Geometry studies form, magnitude, and spatial relation. It asks not only what exists, but how objects extend, intersect, rotate, and occupy space.
Where topology preserves structure under deformation, geometry measures and constrains shape through length, angle, area, and curvature.
Points, Lines, Planes
Geometry begins with undefined primitives whose relationships are fixed by axioms rather than construction.
Axiomatic Systems
Euclid’s postulates, and their later revisions, define what can be constructed and proven.
Metric Assumptions
Distance and angle introduce rigidity, distinguishing geometry from purely topological reasoning.
Euclidean Geometry
Flat-space geometry governed by parallel lines, familiar constructions, and classical measurement.
Non-Euclidean Geometry
Hyperbolic and spherical geometries arise when the parallel postulate is altered.
Differential Geometry
Curvature, geodesics, and smooth manifolds link geometry to physics and spacetime.
Algebraic Geometry
Shapes defined by polynomial equations, where geometry and algebra co-determine structure.
Geometry governs physical law, architectural stability, visual perception, and the structure of equations themselves.
It provides the language through which space becomes measurable, navigable, and computable.
Geometry of flat space governed by rigid distance and angle. Parallel lines never meet, and figures obey classical measurement.
Role: Construction, intuition, and baseline spatial reasoning.
Geometry arising when the parallel postulate is altered. Space may curve positively or negatively.
Role: Cosmology, relativity, and intrinsic curvature.
Geometry of parallelism without distance or angle. Ratios along lines are meaningful; metric notions are discarded.
Role: Linear algebra, vector spaces, coordinate frames.
Geometry with points at infinity added so that all lines intersect. Perspective becomes intrinsic.
Role: Vision, duality, algebraic geometry, invariants.
Geometry of smooth manifolds using calculus. Curvature, geodesics, and tensors describe local structure.
Role: Physics, spacetime, curvature flows.
Geometry defined by polynomial equations. Shape and algebra determine one another.
Role: Number theory, moduli spaces, deep structure.
Geometry of finite objects: graphs, polytopes, triangulations. Algorithms replace smooth analysis.
Role: Computation, graphics, optimization.