Synthetic geometry
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Synthetic geometry is the branch of geometry which makes use of theorems and synthetic observations to draw logical conclusions, as opposed to analytic geometry which uses algebra to perform geometric computations and solve problems.
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[edit] 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, planes and 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.
[edit] 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. Only later did Poincaré discover a physical geometric model of this new geometry, before eventually it became accessible to analysis using Mobius transformations.
The heyday of synthetic geometry can be considered to have been the 19th century, when 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.
Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus , which can be considered synthetic in spirit. The closely related reciprocation expresses analysis of the plane. On the other hand, the theory of special relativity developed analytically with the linear algebra of Lorentz transformation; more confidence in the foundation of spacetime theory may evolve from the synthetic approach given by Lewis and Wilson in 1912 (see reference "Synthetic Spacetime").
[edit] 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.
[edit] 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.
[edit] References
- Synthetic Spacetime
- Hilbert & Cohn-Vossen, Geometry and the imagination.
- Mlodinow, L; Euclid's window.
