Configuration

Configurations are a type of incidence geometry that generalizes the idea of a polygon. Unlike abstract polytopes which generalize polygons to higher ranks but follow the diamond property, configurations are all rank 2 but generalize the diamond property.

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 $$\left(n_2\right)$$. 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 are not dyadic.