Incidence Matrices
A polytope can also be represented by tabulating its incidences. The following incidence matrix is that of a triangle:
| ø | a | b | c | ab | bc | ca | abc | |
|---|---|---|---|---|---|---|---|---|
| ø | • | • | • | • | • | • | • | • |
| a | • | • | • | • | • | |||
| b | • | • | • | • | • | |||
| c | • | • | • | • | • | |||
| ab | • | • | • | • | • | |||
| bc | • | • | • | • | • | |||
| ca | • | • | • | • | • | |||
| abc | • | • | • | • | • | • | • | • |
The table shows a dot wherever a face is a subface of another, or vice versa (so the table is symmetric about the diagonal)- so in fact, the table has redundant information; it would suffice to show only a dot when the row face ≤ the column face.
Since both the body and the empty set are incident with all other elements, the first row and column as well as the last row and column are trivial and can conveniently be omitted.
Further information is gained by counting each occurrence of an incidence as 1 (and hence non-incidence as 0). This numerative usage enables a symmetry grouping, as in the Hasse Diagram of the square pyramid: If vertices B, C, D, and E are considered symmetrically equivalent within the abstract polytope, then edges f, g, h, and j will be grouped together, and also edges k, l, m, and n, And finally also the triangles 'P', 'Q', 'R', and 'S'. Thus the corresponding incidence matrix of this abstract polytope may be shown as:
| A | B,C,D,E | f,g,h,j | k,l,m,n | P,Q,R,S | T | |
|---|---|---|---|---|---|---|
| A | 1 | * | 4 | 0 | 4 | 0 |
| B,C,D,E | * | 4 | 1 | 2 | 2 | 1 |
| f,g,h,j | 1 | 1 | 4 | * | 2 | 0 |
| k,l,m,n | 0 | 2 | * | 4 | 1 | 1 |
| P,Q,R,S | 1 | 2 | 2 | 1 | 4 | * |
| T | 0 | 4 | 0 | 4 | * | 1 |
In this accumulated incidence matrix representation the diagonal entries represent the total counts of either element type.
Elements of different type of the same rank clearly are never incident so the value will always be 0, however to help distinguish such relationships, an asterisk (*) is used instead of 0.
The sub-diagonal entries of each row represent the incidence counts of the relevant sub-elements, while the super-diagonal entries represent the respective element counts of the vertex-, edge- or whatever -figure.
Already this simple square pyramid shows that the symmetry-accumulated incidence matrices are no longer symmetrical. But there is still a simple entity-relation (beside the generalised Euler formulae for the diagonal, respectively the sub-diagonal entities of each row, respectively the super-diagonal elements of each row - those at least whenever no holes or stars etc. are considered), as for any such incidence matrix holds:
Read more about this topic: Abstract Polytope
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