Are you aware of the respective explanatory pages right on the same site?
https://bendwavy.org/klitzing/explain/incmat.htmhttps://bendwavy.org/klitzing/explain/dynkin.htmResp. the following notational compendium?
https://bendwavy.org/klitzing/explain/d ... tation.htmIncidence MatricesCommon are just 0-1-incidence matrices, which do relate just 2 types of elements, say vertices and edges. Their reasoning is obvious.
The clue here his not only to combine
all dimensional elements within a single huge matrix, but to apply symmetry aggregations as well. This is kind like the coset classes in algebra. In fact symmetry does apply an equivalence relation. Thus it is relevant only to consider incidences of general representants each. And you should count the total class size then.
This is what is provided there:
- Iij provides the count of elements of type i, which is incident to elements of type j (i<>j)
- Iii provides the total count of elements of type i
For regular polyhedra this type of matrix already is contained in the first pages of Coxeter's famous book on Regular Polytopes. The same thing thereafter re-occurs therein sporadically within proofs and some historical remarks, even in its application to higher dimensions. Indepted to the topic of this book, therein you'll not find any application of this matrix onto non-regular polytopes. I also have not found any other literature thereon with this extended application. But this is straight Forward non the less: you then just have several to be distinguished element types of the same dimensionality. E.g. the cuboctahedron has 2 types of faces, triangles and squares. This is why I was speaking all the way of "types" instead of dimensionalities.
Thus the incidence matrix for that mentioned polyhedron obviously reads:
- Code: Select all
12 | 4 | 2 2 : vertices \ vertex figure
---+----+----
2 | 24 | 1 1 : edges \ edge figure
---+----+----
3 | 3 | 8 * : triangular faces
4 | 4 | * 6 : square faces
Those numbers contained within the incdence matrices are not only mere counts. There are relations between those individual numbers additionally. In fact you'll have:
- The diagonal elements summed resp. subtracted according to the evenness/oddness of the according dimension would clearly follow to the Euler formula of the total polytope of consideration.
- The subdiagonal elements of every row then would describe nothing but the diagonal elements of the incidence matrix of the considered subelement, and thus would follow the respective Euler relation as well.
- The supediagonal elements of each row will describe nothing but the diagonal elements of the incidence matrix of the considered vertex- / edge- / etc. -figure. Thus there again the according Euler relation does apply.
- There is also a sliding square rule: Iii*Iij=Iji*Ijj (i<j)
Dynkin DiagramsDynkin once came up with a graphical device to represent Lie algebras. Coxeter then found that those also could be used to represent the symmetry groups of polytopes. In fact every node represents a mirror, and every link represents an dihedral angle between those mirrors of the incident nodes. In order to become a repetitive rotational symmetry, these angles then have to be fractions of pi (180 degrees). In general one mostly uses submultiplicatives. Therefore the inverse of this fraction (thus mostly an integrer) is being used to label the according link of the graph.
Because label 2 would have been used very often, such links are simply not drawn at all. And for labels being 3, usually this label is dropped, resulting this time in an unmarked link only.
When going on from the mere symmetry group description towards the polytopal description, then the Wythoffian
kaleidoscopical construction applies. For that purpose a seed point somewhere within or on the boundary of the fundamental region (between those mirror planes) is chosen. In the diagram this position usually is encoded in ringing all those nodes, where the point is not on, resp. leaving the nodes unringed, where the seed point is incident to the corresponding mirror. This seed point then will become one of the vertices of the polytope. The others are obtained by all its mirror Images. Two such points then are to be connected by an edge, whenever those are obtained as direct Images. A closed, planar sequence of edges around an intersection of mirrors then defines a face. Etc.
--- rk