Higher dimension Conway-style operators?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Higher dimension Conway-style operators?

Postby Keiji » Tue Mar 30, 2010 8:06 am

Now, I like Conway operators, since they let you define all the truncations of the 3D seeds using just dual, rectification and truncation (since cantellation is double rectification, and omnitruncation is rectification followed by truncation). But is there a way to do a similar thing in 4D? I've been poring over this for a while, and getting nowhere.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Higher dimension Conway-style operators?

Postby anderscolingustafson » Wed Mar 31, 2010 1:31 am

If you mean is it possible to truncate or rectify a 4d polytope then yes. For instance you can truncate a teserect by expanding each of its 16 corners which are all infinintly small tetrahydrons to get a shape whith 8 cells that are truncated cubes and 16 cells that are perfect tetrahydrons. When a teserect is fully rectified you do not get a shape whith 8 regular cubes and 16 regular tetrahydrons as the cells but rather a shape whith 8 rectified cubes and 16regular tetrahydrons. You can also truncate the 16 celled poytope using 24 perfect octehyrons and when you rectify the 16 celled polytope you get a shape whith 16 cells that are perfect tetrahydrons and 24 cells that are perfect octehydrons. When you rectify a polytope you will not get a shape whith all of its cells perfect polytopes unless the seed polytope has tetrahydral sides. When the 5 celled seed is rectified you get a shape whith 5 cells that are perfect octehydrons and 5 cells that are perfect tetrahydrons. When you truncate the polytope whith 24 cells you get a shape whith 24 cells that are truncated octehydrons and 8 cells that are perfect cubes. When you truncate the polytope whith 120 cells you get a shape whith 120 cells that are truncated dodecahydrons and 600 cells that are perfect tetrahydrons. When you truncate the polytope whith 600 cells you get a shape whith 600 cells that are perfect octehydrons and 120 cells that are perfect isocahydrons.

After you truncate a seed polytope you can then add cells were the edges of the truncated cells are and 2 of the faces of those cells will be the same shape as the angles of the truncating cells and those cells will also have the same number of square/rectangular faces as the number of sides on the 1st 2 faces. These cells that are placed on the edges of the polytope will in fact truncate the truncating faces.

After this you can then place cells on the faces of the truncated cells and the edges of the truncating cells. When you do this you will actualy put faces on the edges of the truncating cells and it will double the number of square sides of on the cells placed were the edges of the polytope were and will also double the number of sides the ends of those cells have.

It is in fact possible to have semy regular polytopes in 4d.
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
anderscolingustafson
Tetronian
 
Posts: 316
Joined: Mon Mar 22, 2010 6:39 pm

Re: Higher dimension Conway-style operators?

Postby Keiji » Wed Mar 31, 2010 7:39 am

Thank you for completely missing the point. Yes, I know you can truncate 4D seeds in various ways. Yes, I've read the Uniform polychoron page many, many times. What I am asking for is a way to reduce all the truncations of the 4D seeds to a small set of operations applied successfully - preferably rectification, truncation and one other operation.

And for the record, do you really expect me to treat anything you say seriously when you can't even spell properly?
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Higher dimension Conway-style operators?

Postby wendy » Wed Mar 31, 2010 8:43 am

Conway-style operators exist in all dimensions. In four dimensions, there are 33 of them, in five dimensions, 65, and so on.

There are very few that arise from the operations of other operators, (apart from the dual). The x-cantellate is the rectified x-rectate, and the x-cantetruncate is the truncated x-rectate, are the only examples that derive from other operators.

The basic operators correspond to wythoff's constructions (ME and MM).
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Higher dimension Conway-style operators?

Postby Keiji » Thu Apr 01, 2010 12:13 pm

So - if I understand your post correctly - you're saying that 3D is a special case and in most dimensions you cannot combine some much smaller set of "fundamental" operations to form the rest?

I had imagined this was probably the case, but it's nice to make sure.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Higher dimension Conway-style operators?

Postby PWrong » Fri Apr 02, 2010 7:40 am

Wendy, where did you get the sequence 3, 33, 65?
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Higher dimension Conway-style operators?

Postby wendy » Sat Apr 03, 2010 8:29 am

There are actually something like 22 operators in three dimensions, but fewer are needed to produce these.

In any case, one should note that Conway's operators assumes any given figure is regular, and that one can go from any regular figure to its dual, or derived archemedian or catalan. So, eg in 3d, the prototype is xoo, the dual is variously oox or moo, and there are then 6 derived archemedian and 6 catalans from these. Add the gyrated and snub, and then the propellor etc to this.

In four dimensions, one has operators with 4 symbols, eg xooo = ooom, and 14 derived elements, and snubs/dual thereof. In five dimensions, the base is xoooo or oooom, each with 30 dependents.

Equities in the singular are of the form d = dual = mooo... = ...ooox . In the compound, the only form is the Conway-Kepler rule, which gives axoo...!..oxo... as .oxaxo.. where a is either x (cantetruncate) or o (cantellate). The other equity is 'd' (x) = m and d(m) = x. That is the dual of xxoo is mmoo.

Geo. Hart used the equities to create things like xox from oxo!oxo, and xxx from xxo!oxo. However, one should not consider something like xxx or xox thus derived. In practice, all of the base operators simply represent vectors in space, implemented from the origin, in the form (1,0,0) + (0,1,0) etc. In order to acheive this, one turns to 'pennant theory'.

A pennant is simply a simplex (triangle, tetrahedron, etc), that is cell of a tiling of such, such that the resulting vertices can be numbered 0,1,2,... consistantly. One way to get this is from the Coxeter-Dynkin graph, another is to use 'flags' or simplices formed by the notional cemtres of the vertex, edge, hedron, choron, ... of a polytope. Of course, the dimension number provides the vertex-order.

Since each triangle is given correct coordinates, it makes sense for eg (1,1,0) ie halfway between v0 and v1, etc. The coordinate is moved by flip (in Wythoff construction, reflection), to every pennant, and edges etc are drawn in as per wythoff construction. The corresponding catalan, comes by removing the walls except where an 'm' retains them.

The cantellate operator, says that if you apply first a single vector vz, and then add v0,v1, then you get v(z-1),v0*vz,v(z+1), which is read either as a vertex (x) or a catalan's face-tangent (m).

Alternation of vertices x, m gives s or g. With Klitzing's rule, you can have any set as alternating, eg ssx (v+e = even, h all), or in a later form even halving by different levels of s/g. An example of such a figure is ssox{3,4,3} = s3s4o3x. This has an assortment of faces as; 24 truncated tetrahedra, 96 triangular cupola, and 24 icosahedra. So there it goes.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia


Return to Other Polytopes

Who is online

Users browsing this forum: No registered users and 11 guests

cron