What Is Geometry?

What is Geometry?

Geometry is the study of shapes and configurations. It attempts to understand and classify spaces in various mathematical contexts. For a space with lots of symmetries, the study naturally focuses on properties which are invariant (remaining the same) under the symmetries.

Euclidean Geometry

An example is the study of flat space, or Euclidean geometry. Between every pair of points there is a unique line segment, which is the shortest curve between those two points. These line segments can be extended to lines. Lines are infinitely long in both directions and, for every pair of points on the line, the segment of the line between them is the shortest curve that can be drawn between them. Furthermore, if you have a line and a point which isn't on the line, there is a second line running through the point, which is parallel to the first line (i.e. never hits it). All of these ideas can be described by a drawing on a flat piece of paper. From the laws of Euclidean Geometry, we get the famous theorem of Pythagoras, and all the formulas you learn in trigonometry. In Euclidean geometry, you also learned how to find the circumference and area of a circle.

Riemannian Geometry

Now suppose we are on the surface of a sphere, which is no longer flat, but curved. A shortest curve between any pair of points on a surface is called a minimal geodesic. You can find a minimal geodesic between two points by stretching a rubber band between them. The first thing that you will notice is that sometimes there is more than one minimal geodesic between two points. For example, there are many minimal geodesics between the north and south poles of a globe. As in Euclidean space, we can look for lines with the property that the segment between every pair of points on the line is a minimal geodesic. Curved surfaces are harder to study than flat surfaces, but there are still theorems that can be used to estimate the length of the hypotenuse of a triangle, the circumference of a circle, and the area inside the circle. These estimates depend on the amount that the surface is curved or bent. One of the basic topics in Riemannian Geometry is the study of such curved surfaces.

Gravitational Lensing

Riemannian Geometers also study higher dimensional spaces. The universe can be described as a three dimensional space. Near the earth, the universe looks roughly like three dimensional Euclidean space. However, near very heavy stars and black holes, the space is curved and bent. The Hubble Telescope has discovered points in space that have more than one minimal geodesic to the telescope. This is called gravitational lensing. The amount that space is curved can be estimated by using theorems from Riemannian Geometry and measurements taken by astronomers. Physicists believe that the curvature of space is related to the gravitational field of a star, according to a partial differential equation called Einstein's Equation. So using the results from the theorems in Riemannian Geometry, they can estimate the mass of the star or black hole which causes the gravitational lensing.

Other Types of Geometry

In general, any mathematical construction which has a notion of curvature falls under the study of geometry. Examples include:

Differential geometry which is a natural extension of calculus and linear algebra, and is known in its simplest form as vector calculus.
Algebraic geometry which studies objects defined by polynomial equations. This was vital to recent solutions of many difficult problems in number theory, such as the finiteness of solutions to the polynomial equations considered in Fermat's Last Theorem.
Semi-Riemannian geometry which Einstein used to study the four dimensional geometry of space and time.
Symplectic geometry which originated with the study of the evolution of simple mechanical systems, but now pervades all aspects of theoretical physics.

Courses

Our geometry courses reflect this immense range from the study of configuration of points and lines in flat and curved spaces to the geometry of objects defined by polynomial equations. Courses we offer beyond vector calculus include differential geometry of curves and surfaces, tensor calculus, differential geometry and general relativity, as well as projective and non-Euclidean geometry.