In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
H. S. M. Coxeter called a
regular polytope a special kind of configuration.
Other
configurations in geometry are something different. These ''polytope configurations'' may be more accurately called
incidence matrices, where like elements are collected together in rows and columns. Regular polytopes will have one row and column per
''k''-face element, while other polytopes will have one row and column for each k-face type by their symmetry classes. A polytope with no symmetry will have one row and column for every element, and the matrix will be filled with 0 if the elements are not connected, and 1 if they are connected. Elements of the same ''k'' will not be connected and will have a "*" table entry.
Every polytope, and
abstract polytope has a
Hasse diagram expressing these connectivities, which can be systematically described with an
incidence matrix.
Configuration matrix for regular polytopes
A configuration for a regular polytope is represented by a matrix where the diagonal element, N
''i'', is the number of ''i''-faces in the polytope. The diagonal elements are also called a polytope's
f-vector. The nondiagonal (''i'' ≠ ''j'') element N
''ij'' is the number of ''j''-faces incident with each ''i''-face element, so that N
''i''N
''ij'' = N
''j''N
''ji''.
The principle extends generally to dimensions, where .
:
Polygons
A
regular polygon,
Schläfli symbol , will have a 2x2 matrix, with the first row for vertices, and second row for edges. The
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
''g'' is 2''q''.
: