The distances

Since the beginning, humans have needed to predict, and therefore to measure. How to evaluate the distance between two buildings, two cities, two mountains, two planets? The GPS integrated into our phones and our cars would almost make us forget that there was a time (not so far away) when estimating a distance was a scientific feat! This began with surveying in a nearby environment, then we became interested in distances that are directly inaccessible, allowing us to find our location on land, at sea or in space. New techniques, often relying on clever mathematics like triangulation, were imagined, as well as a whole range of challenges: finding the shortest paths, determining equidistance curves, placing points as regularly as possible on a sphere…

Since the end of the 19th century, mathematics has embraced the abstract notion of distance by integrating new ways of calculating it in the plane or space: the 'Manhattan' distance, travel time distance... One can then wonder what the corresponding 'circle' looks like. Drawing skills are sometimes necessary to trace the surprising shapes that answer the question. The axiomatization of the concept of metric space also makes it possible to extrapolate distance to combinatorics, analysis, and even algebra. The objects considered are no longer necessarily points in the classical sense. One can then, for example, define a distance between two functions.

After the time of conceptualization came that of applications in very varied fields, very often outside the field of mathematics, such as linguistics, physics or medicine. Many problems are solved there by building an appropriate distance, whose interpretation opens an extraordinary field to issues of our daily life. It becomes possible to measure a distance between words, between curves, between images. Who could believe that bank account numbers, the spell checker in your word processor or the interpretation of your blood test results rely on this concept? Discover some of these applications…