## automated coordinates from CD-symbols

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### automated coordinates from CD-symbols

Hello everyone, I've recently had time to catch up on a project of mine, which was supposed to automate the process of turning Coxeter Dynkin symbols into coordinates, and I am glad to announce that it seems to work!
I'm just trying to make a few improvements, as of yet I can only give coordinates for objects with a symmetry that includes the .2.2.2.-symmetry, or for which there exists a bigger coxeter-group (e.g. .5.3.3.) which includes both the group .2.2.2. and the symmetry of the object as subgroups. Luckily, this means the program can give coordinates for all finite 4d-polytopes with coxeter-dynkin-diagram including only 2,3,4,5-branches, except the .3.2.5.-duoprism symmetry.

This means I can generate coordinates for all interesting polytopes discovered in the "Construction of BT-polytopes via partial Stott-expansion"-thread, as well as generate coordinates for most 3d-polytopes, e.g. when you need these for a lace-tower.

When I'm finished with some of these improvements I will share the program, though the code is quite unreadable (I don't understand everything anymore either), and quite long, but you can then compile it.

Anyhow, I've run the program on some of the EKF-polytopes with .3.3.3.-symmetry, and obtained the following coordinates for xFfxo3xxxof3foxxx3oxfFx&#zx:
Code: Select all
`(-5,0) (0,-5) (-6,2) (-9,-7) (5,0) (0,-5) (-6,2) (-9,-7) (-10,0) (-5,-5) (-1,2) (-4,-7) (-5,0) (0,5) (-6,2) (-9,-7) (-5,0) (-10,-5) (4,2) (1,-7) (5,0) (0,5) (-6,2) (-9,-7) (10,0) (-5,-5) (-1,2) (-4,-7) (-10,0) (5,5) (-1,2) (-4,-7) (-10,-5) (-5,0) (-1,7) (-4,-2) (-5,0) (0,0) (-11,-3) (-4,-7) (5,0) (-10,-5) (4,2) (1,-7) (-5,0) (10,5) (4,2) (1,-7) (-5,-5) (-10,0) (4,7) (1,-2) (5,0) (0,0) (-11,-3) (-4,-7) (10,0) (5,5) (-1,2) (-4,-7) (-10,-5) (5,0) (-1,7) (-4,-2) (-10,0) (0,0) (-6,-8) (-4,-2) (10,5) (-5,0) (-1,7) (-4,-2) (-10,-5) (-5,-5) (-6,2) (1,-2) (5,0) (10,5) (4,2) (1,-7) (-5,-5) (10,0) (4,7) (1,-2) (-5,0) (0,0) (-1,-13) (-4,3) (5,5) (-10,0) (4,7) (1,-2) (-5,-5) (-10,-5) (-1,2) (6,-2) (0,-5) (-5,0) (9,7) (6,-2) (10,0) (0,0) (-6,-8) (-4,-2) (10,5) (5,0) (-1,7) (-4,-2) (-10,-5) (0,-5) (-6,-3) (-4,3) (10,5) (-5,-5) (-6,2) (1,-2) (-10,-5) (5,5) (-6,2) (1,-2) (0,-5) (5,0) (9,7) (6,-2) (5,0) (0,0) (-1,-13) (-4,3) (5,5) (10,0) (4,7) (1,-2) (-5,-5) (0,-5) (-1,-8) (-4,8) (5,5) (-10,-5) (-1,2) (6,-2) (-5,-5) (10,5) (-1,2) (6,-2) (0,5) (-5,0) (9,7) (6,-2) (0,-5) (-10,-5) (4,-3) (6,3) (10,5) (0,-5) (-6,-3) (-4,3) (-10,-5) (0,5) (-6,-3) (-4,3) (10,5) (5,5) (-6,2) (1,-2) (0,5) (5,0) (9,7) (6,-2) (0,-5) (-5,-5) (4,-8) (1,8) (5,5) (0,-5) (-1,-8) (-4,8) (-5,-5) (0,5) (-1,-8) (-4,8) (5,5) (10,5) (-1,2) (6,-2) (0,5) (-10,-5) (4,-3) (6,3) (0,-5) (10,5) (4,-3) (6,3) (10,5) (0,5) (-6,-3) (-4,3) (0,0) (-5,0) (4,7) (11,3) (0,5) (-5,-5) (4,-8) (1,8) (0,-5) (5,5) (4,-8) (1,8) (5,5) (0,5) (-1,-8) (-4,8) (0,0) (5,0) (4,7) (11,3) (0,5) (10,5) (4,-3) (6,3) (0,0) (-10,0) (4,2) (6,8) (0,5) (5,5) (4,-8) (1,8) (0,0) (-5,0) (4,-3) (1,13) (0,0) (10,0) (4,2) (6,8) (0,0) (5,0) (4,-3) (1,13) (-5,-5) (0,0) (-10,5) (-5,-10) (5,5) (0,0) (-10,5) (-5,-10) (-10,-5) (-5,0) (-5,5) (0,-10) (-5,-5) (-5,0) (-10,0) (-10,-5) (-5,0) (-10,-5) (0,10) (5,-5) (5,5) (-5,0) (-10,0) (-10,-5) (10,5) (-5,0) (-5,5) (0,-10) (-10,-5) (5,0) (-5,5) (0,-10) (-10,-5) (-10,0) (-5,0) (-5,-5) (-5,-5) (5,0) (-10,0) (-10,-5) (5,0) (-10,-5) (0,10) (5,-5) (-5,0) (10,5) (0,10) (5,-5) (-5,0) (-15,-5) (0,5) (0,0) (5,5) (5,0) (-10,0) (-10,-5) (10,5) (5,0) (-5,5) (0,-10) (10,5) (-10,0) (-5,0) (-5,-5) (-10,-5) (-5,0) (-5,-5) (-10,0) (-10,-5) (10,0) (-5,0) (-5,-5) (-15,-5) (-5,0) (0,0) (0,-5) (5,0) (10,5) (0,10) (5,-5) (5,0) (-15,-5) (0,5) (0,0) (-5,0) (-5,0) (-5,-10) (-5,10) (-5,0) (15,5) (0,5) (0,0) (-10,0) (-10,-5) (5,5) (5,0) (0,0) (-5,-5) (5,10) (10,-5) (10,5) (-5,0) (-5,-5) (-10,0) (10,5) (10,0) (-5,0) (-5,-5) (-10,-5) (5,0) (-5,-5) (-10,0) (-15,-5) (5,0) (0,0) (0,-5) (15,5) (-5,0) (0,0) (0,-5) (0,0) (5,5) (5,10) (10,-5) (5,0) (-5,0) (-5,-10) (-5,10) (5,0) (15,5) (0,5) (0,0) (-5,0) (5,0) (-5,-10) (-5,10) (-10,0) (10,5) (5,5) (5,0) (10,0) (-10,-5) (5,5) (5,0) (0,0) (-15,-5) (5,0) (0,5) (10,5) (5,0) (-5,-5) (-10,0) (-5,0) (-5,-5) (10,5) (10,0) (-15,-5) (0,0) (0,-5) (-5,0) (15,5) (5,0) (0,0) (0,-5) (0,0) (-10,0) (0,-10) (0,10) (5,0) (5,0) (-5,-10) (-5,10) (-5,0) (5,5) (10,5) (10,0) (-10,0) (0,0) (0,-10) (0,10) (10,0) (10,5) (5,5) (5,0) (0,0) (15,5) (5,0) (0,5) (-5,0) (-10,-5) (10,0) (5,5) (5,0) (-5,-5) (10,5) (10,0) (15,5) (0,0) (0,-5) (-5,0) (0,0) (10,0) (0,-10) (0,10) (-5,0) (-5,0) (5,-10) (5,10) (5,0) (5,5) (10,5) (10,0) (10,0) (0,0) (0,-10) (0,10) (-5,0) (10,5) (10,0) (5,5) (5,0) (-10,-5) (10,0) (5,5) (-5,0) (5,0) (5,-10) (5,10) (5,0) (-5,0) (5,-10) (5,10) (5,0) (10,5) (10,0) (5,5) (5,0) (5,0) (5,-10) (5,10) (0,-5) (-5,0) (-5,5) (-10,-10) (0,5) (-5,0) (-5,5) (-10,-10) (-5,-5) (-10,0) (0,5) (-5,-10) (0,-5) (5,0) (-5,5) (-10,-10) (0,-5) (-5,-5) (-10,0) (-5,-10) (-5,0) (-10,-5) (0,10) (-5,-5) (0,5) (5,0) (-5,5) (-10,-10) (0,5) (-5,-5) (-10,0) (-5,-10) (5,5) (-10,0) (0,5) (-5,-10) (-5,-5) (10,0) (0,5) (-5,-10) (-5,-5) (-10,-5) (-5,0) (0,-10) (-10,-5) (-5,0) (5,5) (0,-10) (0,-5) (0,-5) (-10,-5) (-10,-5) (0,-5) (5,5) (-10,0) (-5,-10) (5,0) (-10,-5) (0,10) (-5,-5) (-5,0) (10,5) (0,10) (-5,-5) (-5,0) (-10,-10) (-5,5) (0,-5) (-10,0) (-5,-5) (5,10) (0,-5) (0,5) (0,-5) (-10,-5) (-10,-5) (0,5) (5,5) (-10,0) (-5,-10) (5,5) (10,0) (0,5) (-5,-10) (5,5) (-10,-5) (-5,0) (0,-10) (-10,-5) (5,0) (5,5) (0,-10) (-5,-5) (0,-5) (-5,-10) (-10,0) (-5,-5) (10,5) (-5,0) (0,-10) (10,5) (-5,0) (5,5) (0,-10) (-10,-5) (-10,-5) (0,-5) (0,-5) (0,-5) (0,5) (-10,-5) (-10,-5) (-10,-10) (-5,0) (0,5) (5,-5) (5,0) (10,5) (0,10) (-5,-5) (5,0) (-10,-10) (-5,5) (0,-5) (-10,0) (5,5) (5,10) (0,-5) (-5,0) (0,-5) (-10,-10) (-5,5) (-5,0) (10,10) (-5,5) (0,-5) (10,0) (-5,-5) (5,10) (0,-5) (-10,0) (-10,-10) (0,0) (0,0) (0,5) (0,5) (-10,-5) (-10,-5) (-10,-5) (-5,-5) (0,10) (5,0) (-5,0) (0,-5) (10,10) (5,-5) (5,5) (0,-5) (-5,-10) (-10,0) (5,5) (10,5) (-5,0) (0,-10) (10,5) (5,0) (5,5) (0,-10) (-10,-5) (-5,-5) (0,-10) (-5,0) (-5,-5) (0,5) (-5,-10) (-10,0) (-10,-10) (5,0) (0,5) (5,-5) (10,5) (-10,-5) (0,-5) (0,-5) (-10,-5) (10,5) (0,-5) (0,-5) (-10,-10) (-10,0) (0,0) (0,0) (10,10) (-5,0) (0,5) (5,-5) (-5,0) (0,5) (10,10) (5,-5) (5,0) (0,-5) (-10,-10) (-5,5) (5,0) (10,10) (-5,5) (0,-5) (10,0) (5,5) (5,10) (0,-5) (-10,0) (-5,-5) (-5,-10) (0,5) (-5,0) (0,5) (-10,-10) (-5,5) (-10,-5) (5,5) (0,10) (5,0) (10,0) (-10,-10) (0,0) (0,0) (-10,0) (10,10) (0,0) (0,0) (-10,-5) (-10,-5) (0,5) (0,5) (10,5) (-5,-5) (0,10) (5,0) (5,0) (0,-5) (10,10) (5,-5) (-5,0) (-10,-10) (5,-5) (0,5) (5,5) (0,5) (-5,-10) (-10,0) (-5,-5) (0,-5) (5,10) (10,0) (10,5) (-5,-5) (0,-10) (-5,0) (-10,-5) (5,5) (0,-10) (-5,0) (-10,-10) (-5,0) (0,-5) (-5,5) (10,10) (5,0) (0,5) (5,-5) (10,5) (10,5) (0,-5) (0,-5) (-10,-10) (10,0) (0,0) (0,0) (10,10) (-10,0) (0,0) (0,0) (5,0) (0,5) (10,10) (5,-5) (-5,0) (-10,-5) (0,-10) (5,5) (5,0) (0,5) (-10,-10) (-5,5) (-5,-5) (0,5) (5,10) (10,0) (10,0) (-5,-5) (-5,-10) (0,5) (-10,0) (5,5) (-5,-10) (0,5) (-10,-5) (-5,0) (-5,-5) (0,10) (10,5) (5,5) (0,10) (5,0) (10,0) (10,10) (0,0) (0,0) (-10,-5) (10,5) (0,5) (0,5) (10,5) (-10,-5) (0,5) (0,5) (5,0) (-10,-10) (5,-5) (0,5) (-5,0) (10,10) (5,-5) (0,5) (-5,-5) (-10,-5) (5,0) (0,10) (5,5) (0,-5) (5,10) (10,0) (10,5) (5,5) (0,-10) (-5,0) (-10,-10) (5,0) (0,-5) (-5,5) (10,10) (-5,0) (0,-5) (-5,5) (0,-5) (0,-5) (10,5) (10,5) (10,10) (10,0) (0,0) (0,0) (5,0) (-10,-5) (0,-10) (5,5) (-5,0) (10,5) (0,-10) (5,5) (-5,-5) (-10,0) (0,-5) (5,10) (5,5) (0,5) (5,10) (10,0) (10,0) (5,5) (-5,-10) (0,5) (-10,-5) (5,0) (-5,-5) (0,10) (10,5) (-5,0) (-5,-5) (0,10) (0,-5) (0,5) (10,5) (10,5) (10,5) (10,5) (0,5) (0,5) (5,0) (10,10) (5,-5) (0,5) (-5,-5) (10,5) (5,0) (0,10) (5,5) (-10,-5) (5,0) (0,10) (0,-5) (-5,-5) (10,0) (5,10) (10,10) (5,0) (0,-5) (-5,5) (0,5) (0,-5) (10,5) (10,5) (5,0) (10,5) (0,-10) (5,5) (-5,-5) (10,0) (0,-5) (5,10) (5,5) (-10,0) (0,-5) (5,10) (0,-5) (-5,0) (5,-5) (10,10) (10,5) (5,0) (-5,-5) (0,10) (0,5) (0,5) (10,5) (10,5) (0,-5) (5,5) (10,0) (5,10) (5,5) (10,5) (5,0) (0,10) (0,5) (-5,-5) (10,0) (5,10) (0,-5) (5,0) (5,-5) (10,10) (5,5) (10,0) (0,-5) (5,10) (0,5) (-5,0) (5,-5) (10,10) (0,5) (5,5) (10,0) (5,10) (0,5) (5,0) (5,-5) (10,10) (-5,0) (-5,0) (-5,10) (-5,-10) (5,0) (-5,0) (-5,10) (-5,-10) (-5,0) (5,0) (-5,10) (-5,-10) (-5,0) (-10,-5) (-10,0) (-5,-5) (0,0) (-10,0) (0,10) (0,-10) (5,0) (5,0) (-5,10) (-5,-10) (5,0) (-10,-5) (-10,0) (-5,-5) (-10,0) (0,0) (0,10) (0,-10) (-5,0) (-5,-5) (-10,-5) (-10,0) (-5,0) (10,5) (-10,0) (-5,-5) (0,0) (10,0) (0,10) (0,-10) (0,0) (-15,-5) (-5,0) (0,-5) (-5,0) (-5,0) (5,10) (5,-10) (5,0) (-5,-5) (-10,-5) (-10,0) (5,0) (10,5) (-10,0) (-5,-5) (10,0) (0,0) (0,10) (0,-10) (-10,0) (-10,-5) (-5,-5) (-5,0) (-5,0) (5,5) (-10,-5) (-10,0) (-15,-5) (0,0) (0,5) (5,0) (-5,0) (5,0) (5,10) (5,-10) (0,0) (-5,-5) (-5,-10) (-10,5) (0,0) (15,5) (-5,0) (0,-5) (5,0) (-5,0) (5,10) (5,-10) (-5,0) (-15,-5) (0,-5) (0,0) (5,0) (5,5) (-10,-5) (-10,0) (-10,-5) (-5,0) (5,5) (10,0) (10,0) (-10,-5) (-5,-5) (-5,0) (-10,0) (10,5) (-5,-5) (-5,0) (-15,-5) (-5,0) (0,0) (0,5) (15,5) (0,0) (0,5) (5,0) (5,0) (5,0) (5,10) (5,-10) (-5,0) (-10,-5) (0,-10) (-5,5) (0,0) (5,5) (-5,-10) (-10,5) (-10,-5) (5,0) (5,5) (10,0) (5,0) (-15,-5) (0,-5) (0,0) (-5,0) (15,5) (0,-5) (0,0) (-10,-5) (-10,0) (5,0) (5,5) (10,5) (-5,0) (5,5) (10,0) (10,0) (10,5) (-5,-5) (-5,0) (-15,-5) (5,0) (0,0) (0,5) (15,5) (-5,0) (0,0) (0,5) (5,0) (-10,-5) (0,-10) (-5,5) (-5,0) (10,5) (0,-10) (-5,5) (-10,-5) (-5,0) (5,-5) (0,10) (10,5) (5,0) (5,5) (10,0) (5,0) (15,5) (0,-5) (0,0) (-10,-5) (10,0) (5,0) (5,5) (10,5) (-10,0) (5,0) (5,5) (-5,-5) (-5,0) (10,0) (10,5) (15,5) (5,0) (0,0) (0,5) (5,0) (10,5) (0,-10) (-5,5) (-10,-5) (5,0) (5,-5) (0,10) (10,5) (-5,0) (5,-5) (0,10) (-5,-5) (5,0) (10,0) (10,5) (10,5) (10,0) (5,0) (5,5) (5,5) (-5,0) (10,0) (10,5) (-5,-5) (0,0) (10,-5) (5,10) (10,5) (5,0) (5,-5) (0,10) (5,5) (5,0) (10,0) (10,5) (5,5) (0,0) (10,-5) (5,10) (0,0) (-5,0) (-4,3) (-1,-13) (0,-5) (-5,-5) (-4,8) (-1,-8) (0,0) (5,0) (-4,3) (-1,-13) (0,0) (-10,0) (-4,-2) (-6,-8) (0,5) (-5,-5) (-4,8) (-1,-8) (0,-5) (5,5) (-4,8) (-1,-8) (0,-5) (-10,-5) (-4,3) (-6,-3) (-5,-5) (0,-5) (1,8) (4,-8) (0,0) (-5,0) (-4,-7) (-11,-3) (0,0) (10,0) (-4,-2) (-6,-8) (0,5) (5,5) (-4,8) (-1,-8) (0,5) (-10,-5) (-4,3) (-6,-3) (-5,-5) (0,5) (1,8) (4,-8) (0,-5) (-5,0) (-9,-7) (-6,2) (0,-5) (10,5) (-4,3) (-6,-3) (5,5) (0,-5) (1,8) (4,-8) (-5,-5) (-10,-5) (1,-2) (-6,2) (0,0) (5,0) (-4,-7) (-11,-3) (-10,-5) (0,-5) (6,3) (4,-3) (-5,0) (0,0) (1,13) (4,-3) (0,5) (-5,0) (-9,-7) (-6,2) (0,5) (10,5) (-4,3) (-6,-3) (5,5) (0,5) (1,8) (4,-8) (-5,-5) (-10,0) (-4,-7) (-1,2) (0,-5) (5,0) (-9,-7) (-6,2) (-10,-5) (0,5) (6,3) (4,-3) (5,5) (-10,-5) (1,-2) (-6,2) (-5,-5) (10,5) (1,-2) (-6,2) (-10,-5) (-5,-5) (6,-2) (-1,2) (10,5) (0,-5) (6,3) (4,-3) (5,0) (0,0) (1,13) (4,-3) (-5,0) (-10,-5) (-4,-2) (-1,7) (0,5) (5,0) (-9,-7) (-6,2) (-10,0) (0,0) (6,8) (4,2) (5,5) (-10,0) (-4,-7) (-1,2) (-5,-5) (10,0) (-4,-7) (-1,2) (-10,-5) (-5,0) (1,-7) (4,2) (10,5) (0,5) (6,3) (4,-3) (5,5) (10,5) (1,-2) (-6,2) (-10,-5) (5,5) (6,-2) (-1,2) (10,5) (-5,-5) (6,-2) (-1,2) (5,0) (-10,-5) (-4,-2) (-1,7) (-5,0) (10,5) (-4,-2) (-1,7) (-10,0) (-5,-5) (1,-2) (4,7) (10,0) (0,0) (6,8) (4,2) (5,5) (10,0) (-4,-7) (-1,2) (-10,-5) (5,0) (1,-7) (4,2) (10,5) (-5,0) (1,-7) (4,2) (-5,0) (0,0) (11,3) (4,7) (10,5) (5,5) (6,-2) (-1,2) (5,0) (10,5) (-4,-2) (-1,7) (-10,0) (5,5) (1,-2) (4,7) (10,0) (-5,-5) (1,-2) (4,7) (-5,0) (0,-5) (6,-2) (9,7) (10,5) (5,0) (1,-7) (4,2) (5,0) (0,0) (11,3) (4,7) (-5,0) (0,5) (6,-2) (9,7) (10,0) (5,5) (1,-2) (4,7) (5,0) (0,-5) (6,-2) (9,7) (5,0) (0,5) (6,-2) (9,7) `
The format is (a,b)=a*x+b*f, and the layers of the &#zx-lacing have been separated by whitespaces. Unfortunately, the edge-length is not a very nice number, I think it is sqrt(5) or 10 or something, the program should be able to handle this as well soon.

Quickfur, could you run these coordinates through your rendering programs? It could be the coordinates are not correct yet, so some testing would be great .
Or you could share your programs such that I can do the testing.
student91
Tetronian

Posts: 328
Joined: Tue Dec 10, 2013 3:41 pm

### Re: automated coordinates from CD-symbols

The coordinates for the various groups run like this.

1.
APAC (all permutations, all changes of sign) full [3,3,4] cubic
EPAC (even permutations, all changes of sign) half = [3,3+,4] pyritochoral
APEC (all permutations, even changes of sign) half [3,3,A] half-cubic
EP+C (even permutations + changes of sign) half [3,3,4]+ rota-cubic
EPEC (even permutations, even changes of sign) [3,3,A+] quarter-cubic
APNC (all permutations, no change of sign) [3,...,3] simplex
EPNC (even permutations, no change of sign) [3,3...3.3]+, rota-simplex

The coordinates of the o3o3...o4o derive from summing these vectors, and applying APAC to the result

x3o3o3o3.. = q,0,0,0,0...
o3x3o3o3.. = q,q,0,0,0...
o3o3x3o3.. = q,q,q,0,0...
o3o3o3o4x = 1,1,1,1,1...

The coordinates of the o3o3o3... oAo, are the same, except as follows, to which APEC is applied

o3o3o3xAo = q,q,q,q,q...q,q,-q
o3o3o3oAx = q,q,q,q,q...,q,q,q.

The coordinates of o3o4o3o, depend on how you choose to sit the two subgroups.

if x3o4o3o = x3o3o4o, the vertices are (q,q,0,0) APAC,
if x3o4o3o = qo3oo3oo4ox, the vertices are variously (1,1,1,1) and (2,0,0,0), APAC.

The vertices of o3x4o3o are thus (q,q,2q,0)/2 APAC giving some 12*8=96 vertices.
The vertices of o3o4x3o are then (3,1,1,1)/2 = 64, and (2,2,2,0)/2 = 32 = 96 vertices.

The vertices of s3s4o3o are then (a, b, c, 0) EPAC gives 12*8 = 96 combinations. a=v, b=1, c=f

The highest shared symmetry between [3,5] and [3,4] is the pyritohedral, that is [3+, 4], the icosahedral is repeated five times in this EPAC.

The highest shared symmetry between [3,3,5] and [3,3,4] is the pyritochoral, the [3,3+,4] is repeated seventy-five times as EPAC.

The simplex group is constructed from taking the octahedral direction of [3,3,..,4], with a sum of zero. The various simplexes are given as (edge given as sqrt(2))
There are n+1 coordinates

x3o3o3o3... [n, -1, -1, -1, ...]
o3x3o3o3... [n-2, n-2, -2, -2, ...]
o3o3x3o3... [n-3, n-3, n-3, -3, ...]
ooooo3x [1,1,1,1,1,1,-n]
&c.

These vectors are added and APNC applied to the result.
The dream you dream alone is only a dream
the dream we dream together is reality.

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wendy
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### Re: automated coordinates from CD-symbols

I see, I can use this to check my program, thanks!
My program calculates coordinates via algebraic rather than geometrical methods, e.g. coordinates for [3,3,3] are found by applying the elements of the corresponding matrix group to the seed coordinate.
Then these coordinates have to be translated to an orthonormal basis, which my program does by searching for [2,2,2] as a subgroup of e.g. [3,3,3] (it fails in this instance, but for e.g. [3,4,3] and [5,3,3] it does work),
and then applies a basis transformation to the mirrors of [2,2,2] as a subset of the mirrors of e.g. [5,3,3].
In the case of [3,3,3] or [5,2,5], the transition can't be direct, but it does work via [5,3,3].

The output of my program for [5,3,3] does correspond nicely to the coordinates you provided for s3s4o3o , only scaled by a factor f.
the coordinates my program gives for .3.4.3. are much uglier than yours, my program gives
Code: Select all
`coordinates of  x3o4o3oformat is qint, (x,q)(1,0) (0,0) (1,0) (0,1) (-1,0) (0,0) (1,0) (0,1) (0,0) (0,1) (0,0) (0,1) (0,0) (0,-1) (0,0) (0,1) (1,0) (0,0) (-1,0) (0,1) (0,0) (0,0) (2,0) (0,0) (-1,0) (0,0) (-1,0) (0,1) (1,0) (0,1) (1,0) (0,0) (-1,0) (0,1) (1,0) (0,0) (1,0) (0,-1) (1,0) (0,0) (-1,0) (0,-1) (1,0) (0,0) (2,0) (0,0) (0,0) (0,0) (1,0) (0,1) (-1,0) (0,0) (-2,0) (0,0) (0,0) (0,0) (-1,0) (0,1) (-1,0) (0,0) (1,0) (0,-1) (-1,0) (0,0) (-1,0) (0,-1) (-1,0) (0,0) (1,0) (0,0) (1,0) (0,-1) (0,0) (0,0) (-2,0) (0,0) (-1,0) (0,0) (1,0) (0,-1) (0,0) (0,1) (0,0) (0,-1) (0,0) (0,-1) (0,0) (0,-1) (1,0) (0,0) (-1,0) (0,-1) (-1,0) (0,0) (-1,0) (0,-1) `
which is way less symmetric, I suppose this is because it takes an ugly embedding of [2,2,2] in [3,4,3] or something?
For the [3,3,3]-symmetry, my program has the slight advantage that it can directly embed it in 4d, instead of in 5d.
and furthermore my program has the advantage that the input is a simple line as x5o3o3o, and it directly outputs all coordinates, which is something I get really excited about.
I've stopped enhancing my program, I uploaded the sources here, if you want to use it, it can simply be compiled using the linux-style makefile, given you have linux on your computer.
The first time you want to use the [5,3,3]-symmetry, it writes the full group in matrix form to a .txt-file, which thus includes 14400 matrices and has a size of 1.8 MB.
I suggest you put it the whole program in a separate directory to keep things tidy.
It should be mostly bug-free, I'd be happy to hear of any bugs.
As of yet, the program only supports values that can be expressed as linear combinations of x, f, q and fq, but other values should be fairly easily implementable.
I hope you are just as excited as I am
student91
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### Re: automated coordinates from CD-symbols

Computing coordinates automatically for simplex / n-cube families is very easy. See this page for a description of how this can be done.
quickfur
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