| Site Map | Polytopes | Dynkin Diagrams | Vertex Figures, etc. | Incidence Matrices | Index |
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 | |
| xn, on | n-fold node, being used in complex polytopes | x3-3-o3 |
|
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 x(12,3) = e 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 1+sqrt(3) = 2.732051 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) esp.: x(n/d) = x(n/d,2) = 2 cos(π d/n), x(n/d,3) = (x(n/d))2-1 other edge lengths, to be defined within a local context ad hoc |
u3x4o v=f-x w=x+q F=ff=x+f e=x+h |
|
*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 |
| Further sporadically used operators | Example | |
| ambo( ... ) | Used to denote the remainder of the figure under the ambification operation. | |
| both( ... ) | Being used within holosnubs to denote both classes of the to be alternated elements. | |
| demi( ... ) | Being used within snubs or other alternated facetings to denote just one class of the to be alternated elements. | |
| holo( ... ) | Used to denote the according holosnub. | |
| hull( ... ) | Convex hull of respective polytope. | |
| mids( ... ) | Being used to denote the new vertex on each former edge under the rectification operation. In fact, when doing ambification instead, it would be the geometrical midpoint of the respective edge by definition. | |
| rect( ... ) | Used to denote the remainder of the figure under the rectification operation. | |
| sefa( ... ) | Short for "sectioning facet underneath", being used to denote the new facet occuring underneath the being omitted element within alternated facetings. It is just the generalization of the usage of verf( ... ) within snubs. | |
| snub( ... ) | Used to denote the according snub. | |
| trops( ... ) | Short for "tropicals", being used to denote the 2 new vertices on each former edge under the truncation operation. | trops( x . . . ) |
| trunc( ... ) | Used to denote the remainder of the figure under the truncation operation. | trunc( x4o . . ) |
| verf( ... ) | Used to denote the vertex figure. | |
| ...:... |
Used to denote (mostly hyperbolic) polytopes which have the elements denoted before the colon solely as its margins (i.e. (d-2)-elements) and further use the bit after the colon as its vertex figure. Here the part before the colon is meant to be read in the first run like a regular polytope: i.e. x3o:... will denote the usual triangle being used for margins, whereas e.g. o3x:... then is a short-cut for rect( x3o:... ) instead, and x3x:... then similarily represents trunc( x3o:... ). | x3o:s3s4s |
© 2004-2024 | top of page |