Synthetic geometry

Synthetic geometry is a way to study shapes and spaces without using numbers to point out locations. This is called using coordinates. Instead, it uses a special way of thinking called the axiomatic method. This method starts with a few simple rules, called axioms or postulates, and builds up proofs from there. People used this method long before they invented coordinate systems.

After René Descartes introduced coordinate methods in the 1600s, which became known as analytic geometry, the older way of studying geometry was named “synthetic geometry.” Felix Klein said that synthetic geometry looks at shapes directly, while analytic geometry uses math formulas after picking a coordinate system.

The first organized system of synthetic geometry is found in Euclid's Elements. Later, people learned that Euclid’s basic rules were not enough to fully explain geometry. The first full set of rules was made by David Hilbert in the late 1800s. Today, both synthetic and analytic methods are used to build geometry. They are proven to be equally good. This was shown by Emil Artin in his book Geometric Algebra. The difference between synthetic and analytic geometry matters mostly for learning the basics or for special kinds of geometry like some finite geometries and non-Desarguesian geometry.

Logical synthesis

Logical synthesis in geometry starts with simple ideas called primitives, such as points, lines, and planes. It also uses rules called axioms. These axioms tell us how the simple ideas relate to each other. For example, one axiom says that any two points are connected by exactly one line.

From these axioms, mathematicians create 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 stayed important for over two thousand years. But in the 1800s, mathematicians like Gauss, Bolyai, Lobachevsky, and Riemann found new types of geometry. This made people think differently about Euclid's ideas.

During this time, some mathematicians wanted to study shapes using only simple 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. 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 using 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 learn about shapes and their properties in a unique way.

Computational synthetic geometry

Computational synthetic geometry is a field that connects with computational geometry. It is closely related to matroid theory. Synthetic differential geometry uses topos theory to study smooth shapes.

Main article: Synthetic differential geometry