Incidence (geometry)

Incidence in is geometry the most simple relationship between the geometric elements such as point , straight line , circle , level can occur etc.. Incidence exists when, for example, a point lies on a straight line , a plane contains a straight line, or vice versa. Mathematically speaking, it is a relation , i.e. H. a subset of the union of the Cartesian products of the set of points with the set of lines, the set of lines with the set of points, the set of planes with the set of straight lines, the set of points with the set of planes, etc.

definition

A geometric structure with incidence relation is a mathematical structure

${\displaystyle {\mathfrak {I}}=\langle M_{1},M_{2},\dots ,{\textbf {F}}\rangle }$

consisting of sets of points, lines, planes etc. together with a relation ${\displaystyle M_{i}}$

${\displaystyle {\textbf {F}}\subseteq M_{1}\times M_{2}\cup M_{1}\times M_{3}\cup \ldots M_{2}\times M_{3}\cup \ldots M_{2}\times M_{1}\cup \ldots ,}$

which defines the incidence. (In the union of Cartesian products on the right, the products of all pairs of sets with belonging to the structure are formed.) The relation is also called the flag set of the structure. ${\displaystyle (M_{j},M_{k})}$ ${\displaystyle j\neq k}$ ${\displaystyle {\textbf {F}}}$

History and meaning

The term incidence has played a role in geometry at least since David Hilbert's axiomatic foundation , since Hilbert's approach no longer attempts to describe the “nature” of geometric objects, but rather these objects are defined solely by their mathematically comprehensible relationships to one another. Hilbert calls his axioms of incidence "axioms of connection" and summarizes them in group I of his system of axioms. The axiom of parallels , which formally also belongs to the incidence axioms, forms a separate group (IV) for Hilbert. If one renounces the axiom of parallels and weakens Hilbert's axiom group III (axioms of congruence), one arrives at absolute geometry , a generalization also for non-Euclidean geometries .

In synthetic geometry , incidence geometry is understood even more generally as a geometric structure that is based solely on incidence axioms (and possibly other richness axioms ).

In the more recent, in particular the Anglo-American literature, the term incidence (as a separately defined relation) is often dispensed with and the relation is largely replaced in terms of content by the "is element of" relation or, more generally, "is a subset of" relation and its inversions. Then the incidence is a generic term for these set- theoretically defined relations. The advantage of the classic incidence relation is that this relation can be defined symmetrically and thus allows more elegant formulations for dualisable statements of the projective geometry . In addition, in this way you can also describe a geometry in which there are different empty objects, such as straight lines that do not intersect with any point. Such applications have proven to be unproductive and hardly survived.

The original, historical purpose of defining a "contain or include" relation that is not based on the element relation and the subset relation, was probably to use as few axioms of set theory as possible when constructing the geometry. From today's point of view, the related objects are to be regarded as sets in the sense of the Zermelo-Fraenkel set theory even if the geometric axioms are formulated with a non-set-theoretical incidence relation (in which, for example, straight lines are not sets of points, but can intersect with points) .

Ways of speaking

In addition to the well-known ways of saying "a point p lies on a straight line G" or "a plane contains a straight line G" for "p incised with G" or "G incised with ", the following forms of speech are also common: ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$

• If two different straight lines intersect with the same point, this is the intersection of the straight lines.
• If two different points intersect with the same straight line, this is the straight line connecting the points.
• If several points intersect with the same line, they are called collinear .
• If several straight lines intersect with the same point, they are called copunctal .

Examples of structures with an incidence relation

• affine and projective planes
• Polygons and polyhedra
• Euclidean space
• Möbius level
• With a suitable definition of the incidence, however, also tables, chairs and beer mugs .

literature

• Hilbert, David: Fundamentals of Geometry, Stuttgart - Leipzig: Teubner (14th edition 1999)
• F.Buekenhout: Handbook of Incidence Geometry. North Holland 1995. ISBN 978-0-444-88355-1
• Jeremy Gray: Worlds out of nothing: a course of the history of geometry of the 19th century . Springer, 2007, ISBN 978-0-85729-059-5 .
• Charles Weibel: Survey of Non-Desarguesian Planes. Notices of the American Mathematical Society , Volume 54 November 2007, pp. 1294-1303. Full text (PDF; 719 kB)

Web links

• Incidence Geometry at PlanetMath

Individual evidence

1. ↑ Weibel (2007)
2. ^ Gray (2007)
3. ^ Gray (2007)