In my particular notation, what is written is actually a 'trace' of the dynkin symbol, which matches in regular cases, but one needs other devices to complete the non-regular symboles. In any case, what is important is that the separate nodes be kept separate (although one can abreviate them). The thing is resolved as an oblique coordinate system, (which is why all of the arithmetic works), and thus while one can effectively drop 0 values or gather together equals under one head, it does no good to drop the coordinate system.
For example,
- Code: Select all
o 5 o 3 o 5 o 3 z o 3 o 3 o 3 o 3 z B z B zz
1 2 3 4 5 1 2 3 4 5 6 7
5 1 o--5--o 2 1 5 6 o-----o 2 7
| | | \ / |
| | | / \ |
4 o--5--o 3 4 o-----o 3
Nodes 5,6 are z, are at the same place as 1
Nodes 7 is zz, is at the same place as 2.
The branch between 5 and 6 is an S-2, so
goes from 5-2 to 6.
The branch between 6 and 7 is also S-2,
so goes 6-2 to 7, ie 5 to 7
One can, for example, write something like _3ox4_&xt is flat, for all values of _, but the essential graph must be kept.
Currently, the nodals are like this. a(A) constructs a short-chord of a polygon (A), which is expanded in the text.
The notation describes a good deal of what's been discussed here.
- a = atom node (these do not add in the way that x-nodes do: the vertex of the rhombo-dodecahedron lie at aq3o4a. Used to show stations in lattices.
- b = bevel node (these create a face perpendicular to this axis: eg the rhombododecahedron = o3b4o
- d = density
- e = reserved open.
- f = a(F) = shortchord of the pentagon the number 1.618033 = sqrt(2.618033)
- g = gyrate node, is the dual of 's', eg the dodecahedron could be written as o4g3g.
- h = a(H) = shortchord of the hexagon, the number 1.7320508 = sqrt(3)
- i = supplement modifier on the branch, so P/D becomes P/(P-D).
- j = overload symbol. It has no specific meaning, one must look up the overload table. currently: j5j2j5j = grand antiprism.
- k = a(K) = shortchord of the octagon: the number 1.847 759 065 = sqrt(3.41421356) [this used to represent sqrt(4/3)]
- m = mirror-node, used to construct duals over x. One could subset m to eg mq to produce the dual of a q-node.
- n = (reserved for generic arrays, eg na - nz, which might be expanded in an accompanying table).
- o = unmarked branch, notionally the vertex is bisected by this mirror.
- p(n) = shortchord of {n}, eg p(10) = 1.90211303259 = sqrt(3.73205080757)
- r = a(R) = shortchord of the digon. This is an infintesimally short edge, used to show a proto-figure as r4o3o = point-like cube
- s = snub node, here alternation and adjustment of a corresponding construction in 'x'
- t = lace-tower, the sections are stated in the proceeding list.
- tz = lace - ring. The lacing proceeds it.
- u = a(U) = shortchord of the horogon (U = W4). 2 = sqrt(4). This is the polygon inscribed in a euclidean straight arc.
- v = a(V) = shortchord of the pentagram (5/2) = 0.618033 = sqrt(0.381966)
- w(n) = sqrt(n), used to directly specify the shortchord of W(n) - mainly used of hyperbolics.
- x = a(S) = shortchord of the triangle, taken as 1 = sqrt(1)
- z = construct-node, representing a repetition of the first node
- zz = construct-node, repetition of the second node.
- : and :: = older varieties of z and zz
- $ (dollar-rune) = vertex-node. - mainly used of vertex-figures of corresponding x-form. eg vf o3x5o = o3$5o.
- *(letter), = construction-node, reperesenting the repetition of the node of that letter (counting A, B, C, ...) [Klitzing]
- +, *, ++ = asterix node, creates a 4/2, 6/3, or 8/4 branch from the previous to next branch. eg /3+3 = stella octangula.
- / = older form of x-node, /q = 'q' etc.
- % = edge needed to join two faces, eg in decorated orbiform notation.
- \ = older form of m node, \q = 'mq', etc.
- # = lace prism or tegum, being arranged according to a simplex arrangement.
The following letters might be used to construct branches. All kinds of branch are used, if one wants to set numbers directly against the symbols, eg 1S2S4. A branch connects S to O (which differ by 1), unless specified in the subject and object nodes.
The original use of this set was to replace Coxeter's use of subscripts at 2_21 etc, to 4B. It was then extended to include non-three branches, such as Q, F, H, V, eg 2F = {3,3,5}. Slashes were implemented to allow node marks, eg /1/1F = x3x3o5o. Extensions made to accomidate the 'second extension', that is hyperbolic groups of finite content, and some open forms (P, W, D). One could write PnS stellates to PnD2D1 and then PnD1D2, means {p,3} stellates to {p/2,p} and then {p,p/2}.
S was added to allow the system to be used as a coordinate system, with the branches as column-separators, eg "1.99 S 2.15 F0".
R and i were allowed to accomidate the socalled 'wythoff' (ie decorated schwarz-triangles) notation. So | 5 3 2 might become /FSR. The lower-case i is used to do supplement angles, so the triangle VSS 5/2 3 3 gives supplements ViSiS 5/3, 3/2, 3 and VSiSi. The slash and its backform allows one to create figures which give mirror-edge and the dual mirror-margin, eg F/SR is the icosahedron, and F\SR is the dodecahedron as a catalan dual. It is actually S/FR also. Note that R\SF is the rhombic tricontahedron o3m5o.
O allows circles and spheres to participate in products. The 'truncates' are read as increasing / or decreasing \ radii, allow ellipses. So /OO sphere, /O/O is a ellipsoid x < y = z. /OO/ is x = y < z. /O&x is a cylinder, \O&m is the corresponding circular tegum. In four dimensions, /O&/O is a duocylinder [(xx)(xx)], \O&\O is that figure's dual (bicurcular tegum) <(xx)(xx)>, /OO&x is a spherinder (spheric prism) = [(xxx)x]. Note that there is no notation for the crind product, eg ([xx]x) has no representation: figures follow notation: you got to watch that!
- A = second-subject node, ie a '3' branch connecting S-1 to O - usually at the tail: E is the second-object node
- B = third-subject node, a '3' branch connectiong S-2 to O - usually at the tail: G is the third-object node
- C = fourth-subject node, a '3' branch connecting S-3 to O
- D = density, a referrent to a previous P or W node. eg P12D5 is {12/5}
- E = second-object node, a '3' branch connecting S to O+1
- F = branch marked '5', representing an angle of 12p
- G = third-object node, a '3' branch connecting S to O+2
- H = branch marked '6', representing an angle of 10p
- I = creates a supplement angle, ie F = 5/1, FI = 5/(5-1): angle is 60p - proceding
- K = branch marked '8', representing an angle of 7p60s
- O = sphere-branch - O = circle, OO = sphere, etc.
- P = polygon, eg P12 is the dodecagon.
- Q = branch marked '4' represent an angle of 15p
- R = branch marked '2' representing an angle of 30p
- S = branch marked '3' representing an angle of 20p
- T = (occasionally used of the hexagram)
- U = branch marked (oo), representing an angle of 0p. This is the horogon W4 only. It is not used of bollogons.
- V = branch marked (5/2)
- W = branch to specify infinitogons one bollogons, giving the shortchord, rather than the number of sides, eg W(4/3) never closes, but is finite.
- & = orthogonal product.
The following do not create a solid per se, but ought be noted.
- ".." = quotes, used to delinearate a figure, eg "o3o5o".
- || = joins two parallel figures by x-lacing, after Klitzing. as in A atop B atop C gives A || B || C . This can be written by multiple node-values too.
- *A etc = return to the first, etc figure of a list, in Klitzing's 'atop' notation eg "o3x || x3x || x3o || o3o || *A" creates a loop (square), in four dimensions. This replicate his *a nodal.
- , = separates values of a general list, eg o3o5o, o3o5x, &c.