Synthetic geometry

Synthetic geometry is a way of studying shapes and spaces without using coordinates. Instead, it uses a special method called the axiomatic method to prove ideas starting from a few basic rules, known as axioms or postulates. This method was used long before the coordinate system was invented.

After René Descartes introduced coordinate methods in the 17th century, which became known as analytic geometry, the older way of studying geometry was given the name "synthetic geometry." According to Felix Klein, synthetic geometry looks at shapes directly, while analytic geometry uses formulas after choosing a coordinate system.

The first organized system of synthetic geometry appears in Euclid's Elements. Later, it was found that Euclid’s basic rules weren’t enough to fully describe geometry. The first complete set of rules was created by David Hilbert at the end of the 19th century. Both synthetic and analytic methods can now be used to build geometry, and they are proven to be equivalent, as shown by Emil Artin in his book Geometric Algebra. Today, the difference between synthetic and analytic geometry is mostly only important for basic learning or for special types of geometry like some finite geometries and non-Desarguesian geometry.

Logical synthesis

Logical synthesis in geometry starts with basic ideas called primitives, such as points, lines, and planes, and rules called axioms. These axioms describe how these basic ideas relate to each other. For example, one axiom states that any two points are connected by exactly one line.

From these axioms, mathematicians build logical arguments to prove new ideas, called theorems. There isn't just one set of axioms for geometry; different sets can lead to different types of geometry, like Euclidean geometry, hyperbolic geometry, or spherical geometry. This shows that geometry can have many forms depending on the axioms chosen.

History

Euclid's way of studying shapes and sizes stayed the main idea for over two thousand years. But in the 1800s, mathematicians like Gauss, Bolyai, Lobachevsky, and Riemann discovered new types of geometry that made people question Euclid's basic ideas.

During the 1800s, some mathematicians preferred to study shapes using only basic ideas and rules, without using coordinates. This was called synthetic geometry. They used these methods to explore projective geometry, which looks at how shapes change when you move or twist them. Even today, studying geometry with basic rules helps us understand many different kinds of spaces.

Proofs using synthetic geometry

Synthetic geometry uses special lines and ideas to prove math facts without coordinates. It looks at things like equal sides, equal angles, and matching shapes. Some famous examples are the Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal incircles theorem, Geometric mean theorem, Heron's formula, Isosceles triangle theorem, and Law of cosines. These theorems help us understand shapes and their properties in a unique way.

Computational synthetic geometry

Computational synthetic geometry is a field that connects with computational geometry. It has strong links with matroid theory. Synthetic differential geometry uses topos theory to study differentiable manifolds.

Main article: Synthetic differential geometry