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Used notational elements of (linearised) Dynkin Diagrams

This page just aims for a short reference. Detailed explanations can be found here.

Link symbols Example
3
4
5
5/2
etc., in general:
n/d
subdivisor of π used as angle between the adjacent mirrors (node symbols) x4/3x3o
2 links marked 2 are usually omitted (left blank) denoting the polytope being the prism-product of the polytopes of the unconnected component-symbols

in cases of snubs they are re-introduced in order to show that the semiation runs across the thus connected components, instead of being a prism-product of (separate) snub components
x x4o


s2s3s
' Refering to the inversion between prograde and retrograde polygons {n/d} ↔ {(n/d)'} = {n/(n-d)}. Wrt. n → ∞ their usages within an euclidean context become equivalent in geometrical representation, i.e. both being aze. Even so as abstract polytopes they still have to be distinguished, and thus esp. their usage as link marks in Dynkin diagrams. x3o3o∞'*a
Ø This is a tribute to the more general Coxeter domains, i.e. non-simplicial fundamental domains. This is the non-intersection symbol (nodes, which represent facets of the domain, which would not intersect, get linked by this symbol). x3xØx4xØ*a5*c
- sometimes introduced on both sides of a link-mark to separate those more clearly from the node symbols x-n/d-o
Node symbols Example
o unringed (real) node

seed-point of construction lies on that mirror
o3o5x
x ringed (real) node

seed-point of construction lies off that mirror, thus demarking (usually) a unit edge
x(2)   = o
x(3)   = x
x(4)   = q
x(5)   = f
x(5/2) = v
x(6)   = h
x(8)   = k
x(∞) = x(6,3) = u
x(8,3) = w
x(10,3) = F
in general:
x(n/d,m)



uppercase letters
edge of length 0 (or no edge at all)
edge of length 1
edge of length sqrt(2) = 1.414214
edge of length (1+sqrt(5))/2 = 1.618034
edge of length (sqrt(5)-1)/2 = 0.618034
edge of length sqrt(3) = 1.732051
edge of length sqrt[2+sqrt(2)] = 1.847759
edge of length 2
edge of length 1+sqrt(2) = 2.414214
edge of length (3+sqrt(5))/2 = 2.618034

edge of length sin(π md/n)/sin(π d/n)
occuring as chord of a regular {n/d}-gon, connecting a vertex with its m-th successor
(if m is omitted as argument, then m=2 is understood)

other edge lengths, to be defined within a local context ad hoc
u3x4o














F=ff=x+f
*a
*b
*c
etc.
virtual nodes, introduced just for linearization of the symbol, refferring to the re-visitation of the first (a-th), second (b-th), third (c-th), etc. real node position (where counting starts at the left of the symbol) x3o3o *b3o3o
x3o3o3*a
s snub node s5/2s5s
β holosnub node β3o3x
m mirror-margin node. Only being used in the o and m combination. Refers then to the dual of that polytope, where the m nodes would be replaced by x nodes. o3m3m3o
Lacings Example
&#. Lace-prisms (or more general lace-simplices) use vertices arranged in several "layers" with some common symmetry. These cross-sections are linked by lacing edges of length ".", i.e. corresponding to that node-symbol; usually "&#x". That lacing mark is attached to the right end of the double load (prisms) resp. multiple load (count of mutually connected layers ≥ 2: simplices) of the symmetry-graph. xx3ox&#x
&#.t Lace-tower, i.e. a stack of lace-prisms xux3oox&#xt
&#.r Lace-rings, i.e. the body encompassed by a circuit of lace-prisms (not the thus described torus) xxoo&#xr
&#z. As preceding qualifier "z" denotes degenerate lace prisms, towers, etc., showing additionally that all respective heights become zero. This symbol moreover becomes required, when all "layer" subsets ought to be pseudo elements solely.
The operant .&#zx is closely related to the tegum sum.
fxo ofx xof&#zx
&#.z Used as suffixing qualifier, "z" implies the figure being designed first as without that suffix. But then apply a scaling onto the lacings, such that the respective segmental heights all become zero. (Aka: telescope view / projection from infinity.)  
: ... : Repetition sign of layers; also used for subelements when those would run across that artificial boarder :x.o: ... :o.x:3*a&#x
&##x Infinit repetion of lace tower stacking. :xoo:3:oxo:3:oox:3*a&##x


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