DISCRETE GEOMETRY

Constructing shapes in discrete geometry is rather like looping string around a pin board. Despite its apparent simplicity, it has found applications in software design and quantum theory.

WHAT IS IT?

Ordinary geometry—the mathematical study of lines, angles, shapes, and solids—is “continuous,” in the sense that all points in space are allowed.

In discrete geometry, however, that all changes. Here, all you get is a well-defined lattice of points, and all other regions are off limits.

PICK’S THEOREM

One remarkable result in discrete geometry was proved in 1899 by the Austrian mathematician George Pick.

He found a formula for calculating the area of an arbitrary-shaped polygon formed by joining points on a discrete grid, each separated by one unit.

If the number of points marking the outside of the shape is x, and the number of points inside the shape (not touching the boundary) is y, then the area inside is given by the formula:

It’s very simple and works for any shape you care to draw. Try it!

APPLICATIONS

POLYGONS

Discrete geometry has applications in the pure study of geometric shapes, such as tessellations and aperiodic tilings.

PACKING PROBLEMS

Attempts to calculate the most efficient way to stack shapes and objects together, such as Kepler’s conjecture, are problems made for discrete geometry.

LATTICE GAUGE THEORIES

Some toy models in quantum physics working by discretizing space and time into a grid of points a fixed distance apart.

COMPUTER GRAPHICS

Less obvious in the high-definition world of today, but computer graphics are also an elaborate construct of discrete geometry.