Editing Incidence structure (section)

===Pictorial representations===
An incidence figure (that is, a depiction of an incidence structure), is constructed by representing the points by dots in a plane and having some visual means of joining the dots to correspond to lines.<ref name=Beth17 /> The dots may be placed in any manner, there are no restrictions on distances between points or any relationships between points. In an incidence structure there is no concept of a point being between two other points; the order of points on a line is undefined. Compare this with [[ordered geometry]], which does have a notion of betweenness. The same statements can be made about the depictions of the lines. In particular, lines need not be depicted by "straight line segments" (see examples 1, 3 and 4 above). As with the pictorial representation of [[Graph (discrete mathematics)|graphs]], the crossing of two "lines" at any place other than a dot has no meaning in terms of the incidence structure; it is only an accident of the representation. These incidence figures may at times resemble graphs, but they aren't graphs unless the incidence structure is a graph.

====Realizability====
Incidence structures can be modelled by points and curves in the [[Euclidean plane]] with the usual geometric meaning of incidence. Some incidence structures admit representation by points and (straight) lines. Structures that can be are called ''realizable''. If no ambient space is mentioned then the Euclidean plane is assumed. The Fano plane (example 1 above) is not realizable since it needs at least one curve. The Möbius–Kantor configuration (example 4 above) is not realizable in the Euclidean plane, but it is realizable in the [[complex plane]].<ref>{{harvnb|Pisanski|Servatius|2013|page=222}}</ref> On the other hand, examples 2 and 5 above are realizable and the incidence figures given there demonstrate this. Steinitz (1894)<ref>E. Steinitz (1894), ''Über die Construction der Configurationen'' {{math|''n''<sub>3</sub>}}, Dissertation, Breslau</ref> has shown that {{nowrap|{{math|''n''<sub>3</sub>}}-configurations}} (incidence structures with {{mvar|n}} points and {{mvar|n}} lines, three points per line and three lines through each point) are either realizable or require the use of only one curved line in their representations.<ref>{{citation|first=Harald|last=Gropp|title=Configurations and their realizations|journal=Discrete Mathematics|year=1997|volume=174|issue=1–3 |pages=137–151|doi=10.1016/s0012-365x(96)00327-5|doi-access=free}}</ref> The Fano plane is the unique ({{math|7<sub>3</sub>}}) and the Möbius–Kantor configuration is the unique ({{math|8<sub>3</sub>}}).