Ehrhart theory

Ehrhart theory studies how the number of lattice points (points with integer coordinates) inside a scaled version of a polytope (a multi-dimensional shape) relates to the scale factor. Specifically, if you have a shape with corners at integer points, multiplying its size by a number t and counting how many lattice points are inside gives a polynomial function of t. This polynomial, called the Ehrhart polynomial, reveals patterns about the shape's geometry and symmetry. The theory helps connect discrete counting problems with continuous geometric properties, offering insight into how shape and size influence lattice point arrangements.