The Dynkin symbol is not just a kaleidoscope for presenting a polytope: it is a generally oblique coordinate system. That is, one is not restricted to using o and x, but can make the values any number: eg
1 [3] 0 [5] 0 is a unit-edge icosahedron, could be written as 1,0,0.
0 [3] 1 [5] 0 is a unit edge ID
0 [3] 0 [5] 1 is a unit edge Dodecahedron
Of course, instead of having 8 octants, you have 120 sectors. You can still do 'even chnage of sign', but instead of four values (1,1,1), (1,-1,-1), (-1,1,-1) and (-1,-1,1), you have sixty. The snub dodecahedron, then is some x3y5,z for some x,y,z with even change of sign.
So,a point P gives not only a posotion-vector (o..P), but a "position polytope", whose edges connect P to P' in each of the mirrors. It may take as many as 120 reflections to get around the whole of space though!
Once you get a method for working with oblique coordinate systems. In particular, the usual dot product of vectors now involve a matrix to do this. The special matrix for this is the Stott matrix. Since the vertex of the polytope 'defines' the polytope, the dot-product of the vector and itself gives the length of the vector, and if the edge is set to 2 (the usual standard), the circum-radius.
We use a simpler example.
The standard presentation of [3,4] (the octahedral group) is to give x,y,z, all change of sign, all permutations. When these are all different and non zero, you get 48 points. The notional point for this coordinate can be set as the sorted value of abs(x), abs(y) abs(z). eg -12, 3, 18 is a reflection of (18,12,3)
One can next write the standard verticex of a polyhedron, of edge 2, which, with all change of sign and permutations give the vertices. This is like in the 'primary octant x,y,z > 0.
O = r2, 0, 0 ; CO = r2, r2, 0 ; C = 1,1,1.
All other values can be constructed from a sum of some O + some CO + some C
in particular these form figures of edge r2. Note these form a parallelohedron in the primary sector, bounded by unit-thickness planes.
tO = O + CO = x3x4o = 2r2, r2, 0
tC = C + CO = o3x4x = r2+1, r2+1, 1
rCO = O + C = x3o4x = r2+1, 1, 1
tCO = O + CO + C = x3x4x = 2r2+1, r2+1, 1
Of course, since this is a coordinate system in the commas of the schläfli symbol, one can just as easily provide varying intensities between the 'construct' and the 'decorations'.
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@---o-4-o Coxeter Dynkin Graph O
@---@-4-o Coxeter Dynkin Graph tO
O tO
@-o4o @-@4-o extra dashses removed
{ 3,4 } -- Schläfli symbol: just the structure
t_0 t_0,1 Coxeter's truncate notation (based on Stott).
{;3,4 } {;3;4} Semicolon for @.
x3o4o x3x4o Inline dynkin symbol (allow measures, duals)
1S0Q0 1S1Q0 Inline dynkin coordinate
/S Q /S/Q Simplified inline
/1 Q /1/Q Further simplified for high dimensions.
(1,0,0) (1,1,0) stott coordinate.
One can see that the Schläfli symbol can not express the truncated octahedron: it's only good for regular figures. Also the standard way of representing the CO is to put 3 over 4, in a coxeter-curtail. However, if you want to run these in line, you need to deal with any mark on any node. Coxeter's solution is to use the t_ (ie t sub ...). However, i prefer full height solutions, of types listed following.
Depending on how much you need to deal with coordinates, etc, you move up and down the list. So if you are doing intense work (eg on a geodesic dome), you may want to use the 1 S 0 Q 0 form. This can take real values, eg 5.8132 S 2.24577 Q 88.12356. If you are wrangling high dimensions, with relatively few nodes, then something like 1/28Q is more in order. This thing has 30 branches and 31 nodes, being in Coxeter's system t_1{3^29,4}, 3 repeated 29 times, not tothe 29th power.
The notation is meant to be a one-to-one substution. That is, you can readily translate between /S/Q asnd x3x4o, because symbols are not generally used in different meanings in different notations. You could write xSxQo for example.
Lace-prisms can as readily be analised in terms of the wythoff process as ordinary figures: one notes that the face of a lace-prism is itself a lace-prism. One needs just to check the "progressions" by removing nodes singularly. We see by removing the first node, the figure becomes ".. ox&#t", that is a point expanding into a line (triangle), and the second node is xo .. &#t (line expanding into a point = triangle). The top and bottom remain the same, ie x.Po. = Pgon while .oP.x is another P-gon.
COMPOUNDS.
Having defined a coordinate system, you can then create compounds by using several vertices, eg this is a P-gonal antiprism.
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1 P 0 polygon compound as tower
\ x. P o.
\ xo P ox &#t
\ .o P .x
0 P 1 dual polygon
The mirrors are vertical through 1, 0. The diagonal creates an edge of an antiprism. When this is view in its full, it zigzags from top to bottom like lacing in a drum: hence the name 'lace-prism'. This figure can be expressed as s2sPs, (alternating vertices of all points at 1, 0), but others can not.
In line, we write these in succession at each coordinate, eg x_1 x_2 x_3 [P] y_1 y_2 y_3. Since we have the construct P is a capital, we use upper-case brackets: [] to designate it. The coordinates are then sequenced between these.
Since it is after all, a coordinate system, we could go from the centre at 0,0. The progression of the line from 0,0 to 0,1 over this height gives rise to a pyramid. Progressing from 1,0 to 1,1 creates a cupola. The square face forms at 1,- to 1,-, and the triangle at -,0, to 0,1. These have no representation in simpler notations. The triangular cupola, or half - cuboctahedron is xo3xx &#x.