In projective geometry

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

 a Desarguesian plane, named after Gérard Desargues

Gérard Desargues

Girard Desargues 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 named in his honour.Born in Lyon, Desargues came from a family devoted to service to the French...

, is a plane in which Desargues' theorem

Desargues' theorem

In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...

 holds. The ordinary real projective plane

Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...

 is a Desarguesian plane. More generally any projective plane over a division ring

Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...

 is Desarguesian, and conversely Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

 showed that any Desarguesian projective plane is the projective plane over some division ring. The division ring is commutative if and only if the projective plane satisfies Pappus's hexagon theorem

Pappus's hexagon theorem

In mathematics, Pappus's hexagon theorem states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear...

. In particular all finite Desarguesian planes satisfy Pappus's theorem, but there are some infinite ones that do not. There are many examples of non-Desarguesian plane

Non-Desarguesian plane

In mathematics, a non-Desarguesian plane, named after Gérard Desargues, is a projective plane that does not satisfy Desargues's theorem, or in other words a plane that is not a Desarguesian plane...

s where Desargues's theorem does not hold, such as the Moulton plane

Moulton plane

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

.