Tree Graphs of A000669

Discussion of shapes with curves and holes in various dimensions.

Tree Graphs of A000669

Postby ICN5D » Sun May 08, 2016 1:48 am

Here's something I put together last couple of days. It's a graphic of toratope notation, along with the equation, and some other stuff.

R^2 to R^4 : http://i.imgur.com/VubABFd.png

R^5 : http://i.imgur.com/wTMGsYV.png
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Tree Graphs of A000669

Postby Marek14 » Sun May 08, 2016 8:38 pm

I like it. The multiindexing is a nice idea.
Marek14
Pentonian
 
Posts: 1102
Joined: Sat Jul 16, 2005 6:40 pm

Re: Tree Graphs of A000669

Postby ICN5D » Sun May 08, 2016 11:51 pm

Yeah, I remember PWrong writing a similar format, and I liked the way it showed the tree height, and which node was branched. I also tried to define them with a fiber bundle notation to describe embeddings.

I left the tigroids blank, since I wasn't sure how to describe them. I'm learning that a tiger will be S^1 x S^1 x S^1 just the same as a 3-torus. A topologist doesn't care about geometrical differences. Maybe the homology groups would be specific enough to distinguish them, but I'm not sure. I plan on doing all of R^6 , and just the 13 'interesting' ones in R^7 , and maybe the 30 in R^8 .

I'm encouraging PWrong to finish his paper on toratopes, when he has time. That way, all of this can (maybe) be placed on wikipedia, which would be cool. I likened it to the case where we only know about cubes and simplices, but none of the cross polytopes, 24-cell, 120-cell, or any CRF. When you look up 'n-dimensional torus' , you don't see anything beyond the n-torus series. We certainly know there's more than that!
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Tree Graphs of A000669

Postby wendy » Mon May 09, 2016 12:51 pm

And to think it's such a tiny range that it beggers belief.

The face in any polytope in 3d, can be turned into a hollow swirl-prism, and these can have imbedded the outcome of a swirled circle (torus), as a hole between the outside and the outside.

One shudders to think what happens when one takes the vertices and edges of a polychoron and 'fatten' them out. The surface is connected, and the hole structure is without compare.
The dream you dream alone is only a dream
the dream we dream together is reality.
User avatar
wendy
Pentonian
 
Posts: 1820
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Tree Graphs of A000669

Postby ICN5D » Thu May 12, 2016 1:06 am

Yes, I've been getting more interested in these possibilities. I wonder how they relate to the tree graphs that we already know? Would they be 3D networks of nodes and branches, or something different? Whatever it is, it would pave the way to some sort of notation we can use to represent them. Perhaps the reverse process would be the way to do it: find a notation, and work towards a tree diagram. Do the Coexeter-Dynkin symbols have a tree graph associated with them? Something different than a lacing diagram?

I think I remember Marek mentioning a 4D tigroid that has a tetrahedral array of 4 inward-facing tori. I think I can define this in 5D, using the general plane of rotation functions, but not in 4D. The equations for such objects must belong to some extension of a generating function like that of a circle sweeping into a torus. Maybe a systematic method of defining polytopes implicitly can lead to the right functions, like how I made the circle embedded into the 1D edges of an empty cube.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Tree Graphs of A000669

Postby Klitzing » Thu May 12, 2016 6:19 am

ICN5D wrote:Do the Coexeter-Dynkin symbols have a tree graph associated with them?

You then might be interested into the associated Hasse diagrams?
Code: Select all
xPoQo    : 3D body   
  |                   
xPo .    : 2D faces   
  |                   
x . .    : 1D edges   
  |                   
. . .    : 0D vertices
  |                   
empty    :-1D nulloid

Code: Select all
    oPxQo        : 3D body   
    /   \                     
oPx .   . xQo    : 2D faces   
    \   /                     
    . x .        : 1D edges   
      |                       
    . . .        : 0D vertices
      |                       
    empty        :-1D nulloid

Code: Select all
        xPxQo        : 3D body   
        /   \                     
    xPx .   . xQo    : 2D faces   
    /   \   /                     
x . .   . x .        : 1D edges   
    \   /                         
    . . .            : 0D vertices
      |                           
    empty            :-1D nulloid

Code: Select all
        xPoQx            : 3D body   
      /   |   \                       
xPo .   x . x   . oQx    : 2D faces   
    \   /   \   /                     
    x . .   . . x        : 1D edges   
        \   /                         
        . . .            : 0D vertices
          |                           
        empty            :-1D nulloid

Code: Select all
        xPxQx            : 3D body   
      /   |   \                       
xPx .   x . x   . xQx    : 2D faces   
  |   X       X   |                   
x . .   . x .   . . x    : 1D edges   
      \   |   /                       
        . . .            : 0D vertices
          |                           
        empty            :-1D nulloid

And note that the e.g. the cuboctahedron could be variously described as o3x4o and as x3o3x. Accordingly it also would have 2 different Hasse diagrams. (The clue then just is whether all its triangles are considered symmetry equivalent (o3x .) or whether there applies a further color symmetry-breaking, which uses alternatingly 2 differently colored triangle types (x3o . and . o3x) instead.)

Similar diagrams could be derived for any dimensions. Even non-uniform polytopes could be represented. This then shows why the empty set as the nulloid was used: then there are different vertex types too. In fact, the Hasse diagram is not related to the Dynkin diagrams directly, it rather is a diagrammal representation of the incidence structure of the polytope of consideration.

--- rk
Klitzing
Pentonian
 
Posts: 1381
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Tree Graphs of A000669

Postby wendy » Tue Jul 12, 2016 9:26 am

The hasse diagram of any polytope that has a dual, is simply the antitegum of the polytope (which is the same as the dual), ie

For a wythoff-polytope, it's a matter of replacing o, x with oo, xm, and then put &#mt at the end. This is of course, to suppose a point expanding into a polytope as a pyramid peak in one direction, and from inside this pyramid peak, a second of the dual expanding. If ye look at the cube, you can colour three faces around a point red, and the opposite three blue. This gives an expansion of triangles from the red vertex, until the triangles meet the blue bits and the section becomes a hexagon (truncated triangle), and then a point (rectified triangle).
The dream you dream alone is only a dream
the dream we dream together is reality.
User avatar
wendy
Pentonian
 
Posts: 1820
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Tree Graphs of A000669

Postby Marek14 » Tue Jul 12, 2016 10:07 am

wendy wrote:The hasse diagram of any polytope that has a dual, is simply the antitegum of the polytope (which is the same as the dual), ie

For a wythoff-polytope, it's a matter of replacing o, x with oo, xm, and then put &#mt at the end. This is of course, to suppose a point expanding into a polytope as a pyramid peak in one direction, and from inside this pyramid peak, a second of the dual expanding. If ye look at the cube, you can colour three faces around a point red, and the opposite three blue. This gives an expansion of triangles from the red vertex, until the triangles meet the blue bits and the section becomes a hexagon (truncated triangle), and then a point (rectified triangle).


Isn't a rectified triangle a dual triangle?
Marek14
Pentonian
 
Posts: 1102
Joined: Sat Jul 16, 2005 6:40 pm

Re: Tree Graphs of A000669

Postby Klitzing » Tue Jul 12, 2016 12:23 pm

The rectification of triangle x3o IS the dual triangle o3x.
The truncation of triangle x3o IS the hexagon x3x.

Wendy was orienting a cube in its diagonal direction, i.e. vertex to vertex.
Then the vertex layers would be: Point o3o, triangle q3o, dual triangle o3q, point o3o.

But midwise between the inner layers occurs a hexagon z3z too (where z=q/2).
Even so, this one is not spanned by vertices, rather by edge-midpoints.

--- rk
Klitzing
Pentonian
 
Posts: 1381
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Tree Graphs of A000669

Postby ICN5D » Sun Oct 09, 2016 6:04 pm

I re-made the two charts, since some of the equations were missing an exponent. Only to be discovered months later, or course!


R^2 to R^4

R^5
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL


Return to Toratopes

Who is online

Users browsing this forum: No registered users and 2 guests