Configuration

Configurations are a type of incidence geometry that generalize the idea of a polygon, allowing edges to connect any fixed number of colinear vertices and vertices to connect to any fixed number of edges. Unlike abstract polytopes which generalize polygons to higher ranks but follow the diamond property, configurations are all rank 2 but generalize the diamond property. Configurations are studied both as abstract combinatorial objects and with realizations in Euclidean and projective space.

From first principles
A configuration is a tuple $$\left(V,E,*\right)$$ where:
 * $V$ is a set whose elements are called vertices.
 * $E$ is a set whose elements are called edges.
 * $$ is a binary relation between edges and vertices.
 * The degree of all vertices are equal, where the degree of a vertex $v$ is $$\mid e\in E\,\,\mathrm{where}\,\,v*e\mid$$.
 * The degree of all edges are equal, where the degree of a edge $e$ is $$\mid v\in V\,\,\mathrm{where}\,\,v*e\mid$$.
 * For vertices $$v_0$$ and $$v_1$$ and edges $$e_0$$ and $$e_1$$, if $$v_0 * e_0$$, $$v_0 * e_1$$, $$v_1 * e_0$$, and $$v_1 * e_1$$ then either $$v_0 = v_1$$ or $$e_0 = e_1$$. In other words, two edges can intersect at most one vertex and two vertices can be connected by at most one edge.

As an incidence geometry
A configuration is an incidence geometry with two types: vertices and edges, such that:
 * Every edge is incident on the same number of vertices.
 * Every vertex is incident on the same number of edges.
 * For vertices $$v_0$$ and $$v_1$$ and edges $$e_0$$ and $$e_1$$, if $$v_0 * e_0$$, $$v_0 * e_1$$, $$v_1 * e_0$$, and $$v_1 * e_1$$ then either $$v_0 = v_1$$ or $$e_0 = e_1$$. In other words, two edges can intersect at most one vertex and two vertices can be connected by at most one edge.

Symbols
A configuration can be given the symbol $$\left(v_\phi,e_\psi\right)$$ where
 * $v$ is the number of vertices
 * $e$ is the number of edges
 * is the degree of each vertex
 * is the degree of each edge

When $$v_\phi = e_\psi$$ the symbol can be abbreviated to $$\left(v_\phi\right)$$.

Symbols are not unique, for example both the hexagon and the hexagram have the symbol.

Not every symbol has a configuration. For example the following conditions must be met for a symbol to have a configuration:
 * $$v\times\phi = e\times \psi$$
 * $$v \geq \phi(\psi-1)+1$$

Realization
While the definition is synthetic, configurations are often realized in Euclidean or projective space. A realization of a configuration in a space $S$ consists of such that
 * An injective mapping $$\eta : V \rightarrow S$$ from vertices to points in $S$
 * An injective mapping $$\sigma : E \rightarrow \mathscr{L}(S)$$ from edges to lines in $S$
 * $$v*e \iff \eta(v) \in \sigma(e)$$

Relationship to polygons
Most polygons are 2-configurations, with an $n$-gon having the symbol. The digon and the monogon which are valid polygons under some definitions are not valid 2-configurations. The monogon is a valid 1-configuration.

Polygon compounds are also valid 2-configurations as there is no requirement of connectivity in the definition of a configuration.

Other configurations are generally exotic polygonoids, as they do not satisfy the diamond property.

Relationship to complex polygons
Like configurations, complex polygons are a generalization of polygons not required to have dyadic edges. The abstract structure of complex polygons is thus a configuration. And many configurations have analygous regular complex polygons. For example the Möbius–Kantor configuration corresponds to the Möbius–Kantor polygon.