Rings of {3,3,3,5}

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

Rings of {3,3,3,5}

Postby wendy » Tue Oct 18, 2005 10:57 pm

The {3,3,3,5} is a hyperbolic tiling of pentachora, five at an edge.

The aim is to find the distribution of vertices in successive shells, beginning with a vertex. This is the latest version, for fourteen rings.

Code: Select all
   ring       count      coordinates ( in ssf, base j7 = 2.61803398875)
------------------------------------------------------------------------
     0,      1       1   0.0.0.0
     1.    120     100   1.0.0.0    {3,3,5,5/2}, {3,5,5/2,5} {5,5/2,5,3}
    10.    600     500   0.0.0.1    {4,3,3,5}, {5/2,5,3,3}, {4,10}
    11.    720     600   0.f.0.0    {5,3,3,5}, {5,10}
    f1.   3600    3000   f.0.1.0    {E,3,3,5}, {10,10}
   100.   1440    1200   10.f.0.0
   101.   6000    5000   0.1.f.0    f0.0.0.1   {3,5,3,3}
   110.   9600    8000   1.f.0.f    ff.1.0.0
   111.   2520    2100   10.0.0.10  10f.0.0.0
   f01.  14400  1.0000   10.10.0.1  f.0.f.f
   f10.   4800    4000   f1.0.0.10  10.f.0.0
   f11.  14400  1.0000   f1.f.0.f   0.f.1.10
   ff1.  28800  2.0000   0.f0.1.1   f0.f.0.10   100.1.f.0   f.1.0.f0
  1000.   3600    3000   10.0.f0.0
  1001.  50400  3.6000   1.0.f.f0   1.11.10.0   10.f.10.1  ff.10.f.0
                         110.f.1.0  10.f1.0.f
---------------------------------------------------------------------------


These coordinates are exact.

The first column, and the coordinates are given in base j7 = 2.61803398875&c. f is a digit representing j3=1.61803398875&c.

The second and third columns represent the count of vertices in that shell, given in decimal and twelfty.

A given shell contains one or more polytopes, expressed in the oblique coordinate system ssf. One treats every reflective region of the group [3,3,5] as a "possitive octant", with axies such that 1.0.0.0, 0.1.0.0, 0.0.1.0 and 0.0.0.1 correspond to x3o3o5o, o3x3o5o, o3o3x5o, and o3o3o5x, of edge 2. Rather like the standard cube is 1,1,1 in the group o2o2o.

One then gets many vertices which form the required polytopes. I put the thing up in the rippler, and it works as expected.
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Postby Batman3 » Sat Oct 22, 2005 9:59 pm

Wendy,
this kind of stuff sounds fascinating, but I don't understand you. I'm afraid I don't even know what 'pentachora' is. Maybe you could give a short, less detailed, descryption of what you are trying to say. (for us mere 3d-mortals).
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Postby wendy » Sun Oct 23, 2005 3:49 am

I'm still learning the stuff myself. But one day, i might write something more. Someone asked me that question, and i had a couple of shots at it.

It's pretty non-standard, because i largely avoid the standard tools like algebra, trig, etc. I use other tools not in general circulation, such as different base notations (such as j7 = 2.61803398875), and lots of stuff that belong to a forgotten mathematics.

For example, {3,3,3,5} is a H4 tiling: that is, it is kind of like the tiling of tesseracts {4,3,3,4}, but it has five pentachora at an edge.

The three-dimensional tiling might go like this: cubic =

0 0.0.0
1 1.0.0
2. 0.1.0
3 0.0.1
4 2.0.0
5 1.1.0
6. 1.0.1
8 0.2.0

These coordinates are stott-vectors of the form

(1,0,0), (1,1,0), (1,1,1) [which gives a figure of 2.
One adds these as indicated in the vector, and then does "all permutatuins
all change of sign".

For example, the vector 1.0.1 is 1.0.0 + 0(1.1.0) + 1(1.1.1)
or (2.1.1). You then can freely rearrange the vertices, and change sign.

The examples i gave for the {3,3,3,5}, apart from being incomplete, contains the figures derived in {3,3,5} in a very oblique frame of reference.

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