Geometry is the branch of mathematics that deals with properties of shapes and spatial relationships. In its modern form, it is the pure mathematics of points and lines and curves and surfaces. It is one of the two most basic branches of mathematics, the other being algebra, the study of relationships between numbers. Primarily, the subject may be divided into six branches:
- Euclidean geometry studies geometry where the parallel postulate is valid.
- Non-Euclidean geometry studies geometry where the parallel postulate is false.
- Differential geometry studies properties of smooth shapes called differentiable manifolds by analyzing their curvature properties.
- Algebraic geometry studies properties of shapes defined by algebraic equations, by making use of the additional structures these objects inherit from their algebraic nature.
- Topology, broadly speaking, studies properties of shapes invariant under continuous functions.