Many of the sections of polytopes come from lace-towers of tilings, or varously, 'stations' and layered tilings. According to conway and slone, layered tilings are the most efficient up to 24 dimensions, except for the coxeter-todd lattice in 12 dimensions.
Suppose you take a layer representing A_n, made of spheres. In 3d, this would be a set of spheres arranged as spheres. You can stack these in different ways to get different 3d lattices. We shall look at the six-dimensional case for this exercise.
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12 5 8 9 8 5 12 5 8 9 8 5 12 5 8 9 8 5
0 x o o o o o 0 x o o a o o 0 x o a o e o
7 o x o o o o 4 o o x o o a 3 o e o x o a
28 o o x o o o 16 o a o o x o 12 x o a o e o
63 o o o x o o 36 x o o a o o 27 o e o x o a
T-basic A_6 Semicubic B_6 Gosset 2_22
12 5 8 9 8 5
0 x o o o o o
7 o x o o o o
28 x o o o o o
63 o x o o o o
Laminate LB5
The above tables show sections of the three trigonal groups against the simplex group A_x. You can read the across row as an unrolled loop group 6:, and the vertical as a linear column. The numbers indicate the nearest point of the same kind, to the origion (0, 12). The lattice edge is sqrt(12), the distances are squares.
In the semi-cubic, the 'a' nodes represent the semuicubic at the 'other end', ie if x represents the vertices of /E4A, then a is that at E4/A, for example. The combination of the two represents a squashed body-centred cubic.
The third shows the 2_22, where the ends of the three tails show as x, a, and e. The cell centres are at a and e, so we can find the cell by drawing a circle of size 8 around it. We can get this by 2 steps on the same row (8, 0), or one step across, and one step up (5, 3). Both of these add to 8. The presentation of the 2_21 down a simplex-first, is xox3ooo3ooo3oxo3ooo&#xt, which shows that it contains the vertices and edges of a simplex-line prism. Note the simplexes point the same way.
The vertex figure 1_22 gives in two rows up (0, 12) = point, one row up = (9, 3) = o3o3x3o3o, same row = x3o3o3o3o3x, and symmetrically below. So it comes to a lace prism ooxoo3ooooo3oxoxo3ooooo3ooxoo&#xt.
The spheres of radius 12 contain the 'eutactuc stars' of the various trigonal groups. Since these represent the normals to the mirror-planes, every polytope that can be written in 5, 4A and 4B, with integer coordinates, have their vertices in these vertices, and no others. So if you can determine the radius and the cell centres, it is possible to use height and spheres (as we did for 2_21 and 1_22), to find the simplex rings of the various figures.
For the simplex group, the base here is 6:, and the stack in the first frame represents 7:. The cell centres of 7: lie at intervals of 6/7 of the row heights in the diagram above (where the rows are evenly spaced). One can determine at what point the centre of any given figure constructed from this symmetry lies, by noting that the stott-vectors correspond to sums of 5,-1,-1,-1,-1,-1, in strict cyclic permutations. So one gets
These are the plane, in 6D, represented by \sum x_i = 0. This gives a perefectly euclidean tiling, giving a lattice of edge sqrt(2).
x3o3o3o3o3o = 5, -1, -1, -1, -1, -1
o3x3o3o3o3o = 4, 4, -2, -2, -2, -2
o3o3x3o3o3o = 3, 3, 3, -3, -3, -3
o3o3o3o3x3o = 2, 2, 2, 2, -4, -4
o3o3o3o3o3x = 1, 1, 1, 1, 1, -5.
x3o3o3o3o3x = 6, 0, 0, 0, 0, 0, -6
Note that the stott vectors are 1, 2, 3, 4, 5, 6 modulo 6, and so if you add these together, the modulo of 6 is still preserved across all coordinates. One is then for any polytope constructed by integer multiples of these numbers, a central cell which is one of these polytopes: that is, the centre lies in one of these cells.