Regular polygons with integer-coordinate vertices: one proof for all n

Are there regular polygons with integer-coordinate vertices?

Discrete geometry is a recent field of research concerned with "discrete objects," such as graphs and tilings. But it also offers a fresh perspective on problems that are difficult to solve in a continuous setting, including the existence of regular polygons with integer-coordinate vertices.

Discrete geometry at the crossroads of several fields of mathematics

Discrete mathematics emerged in the 1950s and remains a relatively recent field of research, encompassing chiefly graph theory, number theory, group theory, discrete geometry and cryptography. Its applications are everywhere in our daily lives, from algorithms and optimization problems to video-game graphics and security coding.

Discrete geometry is concerned with any object that can be described by a finite or countable set of geometric or arithmetic properties. The discrete plane is generally represented by a regular square grid, so that only points with integer coordinates are considered. Discrete geometry studies its own objects, such as plane tilings, discrete lines (whose definition is still under debate; see the article "Discrete lines: the birth of a new geometry?") and particular properties of graphs. Specific forms of reasoning and properties have been developed to study these "discrete objects." One example is the problem of determining which positions in the discrete plane can be reached using a given displacement vector (see the article "Moving around the discrete plane").

Another aspect of discrete geometry is the study of Euclidean objects from a different perspective. A pentagon, for example, can be regarded as five points in the discrete plane together with five line segments. Combinatorics therefore plays a special role, connecting an object of Euclidean geometry with integers and thus forging links with arithmetic. We might ask, for example, whether a given line contains points with integer coordinates (see the article "Integer points on a line"), or, as in this article, whether n-gons with integer-coordinate vertices exist. This last question is easy to understand but extremely difficult to solve within Euclidean geometry, even for small values of n, whereas the tools of discrete geometry make it readily approachable, while allowing all values of n to be handled at once!

A polygon is a collection of points in the plane (the vertices) joined by line segments (the sides) to form a closed polygonal line. It is regular if all its sides have the same length and all its angles are equal.

In discrete geometry (see box), tilings of the plane by polygons have held an important place since antiquity. The existence and construction of different kinds of tilings—regular, semiregular, periodic and so on—continue to generate numerous publications among mathematicians, designers and architects alike. Many such tilings exist, made from one or more different regular polygons.

Although regular polygons with n vertices—which we shall call n-gons—exist for every n > 2, only three can tile the plane on their own: the equilateral triangle, the square and the regular hexagon.

Regular polygons with integer-coordinate vertices: one proof for all n

Are there regular polygons with integer-coordinate vertices?

Polygons on a grid

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On the same theme

Integer points on a line: methods of solution

Discrete lines: the birth of a new geometry

Moving on the discrete plane: generating sets and minimality

So many distances!

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