Dynkin symbols for generalized Coxeter domains

Higher-dimensional geometry (previously "Polyshapes").

Re: Dynkin symbols for generalized Coxeter domains

Postby wendy » Sun Oct 13, 2013 6:59 am

At the moment, i am looking at non-crossing chordals, and whether it is mayly to remove redundant cross-chordals.

Cross-chordals is one of those tar pits that CT has and CD doesn't. Unlike CD, one has to mess with the cross-chordals, because there are things that come only by them. But it needs time to sit by the paper and do all of these archifolds etc.

Another device that is causing me destress, is that the group formed by red x-lines and blue y-lines, exists not just as xx, but as 'n xx', and that for the horogon, the value of n ought be '2', not the indicated '1'. Maybe we're not getting the nature of 'xx' (mirricles) to work properly in the dCT.

Little work has been done on the laws of symmetry, outside things like p p p => 3 p, but i suppose i will have to find conways costings to see if this makes sense too.

W
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Re: Dynkin symbols for generalized Coxeter domains

Postby Klitzing » Sun Oct 13, 2013 10:40 pm

wendy wrote:I do quite agree with you in looking inside the orbifold for interesting things. I just don't agree with you over using a symbol to state that two unconnected symbols have no common interaction. Orbifolds are not CD diagrams. They're way more flexiable. Using something like ø might not actually reflect a zero-angle in any case, and it's inappropriate to connect such a thing to say, a cone.


Ah, I might get your issue, here. "Ø" is not meant as a zero of informatics (crossed zero), but really as empty-sign. Empty because there is no link in the sense of connection of mirrors. Well, I could kind of leave the link out. But in fact this is done already for the links marked 2. And I didn't want to add to babylonean trubbles. So I searched for a different Symbol, which explicitely express that non-linkage.

Thus, using this sign, I no longer am dealing a CD Symbol. It became a generalization thereof.

I suppose my remark was a little harsh, though.

No blessures left. :)

... At the moment, the group 2 2 * is producing some interesting results. You just have to 'colour the edges' and pretend that 2 is really 32 or something.

A recent figure showed up in the previous discussion, but i suppose we already have % to handle this. Something like [13] [24] can described the rulled pages up and down in red, and across in blue, but there are similar things in {6,4} u, {8,4}, so it supposes that the real form is [13][24] 2% or 3% or 4%, where this wrap is entirely unexpected.

There is much to do here. It's nowhere near prime time like CD is.

My Impression too is that you get lot more possibles of color spaces within hyperbolics. And that thes to be added symmetry elements reside mostly in that realm.

--- rk

PS: Wendy, I've been off the last weekend, about 450 km to the north, near Paderborn for a 3 day square dance special. We were about 600 dancers, having a lot of fun. One of the invited callers came from down under, in fact directly from Brisbane (Jason Dean)- the world seems small...
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Re: Dynkin symbols for generalized Coxeter domains

Postby wendy » Mon Oct 14, 2013 7:24 am

The actual space the orbifold does not really change things.

You can generally group the tetrahedron, the antiprisms 2 2 p, and the snubs 2 p q, under the tetrahedron 2 2 2 if you use the same colour for the opposite regions. Anything that is grouped under 'digon', can be replaced by 'polygon', so a red edge becomes variously, a red square, triangle, etc.

There is outside-wrap too. You can cut out a 'p' from two different forms, and glue the two halves together. This is what's essentially happening at the kinds of edges where i use ø-edges. You need to track these edges because they bound active regions, and so if there are three active points inside a block, you get triangles. The behaviour of this is generally understood.

This is the current heirachy of the dCT. It follows the bowers army thing, in that finer detail adds more constraints.

1. Army This us the undecorated CT, like *2 3 5 or 2 2 * 6 4, this is more a mathematical heading, rather than a real pen-on-paper group. None the less JHC has attributed to these a 'cost', which is some kind of area measurement.

2. Regiment, this is an ordered representation of the army, which does not traslate to other ordered representations. So 3 5 8 * and 8 5 3 * are the same regiment, but 5 3 8 * is a different regiment. The regiment is home to the passive elements, which work much the same way as CD things, but there are more possibilities. Where things get different, is that CT allows for 'active regions' and so forth.

3. Company, this is a partition of regiments into areas of common active regions. This is where i hope to find home for your null-set branch. ø, but i am not sure how many different null branches are allowed, ie are there null branches of the archiform [a,b], or are they all {a,b}? It is known that some null branches can be detected from the orbifold.

Paderborn is one of those towns that turns up often in the magazines i read. They're mostly railway magazines, because they have pretty pictures, and one can get involved at varying depths - useful when one is on alert. Apparently there was a railway workshop there. But i am happy that ye had much fun up there.
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Re: Dynkin symbols for generalized Coxeter domains

Postby Klitzing » Tue Oct 15, 2013 10:09 pm

My propagated idea of CD generalization is quite easy and well versatile:

  • Take an arbitrary fundamental domain of reflectional symmetry (not necessary a simplex). To derive the diagram just map its mirroring facets onto nodes of the graph, connect those nodes pairwise with links, and asign to those links either numbers (the submultiples of pi of the dihedral angle) or the symbol Ø if those 2 nodes would not intersect directly ("empty section").

  • Polytopes then can be derived from those generalized CD diagtrams by asigning ringed nodes throughout. Further one can replace ringed nodes by unringed ones, as long as not both sides of a Ø-marked link would become unringed.

Lately I've thus calculated lot of incidence matrices for hyperbolic tilings and honeycombs for symmetries with non-simplicial domains: tetragonal ones, square pyramidal ones, and today even with octahedral domain.

Sure the sheer number of links soon gets unwieldy, esp. in a linearized representation. Nonetheless it works out right as before. For sure you would have to consider in the derivation of the incidence matrices, that Ø-links do not contribute an own polygon.

E.g. the "omnitruncated" member of the symmetry with an octahedral domain with all right dihedral angles comes out to be equivalent to x4o4o3o. Just that the latter unites all edges into one single class, whereas in that former those fall into 8 separate ones: the 8 emanating edges of every vertex here are to be considered colored differently.

--- rk
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Re: Dynkin symbols for generalized Coxeter domains

Postby wendy » Wed Oct 16, 2013 8:10 am

The group x5o3o4o contains a subgroup of order 120, formed by the 12 faces of the dodecahedra. This can be divided into a subgroup of order 5, by dividing the dodecahedron into octants, (ie 2 2), and then a rotation around the vertex-centre axis. The resulting group can be divided into 5 [5,3,4] in two different ways.

The smallest mirror-group of order 15, has a triangular antitegum, somewhat offset. This means that it has six mirrors.

What J Conway is telling us is that polyhedra with preset angles, such as the right-angle octahedron, have no aspects. A uniform figure either works or it diesn't.

One example of this would be o8o3o4o, where the peak opposite the last node might be truncated with an additional mirror.
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Re: Dynkin symbols for generalized Coxeter domains

Postby Klitzing » Tue Oct 22, 2013 8:56 am

Meanwhile I made some progress wrt to that recent mail of mine:

Klitzing wrote:Hy Wendy (or others, which would like to pop in),

meanwhile I've worked out that Ø symbol extension of the Dynkin diagram logics a bit deeper, considered its implications, its to be applied "techniques", the difference to the already in use ∞ symbol. I'll provide below how far I did come.

[...]

So here is how I do introduce that Ø sign in my current private copy of the future update of my Webpage on hyperbolics with Coxeter domains, dealing with 2D cases at that point:
Gnerelized Dynkin diagrams
[...]


No changes within that (snipped) part so far. (For its full version cf. to the original mail or its meanwhile already online version at my website.)

Progress was made in the following part, which thus is given here in its current, now mostly rewised, but not-yet-online version:
Here that word "theory" then will refer to a further outline of its usage, which will follow a bit down on that page, then concerning 3D cases, using for first examplifying toy the general symmetry
Code: Select all
  o---Ø---o 
   \     /   
    p   q   
     \ /     
      o     
     / \     
    s   r   
   /     \   
  o---Ø---o 

To get an understanding what happens in case of 3D Coxeter domains, we consider first explicitely the general case of the to the right being shown symmetry, with has the linearised notation o-p-o-Ø-o-q-*a-r-o-Ø-o-s-*a. The domain here obviously is a (not necessarily straight) square pyramid: the base edges are the 4 not shown connections of pairs of nodes, representing links with mark numbers of 2; the diagonals of that base square then are those links with the non-incidence symbols Ø; the lacing edges finally are those links marked p,q,r,s.

First of all we have to ask: what are the subsymmetries? – For simplicial domains the answer was easy: just omit any single node together with its incident links; the remainder diagram would be the facet symmetry. Here we have to be a bit more careful! The facet symmetries of the to be derived polytopes clearly are that ones, which are provided by the vertex figures of the domain: Those vertex figures do intersect orthogonally with the intersections of domain boundaries, and therefore reduce to a similar problem in one dimension less. So, in our case of a pyramid, those are that of the tip (thus still omitting a single node: that one, which represents the base), and any of the base corners, all resulting in triangles, which each do omit more than just a single node! So we get in our example the subsymmetry o-Ø-o o-Ø-o (which happens to have euclidean curvature and therefore the ominous pseudo bollogons here become pseudo horogons, that is we could replace the Ø symbol here also by the ∞ symbol). Obviously the at the right displayed general symmetry therefore needs to be at least paracompact. The other subsymmetries here are given by o-p-o-r-o, o-q-o-s-o, o-q-o-r-o, and o-p-o-s-o (which, depending on the numbers p,q,r,s might make the total symmetry even a hypercompact one).

Next we have to ask for possible decorations of such generalized Dynkin diagrams. Sure, the omnitruncation always is possible, i.e. applying the x node symbol everywhere. But already in 2D we had seen, that not all applications of the o node symbols where allowed. – Considering first the vertex figure of the tip of the at the right displayed examplifying pyramidal domain, we do know the answer already: this is because of o-Ø-o o-Ø-o = o-∞-o o-∞-o (as described above). Thus we are back to A1×A1 symmetry, i.e. a reducible symmetry group (with disconnected graph). There we have to use at least one x node per component. – And the rules of the theorem of 2D effectively likewise ask, that at least one end of those links, marked by Ø signs, have to bear an x decoration. In other words: at most one end may carry an o decoration!

In order to go even beyond this special case (of our example), we have once more to take refuge to the usual on/off explanation of the decorations for that purpose (of possible decorations). An x node is used in kaleidoscopical construction for a seed point lying off that special mirror plane, while an o node is used, if the seed point is incident to (i.e. on) the mirror. Therefore, the omnitruncate always has all nodes decorated by x nodes, because its seed point is placed completely within the domain, that is off from all mirrors. On the other hand the seed point clearly can be placed for sure at the inner part of any of the bounding facets of the domain. Then it is simply on that specific mirror, but off from all others. This results in the decoration by a single o node. Further the seed point can be placed at the intersection of two neighbouring boundary facets, etc. just as for normal Dynkin diagrams. But obviously with respect to a pair of non-intersecting facets (of the domain boundary), it becomes obvious, that the seed point has to be off from the one, if it wants to be on the other. This is the intrinsic reason for the rule provided at the end of the former paragraph.

Still, there will be an exception to that rule, so. What about placing the seed point at the tip of our pyramidal domain of consideration? This is obviously an allowed position too. Then this seed point is nonetheless on all mirror planes of this domain vertex. That is, the subsymmetry diagram (i.e. restricted to the vertex figure of this vertex) thus has only o nodes, independent of therein contained Ø links.

If finally the question comes to the facets of a polytope with Coxeter domain, then those can be read from the generalized Dynkin diagram in the very same way, as for non-generalized ones: they are the corresponding polytopes according to the provided decorated subgraphs, restricted as in the afore mentioned subsymmetries.


[...]
--- rk


:arrow: I.e. we have 2 major changes in here
(sorry for my privious disorientation :oops: ):
  • subsymmetries are derived from the vertex figures of the domain (not from the facets - sure, in case of simplices those would be equivalent). It is the vertex figure which cuts the dihedral angles of the facet intersections orthogonally, and thus brings that very angle down one dimension.
  • The rule on a-Ø-b, stating so far only that at least one of those node signs a,b has to be an x (ringed node), now gets weakend by the exception, that all nodes of some subsymmetry might get simultanuously decorated by an o sign (unringed node), independently of therein contained Ø links.
    E.g., sticking to the provided example symmetries, x3oØo3*a3oØo3*a would be a valid decoration too.
--- rk
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Re: Dynkin symbols for generalized Coxeter domains

Postby wendy » Tue Oct 22, 2013 9:02 am

I just don't like calling them CD diagrams, because there is no underlying Lie group, which is what the Dynkins part of the name means.

I really have no problems with non-simplex groups. Most of the symmetries in hyperbolic geometry, like {6,3,4}, end up in non-simplex groups. In fact, the cells of {4,3,6} and {3,3,6} form symmetry cells, one is a subgroup of the other, by an order of 5. You can see this, because it is mayly to slice of the alternate corners of a cube, and get five tetrahedra. At this scale, they're all regular. Any subgroup of the tetrahedron automatically generates a corresponding subgroup of the cube.

Beats counting incidence groups, which is how i found the subgroup of order 5.
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Re: generalized Dynkin symbols for Coxeter domains

Postby Klitzing » Tue Oct 22, 2013 10:18 am

So you might want to call those generalized Coxeter diagrams instead?
For he not only was the one who applied Dynkin's graphs onto polytopes,
even those more general domains are associated with his name.

Btw. I've to admit that in the title of this thread the word "generalized" really was missplaced.
It rather should have been situated at the first place instead.
(I've fixed it for this reply. But I don't know how to do that globally.)

--- rk
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