We live in a world with three spatial dimensions and one temporal dimension. This means that to determine an exact location on Earth at any given time, we only need four values: the three coordinates — latitude, longitude, and altitude — and a timestamp. In the mathematical universe, the concept of dimension is much more general and it is common to work with systems of many dimensions or even infinite dimensions.

This notion has nothing to do with the idea – so cultivated in science fiction – of parallel universes, where an alternative reality outside our perception develops. Mathematically, dimensions are a theoretical construct that simply describes various facets of something. In mechanics — the branch of physics that studies motion and the forces that generate it — dimensions are the amount of information needed to describe a system. For example, a pendulum can be viewed as a system – called phase space– Two-dimensional, since to describe the pendulum’s movement at any moment it is only necessary to know the angle that the string makes with respect to the vertical and the speed —or the moment—.

To describe more complex physical systems, with more bodies or with more iterations, it is necessary to resort to more complicated phase spaces, such as those that model a planetary system, for example. These spaces have a specific geometric structure, called symplectic variety. Studying these varieties and their properties has been a topic of interest in various areas of mathematics and physics, including the so-called geometric mechanics.

Geometric mechanics deals with the applications of geometry to mechanics and allows to reduce complicated systems of many dimensions – such as symplectic varieties – to others with which it is easier to work, since, instead of observing each interaction or movement of form Separately, the global characteristics of the system are observed and used to transform the problem into a simpler one.

To do this, the first step is to identify the “symmetries” of the system, that is, the transformations that leave it invariant. With them, it is possible to reduce it and thus study it in a simpler way, as proposed by Emmy Noether in her famous theorem. If Noether’s theorem relates each symmetry of the system to a conserved quantity, the so-called moment application, it incorporates, at the same time, all these relations —symmetry / conserved quantity— of the system.

At the beginning of the 1980s, several researchers realized that, in addition, this application allowed us to translate the phase space into a simpler object, called a polytope, which is the generalization to any dimension of the polygons. These very simple —and discrete— forms therefore made it possible to interpret properties of very complicated systems. They work in a similar way to Google Maps: they offer an image, flat and very easy to understand, to represent a world, complex and with more dimensions, that we face.

An example of a polytope is the rectangle in the image below. It represents a system formed by three objects that move in relation to each other, in an eight-dimensional space. In particular, it indicates when these three objects move in alignment, at right angles, and when the small space The system —that is, the one that we obtain after applying our knowledge about symmetries— is a sphere or other geometric shape. Also, it shows when the three bodies are moving in a stable or more chaotic manner.

This polytope, proposed in recent research results, is related to systems that appear in research in robotics, thermodynamics and multidimensional field theory. Indeed, one of the reasons for the success of geometric mechanics is that it has numerous applications: it is used for the design of interplanetary missions, in computational anatomy, in the design of submarine vehicles, satellites, in robotics, in telecommunications, in the study of global warming … The list is huge.

Amna Shaddad is a Marie Curie researcher at the Institute of Mathematical Sciences (ICMAT)

Timon G Longoria Agate is coordinator of the Mathematical Culture Unit of the ICMAT

Coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”

Editing and coordination: Ágata A. Timón G Longoria (ICMAT).

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