Synthetic geometry

Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.

Logical synthesis

The process of logical synthesis begins with some arbitrary but defined starting point.

• Primitives are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while the fundamental relationship is that of incidence – of one object meeting or joining with another.
• Axioms are statements about these primitives, for example that any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them).

From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. Where a significant result is proved rigorously, it becomes a theorem.

Any given set of axioms leads to a different logical system. In the case of geometry, each distinct set of axioms leads to a different geometry.

Properties of axiom sets

If the axiom set is not categorical (so that there is more than one model) one has the geometry/geometries debate to settle. That's not a serious issue for a modern axiomatic mathematician, since the implication of axiom is now a starting point for theory rather than a self-evident plank in a platform based on intuition. And since the Erlangen program of Klein the nature of any given geometry has been seen as the connection of symmetry and the content of propositions, rather than the style of development.

History

The geometry of Euclid was synthetic, though not all of his books covered topics of pure geometry.

The close axiomatic study of Euclidean geometry in the 19th Century led to the discovery of non-Euclidean geometries having different axioms. Gauss, Bolyai and Lobachevski independently discovered hyperbolic geometry, in which the Euclidean axiom of parallelism is replaced by an alternative. Poincaré soon discovered the first physical geometric model of hyperbolic geometry, in a form known as the Poincaré disc. Eventually, hyperbolic geometry would become accessible to analysis using Mobius transformations.

The heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favour of a purely synthetic development of projective geometry. For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry.

Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus, which can be considered synthetic in spirit. The closely related operation of reciprocation expresses analysis of the plane.

On the other hand, the theory of special relativity was originally developed analytically via the linear algebra of the Lorentz transformation; then in 1912 Lewis and Wilson developed a synthetic approach (see reference), bringing greater confidence in the foundations of spacetime theory.

Computational synthetic geometry

In conjunction with computational geometry, a computational synthetic geometry has been founded, having close connection, for example, with matroid theory. Synthetic differential geometry is an application of topos theory to the foundations of differentiable manifold theory.

References

• Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the American Academy of Arts and Sciences 48:387–507.
• Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite.
• Hilbert & Cohn-Vossen, Geometry and the imagination.
• Mlodinow, L; Euclid's window.