms, e.g. like ox6oo&#x. Here the bases always are true according dimensional facets. But in your description of bilbiro none of those "layers" would be one. In fact all layers will represent pseudo facets there. (The same holds true for your representation of pretasto or Wendy's representation of ex.)


@Wendy:

I first thought about that new suffix "&zx(t)" being nothing but an optional clarification that all according heights there would evaluate as zero. Thus more the kind of being just reader-friendly.

But then I realized that bit mentioned above. Those there mentioned new degenerate structures definitely cannot be described as lace simplices, as not all lacings shall occur. Thus those need to be represented as lace towers at most. But then the 2 base facets ought to be real facets, if it would be a true lace tower, even when having some zero segment heights. But those examples use only pseudo facet layer descriptions!

Thus bilbiro only could be represented (in that advised symmetry representation) as fxo2oxF2ofx&#zxt, where this additional "z" then not only just point out that the heights happen to become zero, but also that all layers in fact are just pseudo facets.

Note that the final "t" still is needed here. Even so the first and third layer are corealmic too, and therefore shall have a height of zero, they are not to be laced. And thus the neglection of that "t" would be wrong.

--- rk

[Klitzing]

would you mind to elaborate on that one?
I.e. on a degenerate (= 0 height) lace tower or lace simplex description of "jak"?

You can make this figure from, eg xoo3oxo2oxo3oox2oox3xoo&#xt. It's kind of like the cuboctahedron, if you look at it in three hedrices. For example, if x3o is 'w' then -w would be o3x, and the jak comes at (w,-w,0) EP3C (where 3C rotates the triangles by 120 degrees in the plane, a complex version of 'change sign'.

Okay. But I more was after a degenerate 2 "layer" representation of jak = 2_12 x3o3o3o *c3o3o. In fact, your degenerate 2 "layer" representation of laq = 2_13 = x3o3o3o *c3o3o3o (by means of suph = /6/ = x3o3o3o3o3o3x and he = 3/3 = o3o3o3x3o3o3o) should need somewhere such jak facets being shown. And I cannot spot those so far.

--- rk

[wendy]

@richard.

Thanks for your input.

The earliest of the lace prisms are the vertex figures of wythoff mirror-edge figures. You simply replace marked nodes with '$' vertex nodes, and each vertex is enumerated as a separate base with all the unmarked nodes, the lacing between the $ nodes being the short-chord of the doubled, eg $3$=h, $4$=p8, $2$=q, so something like $3o4o3$ becomes ox4xo&#q.

The evolution into a striped layer came as a result of lace prisms. This notation is not well suited for what the above does, but it describes sections of the regular figures, and the johnsons quite neatly. Most of the lacing edges here are of a kind not seen in the first type, because $xi$ is a rare thing.

Lace towers are meant to be convex in their own right. For example, xof3ooo5oxo&#xt, is more than the sum of xo3oo5ox&xt, and of3oo5xo&xt, since it involves 12 instances of oxo5ooo&xt over the pentagonal hedra of the middle section. Lace towers, in essence, are already thatched or vertex-convex.

The proposals i am suggesting deal with shapes that can not be presented as a lace-tower, such as qo3oo4ox&#z. In any case, this is not a laced compound, but a thatched one. The height is given, and it is the edges that are calculated, here 1/2 sqrt(3) of the rhombo-dodecahedron.

One can write the bilbiro as fxo2oxF2ofx&#xt or fxo2oxF2ofx&#xz, because both lead to the same figure, but by different constructions.

The first (&#xt) implies that you lace f2o2o to x2x2f to o2F2x , with x-lacing, and then pick the thing up against the height. No one is suggesting that the first must be laced to the third. It's kind of like those christmas lantens that come flat, and form a shape when you tie it up. You still have to make the thing convex, though.

The second (&#z) implies you simply arrange these polytopes f2o2o, x2x2f, and o2F2x together in the same space, and then simply find the convex hull over it.

The fact that &#xt = &#z comes to the same thing, is just like o3x3o4o comes to x3o4o3o. One can not rely on the two falling together.

The proposal for &#xz, is a name for a specific wire-frame projection of a figure. For example, ox5oo&#xz is simply a pentagon, divided into five sectors from the centre. Likewise, ox5xo&#xz is a decagon with two inscribed (unit) pentagons. These are names for the pictures, like quickfur makes, rather than the real things. Note here that the lacing between the shells are different, but this is what the projection does.

[wendy]

Richard asks on /4B "2_21", given that we have written something like 5/B "2_31" as a compound of /6/ + 3/3 = /o3o/3/o.

Code: Select all

   4B1  against 6:.
                                /4B=2_21          4B/ = 1_22

     ,  C  ,  A  ,  B  27      C                  1     4    =  1
     A  ,  B  ,  C  ,  12      .  *  .  C       3+9    2/2   = 20
     ,  C  ,  A  ,  B   3      C                 12    /4/   = 30
     A  ,  B  ,  C  ,   0                       3+9    2/2   = 20
    12  5  8  9  8  5   -      /4B cell.          1     4    =  1



The diagram left shows the distribution of the three 'tails' of 2_22, against the symmetry 6: This is a series of loops of 6-nodes, the order is exactly the same as the nodes fall in x3o3o3o3o3o3z. This gives the holes. The numbers under the columns represent the diameters of the deep holes, where the edge is at the first value (ie 12). The rows of spheres are placed in holes that step three to the left of the previous, as gosset tilings do.

The middle diagram shows the cells of 2_21, as a bi-stratum, here /4 || 3/1 || /4. So, i suppose if you want a flat lace prism for 2_21, you could go for the prism // dual's rectate, ie xo3oo3oo3ox3oo2xo&xt.

The right column shows the lace-tower of the vertex-figure of 2_22 = /4B1. This is 4B/ = "1_22". These are the nearest 'A's to the central figure, for example. The 3+9 and 12 give the relevant radius at that level, the 1,20,30,20,1 gives the vertices in the ring.

Diagrams of this kind are the way one pries data, including lace towers, for the gosset-figures.

Here is the 7d case, being 6C aginsst 7:

Code: Select all

    /6C,  against 7:
                                5B/               5/B

  50  o  x  o  o  o  o  o                    5/    7
  32  o  o  o  o  o  x  o       /5    7     2/3   35
  18  o  o  x  o  c  o  o       4/1  21     /5/   42
   8  o  o  o  o  c  o  x       1/4  21     3/2   35
   2  o  o  o  x  o  o  o       5/    7      /5    7
   0  x  o  o  o  o  o  o
     14  6 10 12 12 10  6            28          1.06



This is the same as /8: + 4/4:, you can see the first as being on the rows that are multiples of 8, the second fall on rows that are not.

/5B, the cells of the tiling, is divisible into 1/5 and 5/1, ie oo3xo3oo3oo3oo3ox3oo&#xt. The second is a lace tower of the /6/ + 3/3, is quite clear in this picture. You can enumerate the coordinates of 5B/ from this diagram too, You need a half-coordinate to handle this. The figure-radius is 24 1/2 by this scale, so you could make the column left read 1/2 <24>, 4 1/2 <20>, 12 1/2 (3/2), and 24 1/2 ( 5). One then gets the good old stott matrix out and finds out what has a radius of 24/14, and 20/14. These are evidently have a vertex-sum of 6 and 2. It's not impossible, and one is not restricted to a single element. But i am not going into this here now.

[student91]

I have decided that there is room to create student91's request,

Hurray

although at a slightly different point. Don't worry, i did this to r.klitzing's notation too, because there are 'global' interests to protect.

I understand that, I hadn't expected my new term to be described perfectly the first time I would try. In fact I made this topic so we can help each other perfecting the new term

[...]
'z' is the appropriate symmetry for a point, since something like 'o' implies that it's a zero-length line (but lives in a 1d space). z is a node that here loops onto itself, and destroys any symmetry. In any case &#z can be read as 'zero-height/dimension' (added here).

Thatching

Ever since Hedrondude rolled out the regiments, i have been trying to find a term to make a non-convex hull. For example, the regiment (figures sharing vertices + edges) that contains {5/2,5} and {3,5/2}, the first of these is the 'most convex' of these. But you can't call it convex, and it's hardly a hull, so i decided instead that it needs a new word. That word is 'thatching', here 'edge-thatching', since the edges are kept proud. Vertex-thatching is the default meaning of thatch, so it does make a convex hull.

For the present case, you can use something like &#z as a suffix, since the edges will come from the stott-differences. If you want to further imply an edge, you can add that after this, ie &#zx.

I have to admit I don't fully understand what a thatching exactly is as far as I get it, an "edge-thatching" makes the most-convex hull of a set of edges, and a vertex-thatching makes the convex hull of a set of vertices. In that respect, vertex-thatching is exactly the same as taking the convex hull, if I'm right.
As far as I get it, the postfix &#zx means: &# there is a lacing, i.e. things get connected, rather than they are forming a compound, then z, which I think means there won't be a higher-dimensional tower nor a higher-dimensional simplex, but everything stays in the same dimension, and then x, which I'm not totaly sure of, but I guess that means the edge-length is x. Then the question arises, what edge-length, is it the edge-length described by definition [2]?

Lace Union

I suppose we could keep this term for a thatched compound, especially if it is expressed as a laced layer. I'm all ears, though.

If I'm right, this is pretty similar to definition [1]. Furthermore you seem to imply that the term "union" isn't used yet

Lace projections

[I guess this is only usefull for projection purposes?]

Lace towers

It is still appropriate to designate a series of layers from a tiling, as a lace tower, since we have ample tools now to determine the height.

Of course, but I just thought it to be superfluous to describe something that is more basic than a lace-tower as a lace-tower.

[...]
Thank you for your help.

Wendy

You're welcome

@student91:

You surely are free to start from existing concepts and build thereon your owns. We generally cannot hinder that. But you should be careful in doing so. Because we don't want any 2nd Babylonian confusion. You should always verify whether your ideas are unique and indeed new, and that those cannot be represented by the so far existing system.

That's why I made this topic, rather that just introducing a new lace-term out of the blue

[...]Just as the stacking of lace prisms already tells, the lacing edges of either segment then can have any arbitrary angle between them. They definitely are not bound to align. It is just the axis, which is assumed to be continued. In fact any lace prism has 2 parallel bases, each being perpendicular to the axis. Accordingly the bases and those intermediate pseudo facets of a tower too will be all perpendicular to the common axis, or, taken differently, are mutually parallel. - This latter wording also then will hold for degenerate towers too. If the segment heights would calculate as zero you clearly have no true axis. But still a base (or sectioning pseudo facet) normal. And therefore, in that special case, parallelity of bases (or sectioning pseudo facets) just reads as coplanarity (of corresponding dimension for sure).

What I was referring to with "angle" and "structure" was unclear. You could see the bilbiro-thing as a lace-city, with a lacing of x between two things, and a "lacing" of f as third connection. All these distances would become 0, so the things become thrown on a point. So the bilbiro-thing could be seen as both a deflated lace-tower as a deflated lace-city. In a lace-city, you can have an angle beween three things. (so there can be an angle between the lacings themselves, rather than between the egdes. It was this angle I was talking about)

But there is also a different feature in real lace towers. It is that not all neighbouring pairs of bases and/or pseudo facets ought to be conected directly by lacings of the assigned length. If the respective segment heights become small enough, there might be edges connecting one vertex layer e.g. to the one but next. And such cases might even intertwine, resulting in cases where neighbouring layers are not being laced at all, each just being connected to other ones. - Such an behaviour for sure becomes more frequent, the lower the respective heights would be. So it is not at all surprising, when you consider degenerate towers, i.e. zero heights.

Ah, okay, so lace towers can be much more complex than just a stack of lace-prisms. I can look at D4.10, and see the lace-tower is much more complex indeed. maybe, to clarify this complexity, we might introduce a "lacing matrix", which would show what things are connected to what. The "lacing matrix" of the bilbiro-compound would the be:

Code: Select all

f2o2o | * | x | f
x2x2f | x | * | x
o2F2x | f | x | *


You can see this matrix is symmetrical. for this particular case it's not that usefull, but for e.g. D4.10, quickfur could just give us such a matrix, instead of the ACSII-art. This is a tentative thing, though.

[...]
@Wendy:

I first thought about that new suffix "&zx(t)" being nothing but an optional clarification that all according heights there would evaluate as zero. Thus more the kind of being just reader-friendly.

But then I realized that bit mentioned above. Those there mentioned new degenerate structures definitely cannot be described as lace simplices, as not all lacings shall occur. Thus those need to be represented as lace towers at most. But then the 2 base facets ought to be real facets, if it would be a true lace tower, even when having some zero segment heights. But those examples use only pseudo facet layer descriptions!

Thus bilbiro only could be represented (in that advised symmetry representation) as fxo2oxF2ofx&#zxt, where this additional "z" then not only just point out that the heights happen to become zero, but also that all layers in fact are just pseudo facets.

Note that the final "t" still is needed here. Even so the first and third layer are corealmic too, and therefore shall have a height of zero, they are not to be laced. And thus the neglection of that "t" would be wrong.

I saw it as if there is a lacing between f2o2o and o2F2x, but this lacing has length f. This f-edge becomes part of bilbiro's pentagons. I hope that's not too creative use of lacings?

As far as I got it now, &#z is something pretty similar to what I wanted, and I can use this for what I need. However, I would still like some explanation on what &#z exactly does, esp. how it differs from &#zx.

Thanks for helping me out

student91

[wendy]

The difference between &#xz and and &z is quite large.

The lace union oo3oo4ox&#xz is from this link http://www.atomium.be . It shows a cube vertex-frist, with all of the vertices laced to the central node. So it's a kind of edge-cage, showing the projected 4d cube-pyramid as a flat thing in 3d. This allows one to see details against the height.

The lace union oo3oo4ox&#z, would just be the convex hull of this, which is the cube.

[Klitzing]

As far as I get it, the postfix &#zx means: &# there is a lacing, i.e. things get connected, rather than they are forming a compound, then z, which I think means there won't be a higher-dimensional tower nor a higher-dimensional simplex, but everything stays in the same dimension, and then x, which I'm not totaly sure of, but I guess that means the edge-length is x. Then the question arises, what edge-length, is it the edge-length described by definition [2]?

"x" indeed provides the length of the lacing edges.

What I was referring to with "angle" and "structure" was unclear. You could see the bilbiro-thing as a lace-city, with a lacing of x between two things, and a "lacing" of f as third connection. All these distances would become 0, so the things become thrown on a point.

Hmmm, again, the edges remain as they are. It is the height (of the respective segment) right by that size of lacing edges evaluates to zero. Thus you well shall write

• ike = fxo ofx xof&#zx
• bilbiro = fxo2ofx2oxF&#zxt
• pretasto = fxo3xoo3ofx *b3oxF&#zxt

Note, that the prefixing "z" here always is required, it is not just additional (hinting simply for zero heights). It is required as None of the "layers" acts as a true element of the respective figure.

So the bilbiro-thing could be seen as both a deflated lace-tower as a deflated lace-city. In a lace-city, you can have an angle beween three things. (so there can be an angle between the lacings themselves, rather than between the egdes. It was this angle I was talking about)

Again, it is not the "lacings" (aka: edges), but rather the axes (aka: position vectors of para-space).

Ah, okay, so lace towers can be much more complex than just a stack of lace-prisms. I can look at D4.10, and see the lace-tower is much more complex indeed. maybe, to clarify this complexity, we might introduce a "lacing matrix", which would show what things are connected to what. The "lacing matrix" of the bilbiro-compound would the be:

Code: Select all

f2o2o | * | x | f
x2x2f | x | * | x
o2F2x | f | x | *

You can see this matrix is symmetrical. for this particular case it's not that usefull, but for e.g. D4.10, quickfur could just give us such a matrix, instead of the ACSII-art. This is a tentative thing, though.

No, it is important to note that that degenerate bilbiro tower was suffixed by "&#zxt". Not only "z" (degenerate = height zero, no "layer" is a real element of the structure), not only "t" (only consecutive layers are laced, not the 1st to the 3rd), but also "x": only x scaled lacing edges are to be used. (In fact, this is what excludes the lacings between 1st and 3rd "layer".)

Note that the final "t" still is needed here. Even so the first and third layer are corealmic too, and therefore shall have a height of zero, they are not to be laced. And thus the neglection of that "t" would be wrong.

I saw it as if there is a lacing between f2o2o and o2F2x, but this lacing has length f. This f-edge becomes part of bilbiro's pentagons. I hope that's not too creative use of lacings?

Not at all. But an introduction of such further f-scaled edges definitively changes the incidence structure! I.e. you would be dealing with a different abstract polytope.

As far as I got it now, &#z is something pretty similar to what I wanted, and I can use this for what I need. However, I would still like some explanation on what &#z exactly does, esp. how it differs from &#zx.

Haha, this is one of Wendy's cryptix. In "&#z" she just neglects the lacings. In her original mail she wrote for that "&#xz". That is, she introduced a single qualifier "z", which has 2 different applications:

• Either it can be used as a prefix on lacings ("&#z.."). Then it just tells that all to be calculated heights would evaluate to zero. And that None of the given "layers" does contribute to the elements of the described structure.
• Or it is used as a postfix on the lacings ("&#..z"). Then it tells to produce the structure in a way, as if that trailing "z" won't be there. You even would be allowed to scale the lacing edge length up if needed, in order to get a true tower or simplex or whatever. And then, afterwards, the trailing "z" comes into action. Implying to take a telescope view, a perspective projection from infinity, i.e. ironing everything down into one single layer space. Thereby shortening the respective lacing edges as required. - That is, here you provide the segment height (to be zero) and re-calculate the corresponding lacing edge length therefrom. (In fact, this might be the reason, after all, that Wendy now neglects that "x" (from "&#xz"), as it contributes only to the first constructional step, but not to the outcome.)

(And further it should be mentioned - be warned! - that Wendy already uses that "z" in non-lacing contexts in the sense of what I would write "*a" for, i.e. in order to connect some linearized Dynkin symbol back to its first node.)

--- rk

[wendy]

Despite what Richard says, the consistancy is kept.

& is a product-marker that resets the counters. So, eg if polytopes A has nodes 1,2,3 and B has nodes 1,2,3, then A.2.B would have to rewrite the nodes to get 1,2,3,4,5,6, whereas A&B allows the bits on each side of the & to keep their own numbers, eg 1,2,3,& 1,2,3

# just means that a mirror is not in operation here. You need this mark to imply that a lace thing exists.

z is just a loop-node. When it occupies the first node, and there is only one node, it designates a symmetry of a point. Note that 'o' has the around-symmetry of a line. When it is the first of several, it creates a Hatch loop, which is a polygon of reflections, eg z2o2o2o2o2o is the rectanuglar group. The only polyhedron that needs a hatch node is the Miller monster, which is a snub.

In &#xz, it just means create a lace tower &#xt, and then project it to onto the plane (t->z).

Despite what richard is saying, i do try very hard to keep it within the original scope of the draft, which means, 'to allow a program to read the symbol and create the polytope.' quickfur's renders is partly in that way.

[Klitzing]

Lace towers

It is still appropriate to designate a series of layers from a tiling, as a lace tower, since we have ample tools now to determine the height. These are to be read as 0=o, 1=x, against o3o3o5o. So the inner shell (marked *), is o3x3o5o. From this to the x3o3x5x, there is a flat lace segmentochoron in &#xt. So the whole 'cage' comes up as a dozen polychora, laced together by unit lacings, but it's still flat. If you replaced it bt &#ft, it would stand upright as a kind of pyramid.

The letters 'a' to 'd' here are differences between layers, is used to verify that thing is flat.

Code: Select all

  1 0 0 0    10.472136
  0 1 0 0    37.885438  *
  0 0 0 1    54.832815  a   0 1 0 -1
  0 0 1 0    82.249223  b   0 0 1 -1
  1 1 0 0    86.249223  c   1 1 -1 0
  1 0 0 1   109.665631  a
  1 0 1 0   147.554175  b
  0 1 0 1   181.442719  d   1 -1 1 -1
  0 1 1 0   229.803398  b
  0 0 1 1   270.164078  a
  1 1 0 1   274.164078  c
  1 1 1 0   332.996894  b
  1 0 1 1   379.829710  a
  0 1 1 1   506.439634
  1 1 1 1   653.993788


This is really a nice coincidence for a rather long stack of degenerate lace prisms only, within that axial around symmetry o3o3o5o.

But it should be pointed out, that those therein mentioned few degenerate lace prisms are not all, what can be found in that very symmetry.
Rather we have the following list:

Code: Select all

hi || rahi       = oo3oo3ox5xo&#x
srahi || xhi     = oo3xx3ox5xo&#x
sidpixhi || srix = xx3oo3ox5xo&#x
prahi || grix    = xx3xx3ox5xo&#x

rox || hi        = oo3xo3oo5ox&#x
xhi || thi       = oo3xo3xx5ox&#x
tex || sidpixhi  = xx3xo3oo5ox&#x
grix || prix     = xx3xo3xx5ox&#x

rahi || sidpixhi = ox3oo3xo5ox&#x
xhi || prahi     = ox3xx3xo5ox&#x

rahi || tex      = ox3ox3xo5oo&#x
thi || prahi     = ox3ox3xo5xx&#x

srix || srahi    = xo3ox3xo5ox&#x


Those are ordered here (individually) according to smaller base atop larger base.
Further they are grouped by mutual Stott expansions.

For sure, by selection, their heights all evaluate to zero.

But note that the ending here is still "&#x", not "&#zx"!
This is because the layers define true facets (cells), rather than pseudo facets.

--- rk

[wendy]

One has the eleven-strata lace tower i give, and also that a+c is allowable, but like the lace-tower that exists in the {3,3,5}, one has extra unit struts that run over two layers. So a+c gives 1,0,-1,1 which has a length of 4 (ie an edge), while b+c gives 8 (sqrt 2).

But then, in the lace tower oxofOfoxo 3 ooooXoooo 3 ooxoOoxoo &xt, (the capitals are just highlights of the middle, to make it easier to read), there are unit edges from the rings 2 to 4 to 6, which is not given in the notation. But i did not pick up a+c gives a flat lace prism too.

[Klitzing]

Just considered "tip" = truncated pentachoron = x3x3o3o.
Its dihedral angles are known to be

• tet-3-tut = 104.478 degrees
• tut-6-tut = 75.522 degrees

On the other hand I considered "octatut", the segmentachoron K-4.52: oct || tut.
Its dihedral angles at the tut base are:

• trip-3-tut = 52.238 degrees
• tricu-6-tut = 75.522 degrees

Thus it well becomes possible to consider the according penta-augmented tip.


- And why do I place that find in this thread?
Well, the 5 octs at the top clearly are also arranged in the same pentachoral symmetry. Thus we have to consider the tegum-sum of x3x3o3o and some o3x3o3A, or in other words that penta-augmentation could be described alternatively by xo3xx3oo3oA&#zx.

A quick lookup by means of the well-known spreadsheet provides A : x = 1,5 here.

For the corresponding incidence matrix one derives

Code: Select all

xo3xx3oo3oA&#zx

o.3o.3o.3o.    | 20  * |  1  3  3  0 |  3  3  3  6  0  0 | 1  6  3 0
.o3.o3.o3.o    |  * 30 |  0  0  2  4 |  0  0  1  4  2  2 | 0  2  2 1
---------------+-------+-------------+-------------------+----------
x. .. .. ..    |  2  0 | 10  *  *  * |  3  0  3  0  0  0 | 0  6  0 0
.. x. .. ..    |  2  0 |  * 30  *  * |  1  2  0  2  0  0 | 1  2  2 0
oo3oo3oo3oo&#x |  1  1 |  *  * 60  * |  0  0  1  2  0  0 | 0  2  1 0
.. .x .. ..    |  0  2 |  *  *  * 60 |  0  0  0  1  1  1 | 0  1  1 1
---------------+-------+-------------+-------------------+----------
x.3x. .. ..    |  6  0 |  3  3  0  0 | 10  *  *  *  *  * | 0  2  0 0
.. x.3o. ..    |  3  0 |  0  3  0  0 |  * 20  *  *  *  * | 1  0  1 0
xo .. .. ..&#x |  2  1 |  1  0  2  0 |  *  * 30  *  *  * | 0  2  0 0
.. xx .. ..&#x |  2  2 |  0  1  2  1 |  *  *  * 60  *  * | 0  1  1 0
.o3.x .. ..    |  0  3 |  0  0  0  3 |  *  *  *  * 20  * | 0  1  0 1
.. .x3.o ..    |  0  3 |  0  0  0  3 |  *  *  *  *  * 20 | 0  0  1 1
---------------+-------+-------------+-------------------+----------
.. x.3o.3o.    |  4  0 |  0  6  0  0 |  0  4  0  0  0  0 | 5  *  * * tet
xo3xx .. ..&#x |  6  3 |  3  3  6  3 |  1  0  3  3  1  0 | * 20  * * tricu
.. xx3oo ..&#x |  3  3 |  0  3  3  3 |  0  1  0  3  0  1 | *  * 20 * trip
.o3.x3.o ..    |  0  6 |  0  0  0 12 |  0  0  0  0  4  4 | *  *  * 5 oct


--- rk

[Klitzing]

Silly me
tricu-6-tut = 75.522 degrees
tut-6-tut = 75.522 degrees
tut-6-tricu = 75.522 degrees
summing up to 226.567 degrees, which clearly is beyond 180 degrees.
Thus the above polychoron is a nice find, but escapes to be CRF.

Well, but a single such augmentation surely is CRF.
But that one then will be just an axial CRF, i.e. it would be in the wrong thread in here...

That one clearly is simply o3x3o || x3x3o || u3o3o || x3o3o  =  oxux-3-xxoo-3-oooo-&#xt
(I'll continue this one therefore there then.)

--- rk

[wendy]

Richard asks about the facets of 2_31 = 5/B, when this is presented as /6/ + 3/3. ie xo3oo3oo3ox3oo3oo3xo&#xz.

The 56 2_21 fall in two sets of 28, these follow the edges of the simplex and the invert of it, viz oo3ox3oo3oo3oo3oo3xo3oo = 1/.4./1

We consider the room formd by removing the first two nodes (ie .3.3 o3o3o3o3o). This is the axial symmetry orthgonal to the edge of a simplex.

/6/ appears here as 3o3o3o3o3x = 4/ || 4/ This is a prism. The 3/3 appears as a central 1/3 margin, the junction between faces from the same simplex. The lace tower 4/ || 1/3 || 4/ or ooo3oxo3ooo3ooo3xox&#xt makes the 2_21. There are 28 on each tetrahedra, and because the 2_21 /4B has 72 simplexs in 36 opposite pairs, the 36*16=4.96 simplexes are then accounted for, and show that there is a 36-fold colouring of the verticies of 1_32 that match the 36 diametric axies of 1_22.

We should also note that the double-bevel of the simplex, ie o3b3o3o3o3o3b3o, can not be uniquely restored to the generating figure, but instead gives rise to 72 distinct simplicies, and can be presented as the common intersection of such a figure. This is similar to the derivation of x3o4o3o from o3x3o4o, but having so derived the 24ch from the rectified 16ch, one can make three different 16ch from the same 24ch.

[Klitzing]

Richard asks about the facets of 2_31 = 5/B, when this is presented as /6/ + 3/3. ie xo3oo3oo3ox3oo3oo3xo&#xz.

Well, had solved that laq representation already, see below:

Code: Select all

xo3oo3oo3ox3oo3oo3xo&#x   → height = 0
(pseudo suph || pseudo he)

o.3o.3o.3o.3o.3o.3o.       | 56  * |  12   20   0 |  30  120   90    0 |  40  120   360  120   0 |  30  40  180  120  240   30   0 |  12  30  120  30   60 |  2 12  30
.o3.o3.o3.o3.o3.o3.o       |  * 70 |   0   16  16 |   0   48  144   48 |   0   32   288  288  32 |   0   8   96  144  288   96   8 |   0  24   96  36   96 |  0 12  32
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x. .. .. .. .. .. ..     & |  2  0 | 336    *   * |   5   10    0    0 |  10   20    30    0   0 |  10  10   30   10   20    0   0 |   5  10   20   5    5 |  1  6   5
oo3oo3oo3oo3oo3oo3oo&#x    |  1  1 |   * 1120   * |   0    6    9    0 |   0    6    36   18   0 |   0   2   18   18   36    6   0 |   0   6   18   9   12 |  0  6   6
.. .. .. .x .. .. ..       |  0  2 |   *    * 560 |   0    0    9    6 |   0    0    18   36   6 |   0   0    6   18   36   18   2 |   0   6   12   9   18 |  0  6   6
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o. .. .. .. .. ..     & |  3  0 |   3    0   0 | 560    *    *    * |   4    4     0    0   0 |   6   4    6    0    0    0   0 |   4   6    4   0    0 |  1  4   1
xo .. .. .. .. .. ..&#x  & |  2  1 |   1    2   0 |   * 3360    *    * |   0    2     6    0   0 |   0   1    6    3    6    0   0 |   0   3    6   3    2 |  0  4   2
.. .. .. ox .. .. ..&#x    |  1  2 |   0    2   1 |   *    * 5040    * |   0    0     4    4   0 |   0   0    2    4    8    2   0 |   0   2    4   4    4 |  0  4   2
.. .. .o3.x .. .. ..     & |  0  3 |   0    0   3 |   *    *    * 1120 |   0    0     0    6   2 |   0   0    0    3    6    6   1 |   0   3    2   3    6 |  0  4   2
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o.3o. .. .. .. ..     & |  4  0 |   6    0   0 |   4    0    0    0 | 560    *     *    *   * |   3   1    0    0    0    0   0 |   3   3    0   0    0 |  1  3   0
xo3oo .. .. .. .. ..&#x  & |  3  1 |   3    3   0 |   1    3    0    0 |   * 2240     *    *   * |   0   1    3    0    0    0   0 |   0   3    3   0    0 |  0  3   1
xo .. .. ox .. .. ..&#x  & |  2  2 |   1    4   1 |   0    2    2    0 |   *    * 10080    *   * |   0   0    1    1    2    0   0 |   0   1    2   2    1 |  0  3   1
.. .. oo3ox .. .. ..&#x  & |  1  3 |   0    3   3 |   0    0    3    1 |   *    *     * 6720   * |   0   0    0    1    2    1   0 |   0   1    1   2    2 |  0  3   1
.. .o3.o3.x .. .. ..     & |  0  4 |   0    0   6 |   0    0    0    4 |   *    *     *    * 560 |   0   0    0    0    0    3   1 |   0   3    0   0    3 |  0  3   1
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o.3o.3o. .. .. ..     & |  5  0 |  10    0   0 |  10    0    0    0 |   5    0     0    0   0 | 336   *    *    *    *    *   * |   2   1    0   0    0 |  1  2   0
xo3oo3oo .. .. .. ..&#x  & |  4  1 |   6    4   0 |   4    6    0    0 |   1    4     0    0   0 |   * 560    *    *    *    *   * |   0   3    0   0    0 |  0  3   0
xo3oo .. ox .. .. ..&#x  & |  3  2 |   3    6   1 |   1    6    3    0 |   0    2     3    0   0 |   *   * 3360    *    *    *   * |   0   1    2   0    0 |  0  2   1
xo .. oo3ox .. .. ..&#x  & |  2  3 |   1    6   3 |   0    3    6    1 |   0    0     3    2   0 |   *   *    * 3360    *    *   * |   0   1    0   2    0 |  0  3   0
xo .. .. ox3oo .. ..&#x  & |  2  3 |   1    6   3 |   0    3    6    1 |   0    0     3    2   0 |   *   *    *    * 6720    *   * |   0   0    1   1    1 |  0  2   1
.. oo3oo3ox .. .. ..&#x  & |  1  4 |   0    4   6 |   0    0    6    4 |   0    0     0    4   1 |   *   *    *    *    * 1680   * |   0   1    0   0    2 |  0  2   1
.o3.o3.o3.x .. .. ..     & |  0  5 |   0    0  10 |   0    0    0   10 |   0    0     0    0   5 |   *   *    *    *    *    * 112 |   0   3    0   0    0 |  0  3   0
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o.3o.3o.3o. .. ..     & |  6  0 |  15    0   0 |  20    0    0    0 |  15    0     0    0   0 |   6   0    0    0    0    0   0 | 112   *    *   *    * |  1  1   0  hix
xo3oo3oo3ox .. .. ..&#x  & |  5  5 |  10   20  10 |  10   30   30   10 |   5   20    30   20   5 |   1   5   10   10    0    5   1 |   * 336    *   *    * |  0  2   0  tac
xo3oo .. ox3oo .. ..&#x  & |  3  3 |   3    9   3 |   1    9    9    1 |   0    3     9    3   0 |   0   0    3    0    3    0   0 |   *   * 2240   *    * |  0  1   1  hix
xo .. oo3ox3oo .. xo&#zx   |  4  6 |   4   24  12 |   0   24   48    8 |   0    0    48   32   0 |   0   0    0   16   16    0   0 |   *   *    * 420    * |  0  2   0  tac
xo .. .. ox3oo3oo ..&#x  & |  2  4 |   1    8   6 |   0    4   12    4 |   0    0     6    8   1 |   0   0    0    0    4    2   0 |   *   *    *   * 1680 |  0  1   1  hix
---------------------------+-------+--------------+--------------------+-------------------------+---------------------------------+-----------------------+----------
x.3o.3o.3o.3o.3o. ..     & |  7  0 |  21    0   0 |  35    0    0    0 |  35    0     0    0   0 |  21   0    0    0    0    0   0 |   7   0    0   0    0 | 16  *   *  hop
xo3oo3oo3ox3oo .. xo&#zx & | 12 15 |  36  120  60 |  40  240  360   80 |  30  120   540  360  30 |  12  30  120  180  240   60   6 |   2  12   40  15   30 |  * 56   *  jak
xo3oo .. ox3oo3oo ..&#x  & |  3  4 |   3   12   6 |   1   12   18    4 |   0    4    18   12   1 |   0   0    6    0   12    3   0 |   0   0    4   0    3 |  *  * 560  hop

--- rk

[Klitzing]

... The lace tower 4/ || 1/3 || 4/ or ooo3oxo3ooo3ooo3xox&#xt makes the 2_21. There are 28 on each tetrahedra, and because the 2_21 /4B has 72 simplexs in 36 opposite pairs, ...

cf.

Code: Select all

xox3ooo3ooo3oxo3ooo&#xt   → both heights = 1/2 = 0.5
(hix || inv pseudo rix || hix)

o..3o..3o..3o..3o..     & | 12  * |  5  10 1  0 | 10  30  30 10  0  0 | 10  30  60  20  10  30  0 |  5 10  30  20  20  5  20 10 0 | 1  5 10  5  5
.o.3.o.3.o.3.o.3.o.       |  * 15 |  0   8 0  8 |  0  12  48  4 12  4 |  0   8  48  48  24  24  8 |  0  2  16  24  24 16  24 12 2 | 0  4  8  6  8
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x.. ... ... ... ...     & |  2  0 | 30   * *  * |  4   6   0  0  0  0 |  6  12  12   0   0   0  0 |  4  6  12   4   4  0   0  0 0 | 1  4  4  1  0
oo.3oo.3oo.3oo.3oo.&#x  & |  1  1 |  * 120 *  * |  0   3   6  1  0  0 |  0   3  12   6   3   6  0 |  0  1   6   6   6  2   6  3 0 | 0  2  3  3  2
o.o3o.o3o.o3o.o3o.o&#x    |  2  0 |  *   * 6  * |  0   0   0 10  0  0 |  0   0   0   0   0  30  0 |  0  0   0   0   0  0  20 10 0 | 0  0  0  5  5
... ... ... .x. ...       |  0  2 |  *   * * 60 |  0   0   6  0  3  1 |  0   0   6  12   6   3  3 |  0  0   2   6   6  6   6  3 1 | 0  2  2  3  3
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x..3o.. ... ... ...     & |  3  0 |  3   0 0  0 | 40   *   *  *  *  * |  3   3   0   0   0   0  0 |  3  3   3   0   0  0   0  0 0 | 1  3  1  0  0
xo. ... ... ... ...&#x  & |  2  1 |  1   2 0  0 |  * 180   *  *  *  * |  0   2   4   0   0   0  0 |  0  1   4   2   2  0   0  0 0 | 0  2  2  1  0
... ... ... ox. ...&#x  & |  1  2 |  0   2 0  1 |  *   * 360  *  *  * |  0   0   2   2   1   1  0 |  0  0   1   2   2  1   2  1 0 | 0  1  1  2  1
ooo3ooo3ooo3ooo3ooo&#xt   |  2  1 |  0   2 1  0 |  *   *   * 60  *  * |  0   0   0   0   0   6  0 |  0  0   0   0   0  0   6  3 0 | 0  0  0  3  2
... ... .o.3.x. ...       |  0  3 |  0   0 0  3 |  *   *   *  * 60  * |  0   0   0   4   0   0  2 |  0  0   0   2   0  4   2  0 1 | 0  2  0  1  2
... ... ... .x.3.o.       |  0  3 |  0   0 0  3 |  *   *   *  *  * 20 |  0   0   0   0   6   0  0 |  0  0   0   0   6  0   0  3 0 | 0  0  2  3  0
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x..3o..3o.. ... ...     & |  4  0 |  6   0 0  0 |  4   0   0  0  0  0 | 30   *   *   *   *   *  * |  2  1   0   0   0  0   0  0 0 | 1  2  0  0  0
xo.3oo. ... ... ...&#x  & |  3  1 |  3   3 0  0 |  1   3   0  0  0  0 |  * 120   *   *   *   *  * |  0  1   2   0   0  0   0  0 0 | 0  2  1  0  0
xo. ... ... ox. ...&#x  & |  2  2 |  1   4 0  1 |  0   2   2  0  0  0 |  *   * 360   *   *   *  * |  0  0   1   1   1  0   0  0 0 | 0  1  1  1  0
... ... oo.3ox. ...&#x  & |  1  3 |  0   3 0  3 |  0   0   3  0  1  0 |  *   *   * 240   *   *  * |  0  0   0   1   0  1   1  0 0 | 0  1  0  1  1
... ... ... ox.3oo.&#x  & |  1  3 |  0   3 0  3 |  0   0   3  0  0  1 |  *   *   *   * 120   *  * |  0  0   0   0   2  0   0  1 0 | 0  0  1  2  0
... ... ... oxo ...&#x    |  2  2 |  0   4 1  1 |  0   0   2  2  0  0 |  *   *   *   *   * 180  * |  0  0   0   0   0  0   2  1 0 | 0  0  0  2  1
... .o.3.o.3.x. ...       |  0  4 |  0   0 0  6 |  0   0   0  0  4  0 |  *   *   *   *   *   * 30 |  0  0   0   0   0  2   0  0 1 | 0  2  0  0  1
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x..3o..3o..3o.. ...     & |  5  0 | 10   0 0  0 | 10   0   0  0  0  0 |  5   0   0   0   0   0  0 | 12  *   *   *   *  *   *  * * | 1  1  0  0  0
xo.3oo.3oo. ... ...&#x  & |  4  1 |  6   4 0  0 |  4   6   0  0  0  0 |  1   4   0   0   0   0  0 |  * 30   *   *   *  *   *  * * | 0  2  0  0  0
xo.3oo. ... ox. ...&#x  & |  3  2 |  3   6 0  1 |  1   6   3  0  0  0 |  0   2   3   0   0   0  0 |  *  * 120   *   *  *   *  * * | 0  1  1  0  0
xo. ... oo.3ox. ...&#x  & |  2  3 |  1   6 0  3 |  0   3   6  0  1  0 |  0   0   3   2   0   0  0 |  *  *   * 120   *  *   *  * * | 0  1  0  1  0
xo. ... ... ox.3oo.&#x  & |  2  3 |  1   6 0  3 |  0   3   6  0  0  1 |  0   0   3   0   2   0  0 |  *  *   *   * 120  *   *  * * | 0  0  1  1  0
... oo.3oo.3ox. ...&#x  & |  1  4 |  0   4 0  6 |  0   0   6  0  4  0 |  0   0   0   4   0   0  1 |  *  *   *   *   * 60   *  * * | 0  1  0  0  1
... ... ooo3oxo ...&#x    |  2  3 |  0   6 1  3 |  0   0   6  3  1  0 |  0   0   0   2   0   3  0 |  *  *   *   *   *  * 120  * * | 0  0  0  1  1
... ... ... oxo3ooo&#x    |  2  3 |  0   6 1  3 |  0   0   6  3  0  1 |  0   0   0   0   2   3  0 |  *  *   *   *   *  *   * 60 * | 0  0  0  2  0
.o.3.o.3.o.3.x. ...       |  0  5 |  0   0 0 10 |  0   0   0  0 10  0 |  0   0   0   0   0   0  5 |  *  *   *   *   *  *   *  * 6 | 0  2  0  0  0
--------------------------+-------+-------------+---------------------+---------------------------+-------------------------------+--------------
x..3o..3o..3o..3o..     & |  6  0 | 15   0 0  0 | 20   0   0  0  0  0 | 15   0   0   0   0   0  0 |  6  0   0   0   0  0   0  0 0 | 2  *  *  *  *  hix
xo.3oo.3oo.3ox. ...&#x  & |  5  5 | 10  20 0 10 | 10  30  30  0 10  0 |  5  20  30  20   0   0  5 |  1  5  10  10   0  5   0  0 1 | * 12  *  *  *  tac
xo.3oo. ... ox.3oo.&#x  & |  3  3 |  3   9 0  3 |  1   9   9  0  0  1 |  0   3   9   0   3   0  0 |  0  0   3   0   3  0   0  0 0 | *  * 40  *  *  hix
xox ... ooo3oxo3ooo&#xt   |  4  6 |  2  24 2 12 |  0  12  48 12  4  4 |  0   0  24  16  16  24  0 |  0  0   0   8   8  0   8  8 0 | *  *  * 15  *  tac
... ooo3ooo3oxo ...&#x    |  2  4 |  0   8 1  6 |  0   0  12  4  4  0 |  0   0   0   8   0   6  1 |  0  0   0   0   0  2   4  0 0 | *  *  *  * 30  hix


--- rk

[wendy]

The real question now is how the group of 36 diameters of 4B/ works. It is the same subgroup that defines how the 36 sets of vertex-diametric simplexes in 5/B work.

I'm trying at the moment to see how it relates to the complex group 3h_3 (the nodes marked '2' in 2(3)2(4)3 3h_3r_3 or 2(4)3(3)3 3h_3c). Coxeter does not identify h_3 as a primitive group in 'regular complex polytopes', but it seems that it is. It has an order of 54, but permutates the 72 vertices of the former, this figure is 1_22 projected onto CE3.

The group is connected to some form of 'tri-centred tri-triangular prism' tiling.

[Klitzing]

The real question now is how the group of 36 diameters of 4B/ works.

So you are asking for the arrangement of the 72 vertices of mo = 1_2,2, ain't you?
For that purpose this lace city display of mo might serve useful:

Code: Select all

        o           -- o3o3o3o3o (point)
                 
    r       R       -- o3o3x3o3o (dot)
                 
P       S       p   -- x3o3o3o3x (scad)
                 
    r       R       -- o3o3x3o3o (dot)
                 
        o           -- o3o3o3o3o (point)

            \       \       \
              \       \       +-   x3o3o *b3o3o (hin)
                \       +-------   o3o3o *b3x3o (rat)
                  +-------------   o3o3x *b3o3o (alt. hin)

where:
o = o3o3o3o (point)
r = o3x3o3o (rap)
R = o3o3x3o (inv. rap)
p = x3o3o3o (pen)
P = o3o3o3x (dual pen)
S = x3o3o3x (spid)

--- rk

[wendy]

Those of us who use the tri-triangular coordinate system, would of course, use this projection of mo=1_22. It shows one of the girthing hexagons.

It's also one of the presentations of the regular complex-polytope 3{3}3{4}2, which has 72 vertices and 54 faces (along with the necessary 216 edges).

This symmetry shares a subgroup with E6, although it does strange things that E6 does not.

Code: Select all

   
         o
     
   o   A   V   o      o = o3o2o3o   point
                     
     V   +   A        A = o3x2o3x   bitriangle prism, 
                      V   x3o2x3o   bitriangle prism, centrally inverted
   o   A   V   o
                      +   m3m2m3m   bihexagonal tegum 
         o
   

[Klitzing]

Oh, nice representation of mo = 1_2,2.

Well, in fact, both its vertical sections, i.e. the thus provided lace towers for dot = o3o3x3o3o = "o A V o" and that for scad = x3o3o3o3x = "V + A", where already known to me. Thus I just should have stacked these ...

"A" = "bitriangular prism" = x3o x3o = triangular duoprism = triddip
"+" = "bihexagonal tegum" = xo3xo ox3ox&#zq

--- rk

[Klitzing]

Thought about oct (octahedron) = xo3ox&#x, height = sqrt(2/3) = 0.816497
Then tridafup (triangular duoantifastegiaprism) = xo3ox xo3ox&#x, height = 1/sqrt(3) = 0.577350
And then I found that xo3ox xo3ox xo3ox&#x would be degenerate, as its height happens to be 0.

But, we have that "new lace term". Accordingly we could investigate xo3ox xo3ox xo3ox&#zx instead.

So far I'm not fully through with it.
But it appears to be quite interesting. - Why?
Well, it is the tegum sum of 2 tri-dual trittips (x3o x3o x3o). - But then?


Reconsider the nice lace city, Wendy depicted just 2 mails before in this thread.
This appealing representation of mo = 1_22 = o3o3o3o3o *c3x can be given as a very interesting 6-dim tegum sum:
mo = hull( x3o x3o x3o,  o3x o3x o3x,  x3x o3o o3o,  o3o x3x o3o,  o3o o3o x3x ) = xoxoo3oxxoo xooxo3oxoxo xooox3oxoox&#zx,
and obviously the thingy considered above just consists of the first to "layers" of that mo representation only.
In other words, it happens to be a diminishing of mo. In fact, exactly the vertices of those 3 mutually orthogonal hexagons ought be choped off!

Thus the lace city of this thingy likewise could be obtained from the provided one of mo:

Code: Select all

  A   V       
              
V       A        A = o3x o3x = triddip, 
                 V = x3o x3o = bidual triddip
  A   V       


--- rk

[wendy]

The figure xo3ox xo3ox xo3ox &#xz + tri-hexagon tegum = 1_22.

[Klitzing]

The figure xo3ox xo3ox xo3ox &#xz + tri-hexagon tegum = 1_22.

as I already said:

This appealing representation of mo = 1_22 = o3o3o3o3o *c3x can be given as a very interesting 6-dim tegum sum:
mo = hull( x3o x3o x3o,  o3x o3x o3x,  x3x o3o o3o,  o3o x3x o3o,  o3o o3o x3x ) = xoxoo3oxxoo xooxo3oxoxo xooox3oxoox&#zx

--- rk