# 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.

## Definition

### From first principles

A configuration is a tuple ${\displaystyle \left(V,E,*\right)}$ 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 ${\displaystyle \mid e\in E\,\,\mathrm {where} \,\,v*e\mid }$.
• The degree of all edges are equal, where the degree of a edge e  is ${\displaystyle \mid v\in V\,\,\mathrm {where} \,\,v*e\mid }$.
• For vertices ${\displaystyle v_{0}}$ and ${\displaystyle v_{1}}$ and edges ${\displaystyle e_{0}}$ and ${\displaystyle e_{1}}$, if ${\displaystyle v_{0}*e_{0}}$, ${\displaystyle v_{0}*e_{1}}$, ${\displaystyle v_{1}*e_{0}}$, and ${\displaystyle v_{1}*e_{1}}$ then either ${\displaystyle v_{0}=v_{1}}$ or ${\displaystyle e_{0}=e_{1}}$. 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 ${\displaystyle v_{0}}$ and ${\displaystyle v_{1}}$ and edges ${\displaystyle e_{0}}$ and ${\displaystyle e_{1}}$, if ${\displaystyle v_{0}*e_{0}}$, ${\displaystyle v_{0}*e_{1}}$, ${\displaystyle v_{1}*e_{0}}$, and ${\displaystyle v_{1}*e_{1}}$ then either ${\displaystyle v_{0}=v_{1}}$ or ${\displaystyle e_{0}=e_{1}}$. 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 ${\displaystyle \left(v_{\phi },e_{\psi }\right)}$ where

• v  is the number of vertices
• e  is the number of edges
• φ  is the degree of each vertex
• ψ  is the degree of each edge

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

Symbols are not unique, for example both the hexagon and the hexagram have the symbol (62).

Not every symbol has a configuration. For example the following conditions must be met for a symbol to have a configuration:

• ${\displaystyle v\times \phi =e\times \psi }$
• ${\displaystyle v\geq \phi (\psi -1)+1}$

## 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

• An injective mapping ${\displaystyle \eta :V\rightarrow S}$ from vertices to points in S
• An injective mapping ${\displaystyle \sigma :E\rightarrow {\mathsf {L}}(S)}$ from edges to lines in S

such that

${\displaystyle v*e\iff \eta (v)\in \sigma (e)}$

## Relationship to polygons

Most polygons are 2-configurations, with an n -gon having the symbol (n2). 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. Regular complex polygons have all vertices of same degree and all edges of same degree, and they are a special kind of configuration where points are in ${\displaystyle \mathbb {C} ^{2}}$ and lines are complex 1-spaces (or equivalently, points are in ${\displaystyle \mathbb {R} ^{4}}$ and lines are geometrically planes). For example, the Möbius–Kantor polygon is a realization of the Möbius–Kantor configuration.