Discrete geometry is the study of discrete arrangements of geometric objects like circles, cubes, lines, rectangles, spheres and triangles. Although these shapes are typically in Euclidean spaces, the study sometimes includes non-Euclidean spaces as well. It includes the theory of polytopes and tilings and the theory of packing and covering (Bezdek, 2010). Before the advent of the computer, discrete geometry was called combinatorial geometry.
Like many types of geometry, problems are usually visual and applied. Solutions in discrete geometry use a wide variety of methods, including:
- Combinatorial methods, concerned with problems of selection, arrangement and operation. Counting and tilings are especially important in discrete geometry. Tilings are a way to decompose a space into lots of smaller pieces.
- Algebraic Coding theory: Transmission of information over less than optimal communication channels. Cryptography is closely related, except that instead of a noisy channel you’re trying to “beat an opponent” (Bierbrauer, 2016).
- Calculus of variations: a generalized calculus which uses small changes in functions, called variations.
- Differential geometry: The application of the calculus of derivatives and integrals as well as linear and multilinear algebra to study geometric problems.
- Geometric analysis: uses tools from differential equations (especially elliptic partial differential equations).
- Group theory: studies algebraic structures called groups.
- Number theory: the study of integers and integer-valued functions.
- Topology: The study of geometric properties and spatial relations unaffected by continuous deformations, like bending, crumpling, stretching, and twisting (but not gluing or tearing).
Example Problem in Discrete Geometry
A well known problem is how to calculate the area of an irregular polygon. The solution is found with Pick’s theorem on grid or lattice planes; A lattice is a grid of points with integer coordinates. Pick’s theorem tells us to count the points on the boundary and the interior of the shape. The formula is:
Area = i + (b/2) – 1,
- i = number of points in the interior,
- b = number of points on the boundary.
For example, the area of the following polygon is A = 7 + (8/2) – 1 = 10.
Bezdek, K. (2010). Classical Topics in Discrete Geometry. Springer New York.
Bierbrauer, J. (2016). Introduction to Coding Theory. CRC Press.