Roger Vilder (FR)

Géométries multiples II

Texts by Samuel Fiorini

Graphs are ubiquitous in our connected world. Subway maps, internet, aerial connections between airports, microprocessors: all these things can be described with graphs. No surprise that graphs are also called networks!

There are several ways to display graphs. What counts is to correctly represent objects (called vertices) and the visual relations (called edges) between said objects. The graph of a cube has eight vertices and twelve edges; each vertex is the number of edge-ends at that vertex.

Whether the graph of a cube is depicted using a one-point perspective, or a central projection, the graph remains the same.

Graphs are separate from their representation. In Roger Vilder’s work, vertices are mobile. Moving the vertices affects the edges, which stretch out or contract. The graph, however, remains the same.

Imagine two different “photos” of the same graph. The elements that help to identify vertices have been erased in both photos. How to know if both drawings represent the same graph?

This question points to one of the most important concepts across mathematical fields: isomorphism. Two graphs are isomorphic if there is a one-to-one correspondence between their vertices in order to create a correspondence between their edges.

Isomorphism can be applied to many other mathematical structures.

To put it simply, mathematicians aim to study the mathematical structure of a set of elements. Regardless of what the elements are called, the structure remains the same. This is exactly what isomorphism refers to.

For example, in algebra, it is possible to show that translations that preserve a frieze pattern and its subgroup of symmetries is isomorphic to a set of integers and their addition.