Editing Projective plane (section)

===Generalized coordinates===
{{main|Planar ternary ring}}

One can construct a coordinate "ring"&mdash;a so-called [[planar ternary ring]] (not a genuine ring)&mdash;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 called [[non-Desarguesian projective plane]]s and are an active area of research.  The [[Cayley plane]] ('''OP'''<sup>2</sup>), a projective plane over the [[octonion]]s, is one of these because the octonions do not form a division ring.{{sfnp|Baez|2002|p=167}}

Conversely, given a planar ternary ring (''R'',&nbsp;''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 operator ''T'' can be used to produce two binary operators on the set ''R'', by:
: ''a'' + ''b'' = ''T''(''a'', 1, ''b''), and
: ''a'' ⋅ ''b'' = ''T''(''a'', ''b'', 0).
The ternary operator is '''''linear''''' if {{nowrap|1=''T''(''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 a [[division ring]], while [[Pappus's hexagon theorem|Pappus's theorem]] corresponds to this ring being obtained from a [[commutative]] field. A projective plane satisfying Pappus's theorem universally is called a ''Pappian plane''. [[Alternative algebra|Alternative]], not necessarily [[associative]], division algebras like the octonions correspond to [[Moufang plane]]s.

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 and [[Wedderburn's little theorem|Wedderburn's theorem]] that finite division rings must be commutative; {{harvtxt|Bamberg|Penttila|2015}} give a proof that uses only more "elementary" algebraic facts about division rings.

To describe a finite projective plane of order ''N''(≥ 2) using non-homogeneous coordinates and a planar ternary ring:
:Let one point be labelled (''∞'').
:Label ''N'' points, (''r'') where ''r'' = 0, ..., (''N''&nbsp;&minus;&nbsp;1).
:Label ''N''<sup>2</sup> points, (''r'', ''c'') where ''r'', ''c'' = 0, ..., (''N''&nbsp;&minus;&nbsp;1).
On these points, construct the following lines:
:One line <nowiki>[</nowiki>''∞''<nowiki>]</nowiki> = { (''∞''), (0), ..., (''N''&nbsp;&minus;&nbsp;1)}
:''N'' lines <nowiki>[</nowiki>''c''<nowiki>]</nowiki> = {(''∞''), (''c'', 0), ..., (''c'', ''N''&nbsp;&minus;&nbsp;1)}, where ''c'' = 0, ..., (''N''&nbsp;&minus;&nbsp;1)
:''N''<sup>2</sup> lines <nowiki>[</nowiki>''r'', ''c''<nowiki>]</nowiki> = {(''r'') and the points (''x'', '''T'''(''x'', ''r'', ''c'')) }, where ''x'', ''r'', ''c'' = 0, ..., (''N''&nbsp;&minus;&nbsp;1) and '''T''' is the ternary operator of the planar ternary ring.

For example, for {{nowrap|1=''N'' = 2}} we can use the symbols {0,&nbsp;1} associated with the finite field of order 2. The ternary operation defined by {{nowrap|1=''T''(''x'', ''m'', ''k'') = ''xm'' + ''k''}} with the operations on the right being the multiplication and addition in the field yields the following:
:One line <nowiki>[</nowiki>''∞''<nowiki>]</nowiki> = { (''∞''), (0), (1)},
:2 lines <nowiki>[</nowiki>''c''<nowiki>]</nowiki> = {(''∞''), (''c'',0), (''c'',1) : ''c'' = 0, 1},
::<nowiki>[</nowiki>0<nowiki>]</nowiki> = {(''∞''), (0,0), (0,1) }
::<nowiki>[</nowiki>1<nowiki>]</nowiki> = {(''∞''), (1,0), (1,1) }
:4 lines <nowiki>[</nowiki>''r'', ''c''<nowiki>]</nowiki>: (''r'') and the points  (''i'', ''ir'' + ''c''), where ''i'' = 0, 1 : ''r'', ''c'' = 0, 1.
::<nowiki>[</nowiki>0,0<nowiki>]</nowiki>: {(0), (0,0), (1,0) }
::<nowiki>[</nowiki>0,1<nowiki>]</nowiki>: {(0), (0,1), (1,1) }
::<nowiki>[</nowiki>1,0<nowiki>]</nowiki>: {(1), (0,0), (1,1) }
::<nowiki>[</nowiki>1,1<nowiki>]</nowiki>: {(1), (0,1), (1,0) }