Inmathematics, aprojective plane is a geometric structure that extends the concept of aplane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thusany two distinct lines in a projective plane intersect at exactly one point.

Renaissance artists, in developing the techniques of drawing inperspective, laid the groundwork for this mathematical topic. The archetypical example is thereal projective plane, also known as theextended Euclidean plane.^[1] This example, in slightly different guises, is important inalgebraic geometry,topology andprojective geometry where it may be denoted variously byPG(2,R),RP², orP₂(R), among other notations. There are many other projective planes, both infinite, such as thecomplex projective plane, and finite, such as theFano plane.

A projective plane is a 2-dimensionalprojective space. Not all projective planes can beembedded in 3-dimensional projective spaces; such embeddability is a consequence of a property known asDesargues' theorem, not shared by all projective planes.

Aprojective plane is a rank 2incidence structure${\displaystyle (\mathcal{P},\mathcal{L},I)}$ consisting of a set ofpoints${\displaystyle \mathcal{P}}$, a set oflines${\displaystyle \mathcal{L}}$, and a symmetric relation${\displaystyle I}$ on the set${\displaystyle \mathcal{P}\cup \mathcal{L}}$ calledincidence, having the following properties:^[2]

The second condition means that there are noparallel lines. The last condition excludes the so-calleddegenerate cases (seebelow). The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression "pointP is incident with lineℓ" is used instead of either "P is onℓ" or "ℓ passes throughP".

It follows from the definition that the number of points${\displaystyle s+1}$ incident with any given line in a projective plane is the same as the number of lines incident with any given point. The (possibly infinite) cardinal number${\displaystyle s}$ is calledorder of the plane.

To turn the ordinary Euclidean plane into a projective plane, proceed as follows:

1. To each parallel class of lines (a maximum set of mutually parallel lines) associate a single new point. That point is to be considered incident with each line in its class. The new points added are distinct from each other. These new points are calledpoints at infinity.
2. Add a new line, which is considered incident with all the points at infinity (and no other points). This line is calledtheline at infinity.

The extended structure is a projective plane and is called theextended Euclidean plane or thereal projective plane. The process outlined above, used to obtain it, is called "projective completion" orprojectivization. This plane can also be constructed by starting fromR³ viewed as a vector space, see§ Vector space construction below.

The points of theMoulton plane are the points of the Euclidean plane, with coordinates in the usual way. To create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, but other lines will remain unchanged. Redefine all the lines with negative slopes so that they look like "bent" lines, meaning that these lines keep their points with negativex-coordinates, but the rest of their points are replaced with the points of the line with the samey-intercept but twice the slope wherever theirx-coordinate is positive.

The Moulton plane has parallel classes of lines and is anaffine plane. It can be projectivized, as in the previous example, to obtain theprojective Moulton plane.Desargues's theorem is not a valid theorem in either the Moulton plane or the projective Moulton plane.

This example has just thirteen points and thirteen lines. We label the points P₁, ..., P₁₃ and the lines m₁, ..., m₁₃. Theincidence relation (which points are on which lines) can be given by the followingincidence matrix. The rows are labelled by the points and the columns are labelled by the lines. A 1 in rowi and columnj means that the point P_i is on the line m_j, while a 0 (which we represent here by a blank cell for ease of reading) means that they are not incident. The matrix is in Paige–Wexler normal form.

Lines Points	m1	m2	m3	m4	m5	m6	m7	m8	m9	m10	m11	m12	m13
P1	1	1	1	1
P2	1				1	1	1
P3	1							1	1	1
P4	1										1	1	1
P5		1			1			1			1
P6		1				1			1			1
P7		1					1			1			1
P8			1		1				1				1
P9			1			1				1	1
P10			1				1	1				1
P11				1	1					1		1
P12				1		1		1					1
P13				1			1		1		1

To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1s appear (every pair of distinct points are on exactly one common line) and that every two columns have exactly one common row in which 1s appear (every pair of distinct lines meet at exactly one point). Among many possibilities, the points P₁, P₄, P₅, and P₈, for example, will satisfy the third condition. This example is known as theprojective plane of order three.

Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace (ageometric line) through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a (geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.^[3]

LetK be anydivision ring (skewfield). LetK³ denote the set of all triplesx =(x₀,x₁,x₂) of elements ofK (aCartesian product viewed as avector space). For any nonzerox inK³, the minimal subspace ofK³ containingx (which may be visualized as all the vectors in a line through the origin) is the subset

${\displaystyle \{kx:k\in K\}}$

ofK³. Similarly, letx andy be linearly independent elements ofK³, meaning thatkx +my = 0 implies thatk =m = 0. The minimal subspace ofK³ containingx andy (which may be visualized as all the vectors in a plane through the origin) is the subset

${\displaystyle \{kx+my:k,m\in K\}}$

ofK³. This 2-dimensional subspace contains various 1-dimensional subspaces through the origin that may be obtained by fixingk andm and taking the multiples of the resulting vector. Different choices ofk andm that are in the same ratio will give the same line.

Theprojective plane overK, denoted PG(2, K) orKP², has a set ofpoints consisting of all the 1-dimensional subspaces inK³. A subsetL of the points of PG(2, K) is aline in PG(2, K) if there exists a 2-dimensional subspace ofK³ whose set of 1-dimensional subspaces is exactlyL.

Verifying that this construction produces a projective plane is usually left as a linear algebra exercise.

An alternate (algebraic) view of this construction is as follows. The points of this projective plane are the equivalence classes of the setK³ \ {(0, 0, 0)} modulo theequivalence relation

x ~kx, for allk inK^×.

Lines in the projective plane are defined exactly as above.

The coordinates(x₀,x₁,x₂) of a point in PG(2, K) are calledhomogeneous coordinates. Each triple(x₀,x₁,x₂) represents a well-defined point in PG(2, K), except for the triple(0, 0, 0), which represents no point. Each point in PG(2, K), however, is represented by many triples.

IfK is atopological space, thenKP² inherits a topology via theproduct,subspace, andquotient topologies.

Thereal projective planeRP² arises whenK is taken to be thereal numbers,R. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology.^[4]

In this construction, consider the unit sphere centered at the origin inR³. Each of theR³ lines in this construction intersects the sphere at two antipodal points. Since theR³ line represents a point ofRP², we will obtain the same model ofRP² by identifying the antipodal points of the sphere. The lines ofRP² will be the great circles of the sphere after this identification of antipodal points. This description gives the standard model ofelliptic geometry.

Thecomplex projective planeCP² arises whenK is taken to be thecomplex numbers,C. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over otherfields (known aspappian planes) serve as fundamental examples inalgebraic geometry.^[5]

Thequaternionic projective planeHP² is also of independent interest.^[6]

ByWedderburn's Theorem, a finite division ring must be commutative and so be a field. Thus, the finite examples of this construction are known as "field planes". TakingK to be thefinite field ofq =pⁿ elements with primep produces a projective plane ofq² +q + 1 points. The field planes are usually denoted by PG(2, q) where PG stands for projective geometry, the "2" is the dimension andq is called theorder of the plane (it is one less than the number of points on any line). The Fano plane, discussed below, is denoted by PG(2, 2). Thethird example above is the projective plane PG(2, 3).

TheFano plane is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines. In the figure at right, the seven points are shown as small balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently consider the balls to be the "lines" and the line segments and circle to be the "points" – this is an example ofduality in the projective plane: if the lines and points are interchanged, the result is still a projective plane (seebelow). A permutation of the seven points that carriescollinear points (points on the same line) to collinear points is called acollineation orsymmetry of the plane. The collineations of a geometry form agroup under composition, and for the Fano plane this group (PΓL(3, 2) = PGL(3, 2)) has 168 elements.

Thetheorem of Desargues is universally valid in a projective plane if and only if the plane can be constructed from a three-dimensional vector space over a skewfield asabove.^[7] These planes are calledDesarguesian planes, named afterGirard Desargues. The real (or complex) projective plane and the projective plane of order 3 givenabove are examples of Desarguesian projective planes.

The projective planes that can not be constructed in the preceding manner are callednon-Desarguesian planes, and theMoulton plane givenabove is an example of one. Many such finite and infinite planes are known.

The PG(2, K) notation is reserved for Desarguesian planes. WhenK is afield, a very common case, they are also known asfield planes and if the field is afinite field they can be calledGalois planes.

Asubplane of a projective plane${\displaystyle (\mathcal{P},\mathcal{L},I)}$ is a pair of subsets${\displaystyle ({\mathcal{P}}^{\prime },{\mathcal{L}}^{\prime })}$ where${\displaystyle {\mathcal{P}}^{\prime }\subseteq \mathcal{P}}$,${\displaystyle {\mathcal{L}}^{\prime }\subseteq \mathcal{L}}$ and${\displaystyle ({\mathcal{P}}^{\prime },{\mathcal{L}}^{\prime },I^{\prime })}$ is itself a projective plane with respect to the restriction${\displaystyle I^{\prime }}$ of the incidence relation${\displaystyle I}$ to${\displaystyle ({\mathcal{P}}^{\prime }\cup {\mathcal{L}}^{\prime })\times ({\mathcal{P}}^{\prime }\cup {\mathcal{L}}^{\prime })}$.

(Bruck 1955) proves the following theorem. Let Π be a finite projective plane of orderN with a proper subplane Π₀ of orderM. Then eitherN =M² orN ≥M² +M.

A subplane${\displaystyle ({\mathcal{P}}^{\prime },{\mathcal{L}}^{\prime })}$ of${\displaystyle (\mathcal{P},\mathcal{L},I)}$ is aBaer subplane if every line in${\displaystyle \mathcal{L}\setminus {\mathcal{L}}^{\prime }}$ is incident with exactly one point in${\displaystyle {\mathcal{P}}^{\prime }}$ and every point in${\displaystyle \mathcal{P}\setminus {\mathcal{P}}^{\prime }}$ is incident with exactly one line of${\displaystyle {\mathcal{L}}^{\prime }}$.

A finite Desarguesian projective plane of order${\displaystyle q}$ admits Baer subplanes (all necessarily Desarguesian) if andonly if${\displaystyle q}$ is square; in thiscase the order of the Baer subplanes is${\displaystyle \sqrt{q}}$.

In the finite Desarguesian planes PG(2, pⁿ), the subplanes have orders which are the orders of the subfields of the finite field GF(pⁿ), that is,pⁱ wherei is a divisor ofn. In non-Desarguesian planes however, Bruck's theorem gives the only information about subplane orders. The case of equality in the inequality of this theorem is not known to occur. Whether or not there exists a subplane of orderM in a plane of orderN withM² +M =N is an open question. If such subplanes existed there would be projective planes of composite (non-prime power) order.

AFano subplane is a subplane isomorphic to PG(2, 2), the unique projective plane of order 2.

If you consider aquadrangle (a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane. The remaining three points (called thediagonal points of the quadrangle) are the points where the lines that do not intersect at a point of the quadrangle meet. The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle).

In finite desarguesian planes, PG(2, q), Fano subplanes exist if and only ifq is even (that is, a power of 2). The situation in non-desarguesian planes is unsettled. They could exist in any non-desarguesian plane of order greater than 6, and indeed, they have been found in all non-desarguesian planes in which they have been looked for (in both odd and even orders).

An open question, apparently due toHanna Neumann though not published by her, is: Does every non-desarguesian plane contain a Fano subplane?

A theorem concerning Fano subplanes due to (Gleason 1956) is:

If every quadrangle in a finite projective plane has collinear diagonal points, then the plane is desarguesian (of even order).

Projectivization of the Euclidean plane produced the real projective plane. The inverse operation—starting with a projective plane, remove one line and all the points incident with that line—produces anaffine plane.

More formally anaffine plane consists of a set oflines and a set ofpoints, and a relation between points and lines calledincidence, having the following properties:

The second condition means that there areparallel lines and is known asPlayfair's axiom. The expression "does not meet" in this condition is shorthand for "there does not exist a point incident with both lines".

The Euclidean plane and the Moulton plane are examples of infinite affine planes. A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. Theorder of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes). The affine planes which arise from the projective planes PG(2, q) are denoted by AG(2, q).

There is a projective plane of orderN if and only if there is anaffine plane of orderN. When there is only one affine plane of orderN there is only one projective plane of orderN, but the converse is not true. The affine planes formed by the removal of different lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane. These statements hold for infinite projective planes as well.

The affine planeK² overK embeds intoKP² via the map which sends affine (non-homogeneous) coordinates tohomogeneous coordinates,

${\displaystyle (x_1,x_2)\mapsto (1,x_1,x_2).}$

The complement of the image is the set of points of the form(0,x₁,x₂). From the point of view of the embedding just given, these points are thepoints at infinity. They constitute a line inKP²—namely, the line arising from the plane

${\displaystyle \{k(0,0,1)+m(0,1,0):k,m\in K\}}$

inK³—called theline at infinity. The points at infinity are the "extra" points where parallel lines intersect in the construction of the extended real plane; the point (0,x₁,x₂) is where all lines of slopex₂ /x₁ intersect. Consider for example the two lines

${\displaystyle u=\{(x,0):x\in K\}}$

${\displaystyle y=\{(x,1):x\in K\}}$

in the affine planeK². These lines have slope 0 and do not intersect. They can be regarded as subsets ofKP² via the embedding above, but these subsets are not lines inKP². Add the point(0, 1, 0) to each subset; that is, let

${\displaystyle \overline{u}=\{(1,x,0):x\in K\}\cup \{(0,1,0)\}}$

${\displaystyle \overline{y}=\{(1,x,1):x\in K\}\cup \{(0,1,0)\}}$

These are lines inKP²; ū arises from the plane

${\displaystyle \{k(1,0,0)+m(0,1,0):k,m\in K\}}$

inK³, while ȳ arises from the plane

${\displaystyle k(1,0,1)+m(0,1,0):k,m\in K.}$

The projective lines ū and ȳ intersect at(0, 1, 0). In fact, all lines inK² of slope 0, when projectivized in this manner, intersect at(0, 1, 0) inKP².

The embedding ofK² intoKP² given above is not unique. Each embedding produces its own notion of points at infinity. For example, the embedding

${\displaystyle (x_1,x_2)\to (x_2,1,x_1),}$

has as its complement those points of the form(x₀, 0,x₂), which are then regarded as points at infinity.

When an affine plane does not have the form ofK² withK a division ring, it can still be embedded in a projective plane, but the construction used above does not work. A commonly used method for carrying out the embedding in this case involves expanding the set of affine coordinates and working in a more general "algebra".

One can construct a coordinate "ring"—a so-calledplanar ternary ring (not a genuine ring)—corresponding to any projective plane. A planar ternary ring need not be a field or division ring, and there are many projective planes that are not constructed from a division ring. They are callednon-Desarguesian projective planes and are an active area of research. TheCayley plane (OP²), a projective plane over theoctonions, is one of these because the octonions do not form a division ring.^[8]

Conversely, given a planar ternary ring (R, T), a projective plane can be constructed (see below). The relationship is not one to one. A projective plane may be associated with several non-isomorphic planar ternary rings. The ternary operatorT can be used to produce two binary operators on the setR, by:

a +b =T(a, 1,b), and

a ⋅b =T(a,b, 0).

The ternary operator islinear ifT(x,m,k) =x⋅m +k. When the set of coordinates of a projective plane actually form a ring, a linear ternary operator may be defined in this way, using the ring operations on the right, to produce a planar ternary ring.

Algebraic properties of this planar ternary coordinate ring turn out to correspond to geometric incidence properties of the plane. For example,Desargues' theorem corresponds to the coordinate ring being obtained from adivision ring, whilePappus's theorem corresponds to this ring being obtained from acommutative field. A projective plane satisfying Pappus's theorem universally is called aPappian plane.Alternative, not necessarilyassociative, division algebras like the octonions correspond toMoufang planes.

There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane (finite Desarguesian planes are Pappian). (The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian.) The most common proof uses coordinates in a division ring andWedderburn's theorem that finite division rings must be commutative;Bamberg & Penttila (2015) give a proof that uses only more "elementary" algebraic facts about division rings.

To describe a finite projective plane of orderN(≥ 2) using non-homogeneous coordinates and a planar ternary ring:

Let one point be labelled (∞).

LabelN points, (r) wherer = 0, ..., (N − 1).

LabelN² points, (r,c) wherer,c = 0, ..., (N − 1).

On these points, construct the following lines:

One line [∞] = { (∞), (0), ..., (N − 1)}

N lines [c] = {(∞), (c, 0), ..., (c,N − 1)}, wherec = 0, ..., (N − 1)

N² lines [r,c] = {(r) and the points (x,T(x,r,c)) }, wherex,r,c = 0, ..., (N − 1) andT is the ternary operator of the planar ternary ring.

For example, forN = 2 we can use the symbols {0, 1} associated with the finite field of order 2. The ternary operation defined byT(x,m,k) =xm +k with the operations on the right being the multiplication and addition in the field yields the following:

One line [∞] = { (∞), (0), (1)},

2 lines [c] = {(∞), (c,0), (c,1) :c = 0, 1},

[0] = {(∞), (0,0), (0,1) }

[1] = {(∞), (1,0), (1,1) }

4 lines [r,c]: (r) and the points (i,ir +c), wherei = 0, 1 :r,c = 0, 1.

[0,0]: {(0), (0,0), (1,0) }

[0,1]: {(0), (0,1), (1,1) }

[1,0]: {(1), (0,0), (1,1) }

[1,1]: {(1), (0,1), (1,0) }

Degenerate planes do not fulfill thethird condition in the definition of a projective plane. They are not structurally complex enough to be interesting in their own right, but from time to time they arise as special cases in general arguments. There are seven kinds of degenerate plane according to (Albert & Sandler 1968). They are:

1. the empty set;
2. a single point, no lines;
3. a single line, no points;
4. a single point, a collection of lines, the point is incident with all of the lines;
5. a single line, a collection of points, the points are all incident with the line;
6. a pointP incident with a linem, an arbitrary collection of lines all incident withP and an arbitrary collection of points all incident withm;
7. a pointP not incident with a linem, an arbitrary (can be empty) collection of lines all incident withP and all the points of intersection of these lines withm.

These seven cases are not independent, the fourth and fifth can be considered as special cases of the sixth, while the second and third are special cases of the fourth and fifth respectively. The special case of the seventh plane with no additional lines can be seen as an eighth plane. All the cases can therefore be organized into two families of degenerate planes as follows (this representation is for finite degenerate planes, but may be extended to infinite ones in a natural way):

1) For any number of pointsP₁, ...,P_n, and linesL₁, ...,L_m,

L₁ = {P₁,P₂, ...,P_n}

L₂ = {P₁ }

L₃ = {P₁ }

...

L_m = {P₁ }

2) For any number of pointsP₁, ...,P_n, and linesL₁, ...,L_n, (same number of points as lines)

L₁ = {P₂,P₃, ...,P_n }

L₂ = {P₁,P₂ }

L₃ = {P₁,P₃ }

...

L_n = {P₁,P_n }

Acollineation of a projective plane is abijective map of the plane to itself which maps points to points and lines to lines that preserves incidence, meaning that ifσ is a bijection and pointP is on linem, thenP^σ is onm^σ.^[9]

Ifσ is a collineation of a projective plane, a pointP withP =P^σ is called afixed point ofσ, and a linem withm =m^σ is called afixed line of σ. The points on a fixed line need not be fixed points, their images underσ are just constrained to lie on this line. The collection of fixed points and fixed lines of a collineation form aclosed configuration, which is a system of points and lines that satisfy the first two but not necessarily the third condition in thedefinition of a projective plane. Thus, the fixed point and fixed line structure for any collineation either form a projective plane by themselves, or adegenerate plane. Collineations whose fixed structure forms a plane are calledplanar collineations.

Ahomography (orprojective transformation) of PG(2, K) is a collineation of this type of projective plane which is a linear transformation of the underlying vector space. Using homogeneous coordinates they can be represented by invertible3 × 3 matrices overK which act on the points of PG(2, K) byy =Mx^T, wherex andy are points inK³ (vectors) andM is an invertible3 × 3 matrix overK.^[10] Two matrices represent the same projective transformation if one is a constant multiple of the other. Thus the group of projective transformations is the quotient of thegeneral linear group by the scalar matrices called theprojective linear group.

Another type of collineation of PG(2, K) is induced by anyautomorphism ofK, these are calledautomorphic collineations. Ifα is an automorphism ofK, then the collineation given by(x₀,x₁, x₂) → (x₀^α,x₁^α,x₂^α) is an automorphic collineation. Thefundamental theorem of projective geometry says that all the collineations of PG(2, K) are compositions of homographies and automorphic collineations. Automorphic collineations are planar collineations.

A projective plane is defined axiomatically as anincidence structure, in terms of a setP of points, a setL of lines, and anincidence relationI that determines which points lie on which lines. AsP andL are only sets one can interchange their roles and define aplane dual structure.

By interchanging the role of "points" and "lines" in

C = (P,L,I)

we obtain the dual structure

C* = (L,P,I*),

whereI* is theconverse relation ofI.

In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called theplane dual statement of the first. The plane dual statement of "Two points are on a unique line." is "Two lines meet at a unique point." Forming the plane dual of a statement is known asdualizing the statement.

If a statement is true in a projective planeC, then the plane dual of that statement must be true in the dual planeC*. This follows since dualizing each statement in the proof "inC" gives a statement of the proof "inC*."

In the projective planeC, it can be shown that there exist four lines, no three of which are concurrent. Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structureC* is also a projective plane, called thedual plane ofC.

IfC andC* are isomorphic, thenC is calledself-dual. The projective planes PG(2, K) for any division ringK are self-dual. However, there arenon-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as theHughes planes.

ThePrinciple of plane duality says that dualizing any theorem in a self-dual projective planeC produces another theorem valid inC.

Aduality is a map from a projective planeC = (P,L,I) to its dual planeC* = (L,P,I*) (seeabove) which preserves incidence. That is, a dualityσ will map points to lines and lines to points (P^σ =L andL^σ =P) in such a way that if a pointQ is on a linem (denoted byQIm) thenQ^σI*m^σ ⇔m^σIQ^σ. A duality which is an isomorphism is called acorrelation.^[11] If a correlation exists then the projective planeC is self-dual.

In the special case that the projective plane is of thePG(2, K) type, withK a division ring, a duality is called areciprocity.^[12] These planes are always self-dual. By thefundamental theorem of projective geometry a reciprocity is the composition of anautomorphic function ofK and ahomography. If the automorphism involved is the identity, then the reciprocity is called aprojective correlation.

A correlation of order two (aninvolution) is called apolarity. If a correlationφ is not a polarity thenφ² is a nontrivial collineation.

It can be shown that a projective plane has the same number of lines as it has points (infinite or finite). Thus, for every finite projective plane there is anintegerN ≥ 2 such that the plane has

N² +N + 1 points,

N² +N + 1 lines,

N + 1 points on each line, and

N + 1 lines through each point.

The numberN is called theorder of the projective plane.

The projective plane of order 2 is called theFano plane. See also the article onfinite geometry.

Using the vector space construction with finite fields there exists a projective plane of orderN =pⁿ, for eachprime powerpⁿ. In fact, for all known finite projective planes, the orderN is a prime power.^[13]

The existence of finite projective planes of other orders is an open question. The only general restriction known on the order is theBruck–Ryser–Chowla theorem that if the orderN iscongruent to 1 or 2 mod 4, it must be the sum of two squares. This rules outN = 6. The next caseN = 10 has been ruled out by massive computer calculations.^[14] Nothing more is known; in particular, the question of whether there exists a finite projective plane of orderN = 12 is still open.^[15]

Another longstanding open problem is whether there exist finite projective planes ofprime order which are not finite field planes (equivalently, whether there exists a non-Desarguesian projective plane of prime order).^[16]

A projective plane of orderN is a SteinerS(2,N + 1,N² +N + 1) system(seeSteiner system). Conversely, one can prove that all Steiner systems of this form (λ = 2) are projective planes.

Automorphisms for PG(n,k), withk=p^m,p=prime is (m!)(kⁿ⁺¹ − 1)(kⁿ⁺¹ −k)(kⁿ⁺¹ −k²)...(kⁿ⁺¹ −kⁿ)/(k − 1).

The number of mutuallyorthogonal Latin squares of orderN is at mostN − 1.N − 1 exist if and only if there is a projective plane of orderN.

While the classification of all projective planes is far from complete, results are known for small orders:

• 2 : all isomorphic toPG(2, 2),configuration (7₃) with 168 automorphisms
• 3 : all isomorphic toPG(2, 3), configuration (13₄) with 1516 automorphisms
• 4 : all isomorphic to PG(2, 4), configuration (21₅) with 120,960 automorphisms, doubled by prime power, 4=2²
• 5 : all isomorphic to PG(2, 5), configuration (31₆) with 372,000 automorphisms
• 6 : impossible as the order of a projective plane, proved byTarry who showed thatEuler'sthirty-six officers problem has no solution. However, the connection between these problems was not known untilBose proved it in 1938.^[17]
• 7 : all isomorphic to PG(2, 7), configuration (57₈) with 5,630,688 automorphisms
• 8 : all isomorphic to PG(2, 8), configuration (73₉) with 98,896,896 automorphisms, 6 higher by prime power, 8=2³
• 9 : PG(2, 9), and three more different (non-isomorphic)non-Desarguesian planes: aHughes plane, aHall plane, and the dual of this Hall plane. All are described in (Room & Kirkpatrick 1971).
• 10 : impossible as an order of a projective plane, proved by heavy computer calculation.^[14]
• 11 : at least PG(2, 11), others are not known but possible.
• 12 : it is conjectured to be impossible as an order of a projective plane.^[18]

Projective planes may be thought of asprojective geometries of dimension two.^[19]Higher-dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane.

The smallest projective space of dimension 3 isPG(3,2).

These turn out to be "tamer" than the projective planes since the extra degrees of freedom permitDesargues' theorem to be proved geometrically in the higher-dimensional geometry. This means that the coordinate "ring" associated to the geometry must be adivision ring (skewfield)K, and the projective geometry is isomorphic to the one constructed from the vector spaceK^d+1, i.e. PG(d, K). As in the construction given earlier, the points of thed-dimensionalprojective space PG(d, K) are the lines through the origin inK^d+1 and a line in PG(d, K) corresponds to a plane through the origin inK^d+1. In fact, eachi-dimensional object in PG(d, K), withi <d, is an(i + 1)-dimensional (algebraic) vector subspace ofK^d+1 ("goes through the origin"). The projective spaces in turn generalize to theGrassmannian spaces.

It can be shown that if Desargues' theorem holds in a projective space of dimension greater than two, then it must also hold in all planes that are contained in that space. Since there are projective planes in which Desargues' theorem fails (non-Desarguesian planes), these planes can not be embedded in a higher-dimensional projective space. Only the planes from the vector space construction PG(2, K) can appear in projective spaces of higher dimension. Some disciplines in mathematics restrict the meaning of projective plane to only this type of projective plane since otherwise general statements about projective spaces would always have to mention the exceptions when the geometric dimension is two.^[20]

• Block design – a generalization of a finite projective plane.
• Combinatorial design
• Difference set
• Incidence structure
• Generalized polygon
• Projective geometry
• Non-Desarguesian plane
• Smooth projective plane
• Transversals in finite projective planes
• Truncated projective plane – a projective plane with one vertex removed.
• VC dimension of a finite projective plane

• Albert, A. Adrian; Sandler, Reuben (1968),An Introduction to Finite Projective Planes, New York: Holt, Rinehart and Winston
• Baez, John C. (2002),"The octonions",Bull. Amer. Math. Soc.,39 (2):145–205,arXiv:math/0105155,doi:10.1090/S0273-0979-01-00934-X,S2CID 586512
• Bagchi, Bhaskar (2019), "A coding theoretic approach to the uniqueness conjecture for projective planes of prime order",Designs, Codes and Cryptography,87 (10):2375–2389,arXiv:1801.07038,doi:10.1007/s10623-019-00623-y,MR 4001798
• Bamberg, John; Penttila, Tim (2015),"Completing Segre's proof of Wedderburn's little theorem"(PDF),Bulletin of the London Mathematical Society,47 (3):483–492,doi:10.1112/blms/bdv021,S2CID 123036578
• Bredon, Glen E. (1993),Topology and Geometry, Springer-Verlag,ISBN 0-387-97926-3
• Bruck, R. H. (1955), "Difference Sets in a Finite Group",Trans. Amer. Math. Soc.,78 (2):464–481,doi:10.1090/s0002-9947-1955-0069791-3
• Bruck, R. H.;Bose, R. C. (1964),"The Construction of Translation Planes from Projective Spaces"(PDF),J. Algebra,1:85–102,doi:10.1016/0021-8693(64)90010-9
• Casse, Rey (2006),Projective Geometry: An Introduction, Oxford: Oxford University Press,ISBN 0-19-929886-6
• Cook, David II; Migliore, Juan; Nagel, Uwe; Zanello, Fabrizio (2016), "An algebraic approach to finite projective planes",Journal of Algebraic Combinatorics,43 (3):495–519,arXiv:1503.04335,doi:10.1007/s10801-015-0644-8,MR 3482438
• Dembowski, Peter (1968),Finite geometries,Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York:Springer-Verlag,ISBN 3-540-61786-8,MR 0233275
• Gleason, Andrew M. (1956), "Finite Fano planes",American Journal of Mathematics,78 (4):797–807,doi:10.2307/2372469,JSTOR 2372469,MR 0082684
• Gorodkov, Denis (2019), "A 15-vertex triangulation of the quaternionic projective plane",Discrete & Computational Geometry,62 (2):348–373,arXiv:1603.05541,doi:10.1007/s00454-018-00055-w
• Hall, Marshall (1943), "Projective planes",Transactions of the American Mathematical Society,54 (2), American Mathematical Society:229–277,doi:10.2307/1990331,ISSN 0002-9947,JSTOR 1990331,MR 0008892
• Hughes, D.; Piper, F. (1973),Projective Planes, Springer-Verlag,ISBN 0-387-90044-6
• Kárteszi, F. (1976),Introduction to Finite Geometries, Amsterdam: North-Holland,ISBN 0-7204-2832-7
• Kiss, György; Szőnyi, Tamás (2020),Finite Geometries, Boca Raton, Florida: CRC Press,ISBN 978-1-4987-2165-3,MR 4242245
• Lam, Clement W. H. (1991),"The Search for a Finite Projective Plane of order 10"(PDF),The American Mathematical Monthly,98 (4):305–318,doi:10.1080/00029890.1991.12000759,JSTOR 2323798, retrieved2021-11-02
• Lindner, Charles C.; Rodger, Christopher A., eds. (October 31, 1997),Design Theory (2st ed.), CRC Press,ISBN 0-8493-3986-3
• Lüneburg, Heinz (1980),Translation Planes, Berlin: Springer Verlag,ISBN 0-387-09614-0
• Moulton, Forest Ray (1902), "A Simple Non-Desarguesian Plane Geometry",Transactions of the American Mathematical Society,3 (2):192–195,doi:10.2307/1986419,ISSN 0002-9947,JSTOR 1986419
• Room, T. G.; Kirkpatrick, P. B. (1971),Miniquaternion Geometry, Cambridge: Cambridge University Press,ISBN 0-521-07926-8
• Shafarevich, I. R. (1994),Basic Algebraic Geometry, Springer-Verlag,ISBN 0-387-54812-2
• Stevenson, Frederick W. (1972),Projective Planes, San Francisco: W.H. Freeman and Company,ISBN 0-7167-0443-9
• Suetake, Chihiro (2004), "The nonexistence of projective planes of order 12 with a collineation group of order 16",Journal of Combinatorial Theory, Series A,107 (1):21–48,doi:10.1016/j.jcta.2004.03.006,MR 2063952
• Weintraub, Steven H. (1978), "Group actions on homology quaternionic projective planes",Proceedings of the American Mathematical Society,70 (1):75–82,doi:10.2307/2042588,JSTOR 2042588

Wikimedia Commons has media related toProjective plane.

• G. Eric Moorhouse,Projective Planes of Small Order, (2003)
• Ch. Weibel: Survey of Nondesarguesian planes
• Weisstein, Eric W.,"Projective plane",MathWorld
• "Projective plane" atPlanetMath.

• Projective geometry
• Incidence geometry
• Euclidean plane geometry
• Algebraic geometry
• Hypergraphs
• Computer-assisted proofs

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