Projective geometry is a venerable subject that has its origins in the study of perspective by Renaissance painters. As seen by the eye, parallel lines — e.g., train tracks — appear to meet at a 'point at infinity'. When one changes ones viewpoint, distances and angles appear to change, but points remain points and lines remain lines. These facts suggest a modification of Euclidean plane geometry, based on a set of points, a set of lines, and relation whereby a point 'lies on' a line, satisfying the following axioms:
We have already met one example of a projective plane in Section 2.1: the smallest one of all, the Fano plane. The example relevant to perspective is the real projective plane, . Here the points are lines through the origin in , the lines are planes through the origin in , and the relation of 'lying on' is taken to be inclusion. Each point determines a point in , namely the line in containing the origin and the point :
There are also other points in , the 'points at infinity', corresponding to lines through the origin in that do not intersect the plane . For example, any point on the horizon in the above picture determines a point at infinity.
Projective geometry is also interesting in higher dimensions. One can define a projective space by the following axioms:
If is any field, there is an -dimensional projective space
called where the points are lines through the origin in
, the lines are planes through the origin in , and
the relation of 'lying on' is inclusion. In fact, this construction
works even when is a mere skew field: a ring such that every
nonzero element has a left and right multiplicative inverse. We just
need to be a bit careful about defining lines and planes through the
origin in . To do this, we use the fact that is a
-bimodule in an obvious way. We take a line through the origin to
be any set
Given this example, the question naturally arises whether every projective -space is of the form for some skew field . The answer is quite surprising: yes, but only if . Projective planes are more subtle . A projective plane comes from a skew field if and only if it satisfies an extra axiom, the 'axiom of Desargues', which goes as follows. Define a triangle to be a triple of points that don't all lie on the same line. Now, suppose we have two triangles and . The sides of each triangle determine three lines, say and . Sometimes the line through and , the line through and , and the line through and will all intersect at the same point:
This axiom holds automatically for projective spaces of dimension 3 or more, but not for projective planes. A projective plane satisfying this axiom is called Desarguesian.
The axiom of Desargues is pretty, but what is its connection to skew fields? Suppose we start with a projective plane and try to reconstruct a skew field from it. We can choose any line , choose three distinct points on this line, call them , and , and set . Copying geometric constructions that work when , we can define addition and multiplication of points in . In general the resulting structure will not be a skew field. Even worse, it will depend in a nontrivial way on the choices made. However, if we assume the axiom of Desargues, these problems go away. We thus obtain a one-to-one correspondence between isomorphism classes of skew fields and isomorphism classes of Desarguesian projective planes.
Projective geometry was very fashionable in the 1800s, with such worthies as Poncelet, Brianchon, Steiner and von Staudt making important contributions. Later it was overshadowed by other forms of geometry. However, work on the subject continued, and in 1933 Ruth Moufang constructed a remarkable example of a non-Desarguesian projective plane using the octonions . As we shall see, this projective plane deserves the name .
The 1930s also saw the rise of another reason for interest in projective
geometry: quantum mechanics! Quantum theory is distressingly different
from the classical Newtonian physics we have learnt to love. In
classical mechanics, observables are described by real-valued functions.
In quantum mechanics, they are often described by hermitian complex matrices. In both cases, observables are closed under
addition and multiplication by real scalars. However, in quantum
mechanics, observables do not form an associative algebra. Still,
one can raise an observable to a power, and from squaring one
can construct a commutative but nonassociative product:
In 1934, Jordan published a paper with von Neumann and Wigner classifying all formally real Jordan algebras . The classification is nice and succinct. An ideal in the Jordan algebra is a subspace such that implies for all . A Jordan algebra is simple if its only ideals are and itself. Every formally real Jordan algebra is a direct sum of simple ones. The simple formally real Jordan algebras consist of 4 infinite families and one exception.
The paper by Jordan, von Neumann and Wigner appears to have been uninfluenced by Moufang's discovery of , but in fact they are related. A projection in a formally real Jordan algebra is defined to be an element with . In the familiar case of , these correspond to hermitian matrices with eigenvalues and , so they are used to describe observables that assume only two values — e.g., 'true' and 'false'. This suggests treating projections in a formally real Jordan algebra as propositions in a kind of 'quantum logic'. The partial order helps us do this: given projections and , we say that 'implies' if .
The relation between Jordan algebras and quantum logic is already
interesting , but the real fun starts when we note
that projections in correspond to subspaces of . This
sets up a relationship to projective geometry , since
the projections onto 1-dimensional subspaces correspond to points in
, while the projections onto 2-dimensional subspaces correspond
to lines. Even better, we can work out the dimension of a subspace
from the corresponding projection
using only the partial order on projections: has dimension iff
the longest chain of distinct projections
If we try this starting with , or , we succeed when , and we obtain the projective spaces , and , respectively. If we try this starting with the spin factor we succeed when , and obtain a series of 1-dimensional projective spaces related to Lorentzian geometry. Finally, in 1949 Jordan  discovered that if we try this construction starting with the exceptional Jordan algebra, we get the projective plane discovered by Moufang: .
In what follows we describe the octonionic projective plane
and exceptional Jordan algebra in more detail. But first let us
consider the octonionic projective line, and the Jordan algebra
© 2001 John Baez