# Incidence geometry

An **incidence geometry** or **Buekenhout geometry** is a type of combinatorial structure comprising a typed set of elements with an incidence relation describing how they are connected. Incidence geometries are highly general and encompass abstract polytopes, configurations, hypertopes, generalized polygons, and many other concepts.

## Definition[edit | edit source]

### Incidence system[edit | edit source]

An **incidence system** is a tuple where

- X is a set whose members are called
**elements**. - I is a finite set whose members are called
**types**. (Types generalize the notion of rank in polytopes, or the distinction between nodes and edges in a graph.) - is a
**type function**, mapping each element to a type. - * is a binary relation on X called
**incidence**. If is true, we say that x and y are**incident**. The relation must obey three properties:- (reflexivity)
- (symmetry)
- . This states that two distinct elements of the same type cannot be incident (for example, in a graph, two vertices are not considered incident).

In the context of incidence system a **flag** is a set of elements, F , such that any two elements in F are incident. This is a distinct notion from the flag of a polytope. It follows from the final condition on the incidence relation that all the types of elements in a flag must be distinct.

A **chamber** is a flag, F , such that (using the image). In a chamber, every type is represented exactly once.

### Incidence geometry[edit | edit source]

An **incidence geometry** is an incidence system such that every flag is a subset of some chamber. In other words, every maximal clique in the incidence graph has type . A necessary but not sufficient condition is that every element must be incident on elements of every other type.

## Incidence structure[edit | edit source]

A incidence system with two types is called an **incidence structure**. Incidence structures in general are too broad of a category to make them interesting objects of study on their own, but several subclasses of incidence structures have interesting behavior:

- Graphs (including definitions of graphs that permit loops and multiple edges)
- hypergraphs
- Block designs
- Configurations
- Generalized polygons, which include projective planes
- Near polygons

If every element in an incidence structure is incident on at least one other distinct element, then it is also an incidence geometry. For example, a graph is an incidence geometry iff every vertex has a degree of at least 1.

An incidence structure may be defined more simply as a triple where P is a set of vertices, L is a set of edges and I is an incidence relation. Some authors require that I is symmetric, however this is only required of you do not require I to be typed.

## Examples[edit | edit source]

### Abstract polytopes[edit | edit source]

Every abstract polytope is an incidence geometry.

- The element set are the elements of the abstract polytope.
- The incidence relation is incidence.
- The type function is the rank of each element.
- The chambers are flags of the abstract polytope.

## External links[edit | edit source]

- Wikipedia contributors. "Buekenhout geometry".
- Wikipedia contributors. "Incidence structure".

## Bibliography[edit | edit source]

- Fernandes, Maria (2014). "Regular and chiral hypertopes" (PDF).
- Fernandes, Maria; Leemans, Dmitri; Weiss, Asia (2016). "Highly symmetric hypertopes".
*Aequationes mathematicae*.