Discrete geometry, at the crossroads of disciplines
Discrete geometry is a field at the crossroads of several disciplines, such as geometry, arithmetic or combinatorics. To get an idea, let us replace the continuous Euclidean plane with a grid, made up of points with integer coordinates. The geometric objects we encounter there are, for example, polygons whose vertices lie on these points. Can we define a line, a circle? These questions arose with particular acuity from the 1950s onwards, when it became necessary to represent images on screens, discrete sets of pixels. Many problems in discrete geometry are still open and lead to exciting developments.
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Regular polygons with integer-coordinate vertices: one proof for all n
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.
Integer points on a line: methods of solution
Many problems in discrete geometry are simply stated and can be used in teaching to make certain concepts easier to understand. Searching for points with integer coordinates on a line naturally leads to applications of theorems from number theory.
Discrete lines: the birth of a new geometry
Problems involving representation on a pixelated screen belong to discrete geometry. But beyond mere aesthetic considerations, are these new subjects of study—including discrete lines—tools for exploring Euclidean geometry more deeply, or objects of a new geometry?
Moving on the discrete plane: generating sets and minimality
Exploring how one can move around a grid leads to the concepts of generating sets and minimality. This example shows the power of discrete geometry as a tool for better visualizing and exploring complex ideas from classical geometry or algebra.