Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Incidence geometry
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Fundamental geometric examples == Some of the basic concepts and terminology arises from geometric examples, particularly [[projective plane]]s and [[Affine plane (incidence geometry)|affine planes]]. ===Projective planes=== {{main|projective plane}} A ''projective plane'' is a linear space in which: * Every pair of distinct lines meet in exactly one point, and that satisfies the non-degeneracy condition: * There exist four points, no three of which are [[collinear]]. There is a [[bijection]] between {{math|''P''}} and {{math|''L''}} in a projective plane. If {{math|''P''}} is a finite set, the projective plane is referred to as a ''finite'' projective plane. The '''order''' of a finite projective plane is {{math|1=''n'' = ''k'' – 1}}, that is, one less than the number of points on a line. All known projective planes have orders that are [[prime power]]s. A projective plane of order {{math|''n''}} is an {{math|((''n''<sup>2</sup> + ''n'' + 1)<sub>''n'' + 1</sub>)}} configuration. The smallest projective plane has order two and is known as the ''Fano plane''. [[File:Fano plane.svg|thumb|{{center|Projective plane of order 2 <br> the Fano plane}}]] ==== Fano plane ==== {{main|Fano plane}} This famous incidence geometry was developed by the Italian mathematician [[Gino Fano]]. In his work<ref>{{citation|first=G.|last=Fano|title=Sui postulati fondamentali della geometria proiettiva|year=1892|journal=Giornale di Matematiche|volume= 30|pages=106–132}}</ref> on proving the independence of the set of axioms for [[Projective space|projective ''n''-space]] that he developed,<ref>{{harvnb|Collino|Conte|Verra|2013|loc=p. 6}}</ref> he produced a finite three-dimensional space with 15 points, 35 lines and 15 planes, in which each line had only three points on it.<ref>{{harvnb|Malkevitch}} Finite Geometries? an AMS Featured Column</ref> The planes in this space consisted of seven points and seven lines and are now known as [[Fano plane]]s. The Fano plane cannot be represented in the [[Euclidean plane]] using only points and straight line segments (i.e., it is not realizable). This is a consequence of the [[Sylvester–Gallai theorem]], according to which every realizable incidence geometry must include an ''ordinary line'', a line containing only two points. The Fano plane has no such line (that is, it is a [[Sylvester–Gallai configuration]]), so it is not realizable.{{sfnp|Aigner|Ziegler|2010}} A [[complete quadrangle]] consists of four points, no three of which are collinear. In the Fano plane, the three points not on a complete quadrangle are the diagonal points of that quadrangle and are collinear. This contradicts the ''Fano axiom'', often used as an axiom for the Euclidean plane, which states that the three diagonal points of a complete quadrangle are never collinear. ===Affine planes=== {{main|affine plane (incidence geometry)}} An ''affine plane'' is a linear space satisfying: * For any point {{math|''A''}} and line {{math|''l''}} not incident with it (an '''anti-flag''') there is exactly one line {{math|''m''}} incident with {{math|''A''}} (that is, {{math|''A'' I ''m''}}), that does not meet {{math|''l''}} (known as [[Playfair's axiom]]), and satisfying the non-degeneracy condition: * There exists a triangle, i.e. three non-collinear points. The lines {{math|''l''}} and {{math|''m''}} in the statement of Playfair's axiom are said to be ''parallel''. Every affine plane can be uniquely extended to a projective plane. The ''order'' of a finite affine plane is {{math|''k''}}, the number of points on a line. An affine plane of order {{math|''n''}} is an {{math|((''n''<sup>2</sup>)<sub>''n'' + 1</sub>, (''n''<sup>2</sup> + ''n'')<sub>''n''</sub>)}} configuration. [[File:Hesse configuration.svg|thumb|left|{{center|Affine plane of order 3 <br> (Hesse configuration)}}]] ====Hesse configuration==== {{main|Hesse configuration}} The affine plane of order three is a {{math|(9<sub>4</sub>, 12<sub>3</sub>)}} configuration. When embedded in some ambient space it is called the '''[[Hesse configuration]]'''. It is not realizable in the Euclidean plane but is realizable in the [[complex projective plane]] as the nine [[inflection point]]s of an [[elliptic curve]] with the 12 lines incident with triples of these. The 12 lines can be partitioned into four classes of three lines apiece where, in each class the lines are mutually disjoint. These classes are called ''parallel classes'' of lines. Adding four new points, each being added to all the lines of a single parallel class (so all of these lines now intersect), and one new line containing just these four new points produces the projective plane of order three, a {{math|(13<sub>4</sub>)}} configuration. Conversely, starting with the projective plane of order three (it is unique) and removing any single line and all the points on that line produces this affine plane of order three (it is also unique). Removing one point and the four lines that pass through that point (but not the other points on them) produces the {{math|(8<sub>3</sub>)}} [[Mobius-Kantor configuration|Möbius–Kantor configuration]].
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)