In mathematics, a non-Desarguesian plane is a

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...

that does not satisfy

Desargues' theorem In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...

(named after

Girard Desargues Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are ...

), or in other words a plane that is not a

Desarguesian plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...

. The theorem of Desargues is true in all

projective spaces In mathematics, the concept of a projective space originated from the visual effect of perspective (graphical), perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean s ...

of dimension not 2; in other words, the only projective spaces of dimension not equal to 2 are the classical projective geometries over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

(or

division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...

). However,

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

found that some projective planes do not satisfy it. The current state of knowledge of these examples is not complete.

Examples

There are many examples of both

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

and infinite non-Desarguesian planes. Some of the known examples of infinite non-Desarguesian planes include: *The

Moulton plane In incidence geometry, the Moulton plane is an example of an affine plane in which Desargues's theorem does not hold. It is named after the American astronomer Forest Ray Moulton. The points of the Moulton plane are simply the points in the real pl ...

. *

Moufang plane In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a ''translation line'', that is, a line with the property ...

s over alternative division rings that are not associative, such as the projective plane over the octonions. Since all finite alternative division rings are fields (

Artin–Zorn theorem In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Arti ...

), the only non-Desarguesian Moufang planes are infinite. Regarding finite non-Desarguesian planes, every projective plane of order at most 8 is Desarguesian, but there are three non-Desarguesian examples of order 9, each with 91 points and 91 lines. They are: *The

Hughes plane In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by . There are examples of order ''p''2''n'' for every odd prime ''p'' and every positive integer ''n''. Construction The construction of a Hughes plane is based ...

of order 9. *The

Hall plane of order 9 In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''2''n'' > 4. Algebraic cons ...

. Initially discovered by Veblen and Wedderburn, this plane was generalized to an infinite family of planes by Marshall Hall. Hall planes are a subclass of the more general André planes. *The

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of the Hall plane of order 9. Numerous other constructions of both finite and infinite non-Desarguesian planes are known, see for example . All known constructions of finite non-Desarguesian planes produce planes whose order is a proper prime power, that is, an integer of the form p^e, where p is a prime and e is an integer greater than 1.

Classification

Hanfried Lenz gave a classification scheme for projective planes in 1954, which was refined by Adriano Barlotti in 1957. This classification scheme is based on the types of point–line transitivity permitted by the

collineation group In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus ...

of the plane and is known as the ''Lenz–Barlotti classification of projective planes''. The list of 53 types is given in and a table of the then known existence results (for both collineation groups and planes having such a collineation group) in both the finite and infinite cases appears on page 126. As of 2007, "36 of them exist as finite groups. Between 7 and 12 exist as finite projective planes, and either 14 or 15 exist as infinite projective planes." Other classification schemes exist. One of the simplest is based on special types of

planar ternary ring In mathematics, an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping T \colon R^3 \to R \, may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marsh ...

(PTR) that can be used to coordinatize the projective plane. These types are

fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...

, skewfields,

alternative division ring In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is o ...

semifield In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. Overview The term semifield has two conflicting meanings, both of which incl ...

s, nearfields, right nearfields,

quasifield In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields. Definition A qua ...

s and right quasifields.

Conics and Ovals

In a Desarguesian projective plane a

conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...

can be defined in several different ways that can be proved to be equivalent. In non-Desarguesian planes these proofs are no longer valid and the different definitions can give rise to non-equivalent objects. Theodore G. Ostrom had suggested the name ''conicoid'' for these conic-like figures but did not provide a formal definition and the term does not seem to be widely used. There are several ways that conics can be defined in Desarguesian planes: # The set of absolute points of a polarity is known as a

von Staudt conic In projective geometry, a von Staudt conic is the point set defined by all the absolute points of a polarity that has absolute points. In the real projective plane a von Staudt conic is a conic section in the usual sense. In more general projective ...

. If the plane is defined over a

of characteristic two, only

degenerate conic In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers (or more ...

s are obtained. # The set of points of intersection of corresponding lines of two pencils which are projectively, but not perspectively, related is known as a

Steiner conic The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field. The usual def ...

. If the pencils are perspectively related, the conic is degenerate. # The set of points whose coordinates satisfy an irreducible homogeneous equation of degree two. Furthermore, in a finite Desarguesian plane: #

A set of + 1 points, no three collinear in PG(2,) is called an ''oval''. If is odd, by

Segre's theorem In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement: *Any oval in a ''finite pappian'' projective plane of ''odd'' order is a nondegenerate projective conic section. This statement was ...

, an oval in PG(2,) is a conic, in sense 3 above. # An ''Ostrom conic'' is based on a generalization of harmonic sets. Artzy has given an example of a Steiner conic in a Moufang plane which is not a von Staudt conic. Garner gives an example of a von Staudt conic that is not an Ostrom conic in a finite semifield plane.

Notes

References

* * * * * * * * * * *{{Citation , last1=Weibel , first1=Charles , title=Survey of Non-Desarguesian Planes , url=https://www.ams.org/notices/200710/ , year=2007 , journal= Notices of the AMS , volume= 54 , issue=10 , pages=1294–1303 Projective geometry Finite geometry