Just imagine that you have attended the SIAM Conference on Industrial and Applied Geometry (GD29) in San Francisco and you are aboard a plane that is taking you back home to your research institute in Berlin. You see on the monitor in front of you that the flight route from the San Francisco airport to the recently-opened Berlin Brandenburg airport is a crooked curve passing over Iceland.
Why are airlines not using world maps on which the flight route between two cities is a straight line segment? Would life be not easier if we all used gnomonic map projections?
Left: Mercator projection Right: Gnomonic projection. Image credit: Wikimedia Commons
The life of sailors in the early modern period would have certainly been much harder. At this time, they had no modern devices such as GPS to navigate their ships. In order to determine their position on the open sea, they used a compass and Jacob’s staff, later an octant or a sextant. These instruments enabled them to measure angles with respect to the North Pole, fixed stars, or coastlines. Hence, the angles of their nautical charts had to coincide with the angles on the earth. A map from one surface—such as the surface of the earth, to another, such as a piece of paper—that preserves all angles is called conformal. The first such map was presented in 1569 by the cartographer Gerardus Mercator and is nowadays used by most web mapping services.
Conformal maps preserve angles – picture taken from the film conform! by Alexander Bobenko and Charles Gunn
On a conformal map of the earth, the shortest path between two points is almost never straight. Instead, conformal maps possess the remarkable property that at each point, any direction is scaled by the same factor, so such maps are not distorting shapes on the small scale. However, this is not true globally: On the Mercator projection, Greenland seems to be as large as Africa, even though it is smaller than the Congo Basin. For this reason, organizations such as the UNESCO prefer equal-area map projections. For applications in computer graphics conformal maps are the tool to use — for example, when two-dimensional images are projected onto virtual three-dimensional objects.
Conformal projections of two-dimensional patterns onto surfaces of three-dimensional objects. Image credit: Peter Schröder
However, the abilities of computers are limited. That means that it is necessary to describe conformal maps using a manageable amount of data. Mathematicians here speak of a discretization. In the case of the earth’s surface, we may take a lattice composed of meridians and circles of latitudes. Apart from the poles, this discrete earth is a quadrilateral lattice. A map from one quadrilateral lattice to another is then called discrete conformal if the angles under which the diagonals of a quadrilateral intersect as well as the ratio of their lengths are preserved. Remember that in the continuous world, equal scaling was a consequence of the preservation of angles, in contrast to the discrete setting. This notion of discrete conformal maps, which is just one out of several possible discretizations, was first studied in the forties of the last century on square lattices in the plane. In the late sixties, lattices composed of rhombs were studied, and recently, comprehensive theories on general lattices in the plane or on surfaces were presented.
Discrete conformal map from one quadrilateral to another. Image credit: Felix Günther
In the last decade, interest in that theory of discrete conformality arose from its applications in statistical physics. A notable protagonist is the mathematician Stanislav Smirnov who has been awarded the Fields Medal for his striking contributions to the proof of conformal invariance of the planar Ising model that is one of the most popular mathematical models of ferromagnetism. The critical temperature at which the magnet loses its permanent magnetic properties is known as the Curie temperature. Of particular interest is the behaviour of the model at this phase transition. If one investigates the Ising model on planar domains that are transformed to each other by conformal maps, then conformal invariance states that the shapes of the interfaces between areas of the same magnetization transform by the very same conformal transformations (to be precise, one has to consider their probability distributions). Conformal invariance is the central pillar of conformal field theories that find applications in different fields of physics.
Zoom of a 2D Ising model at the critical temperature
The reason why I took discrete conformal maps to heart are the impressing analogies to the classical theory. Several theorems that are known for continuous conformal maps can be adapted to their discrete counterparts. One of my favourite theorems is Cauchy’s integral formula. It describes how the behaviour of a (continuous or discrete) conformal map in the interior of a planar domain can be computed out of its image at the boundary of that domain. That means that the outward appearance of a conformal map suggests all its inner values — is that not love at first sight?
Watch Felix’s presentation at the Falling Walls Lab 2014 Finale in Berlin here. He is an active science slammer.
This article has been adapted from the author’s article on the Falling Walls Fragments blog.
||Felix Günther is a postdoctoral researcher in the Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics” at TU Berlin. As an active science slammer, he is regularly presenting his research on discrete complex analysis to a broad audience. In his spare time, he is a passionate tango and salsa dancer.