The Discrete Charm of Geometry


A team of mathematicians is working together on a big project. Excitement of discovery, hope and disappointment, competition and recognition are shown from an infinitely close distance. Scientists united by the idea of discretization, which, in short, means: constructing continuous objects from basic building blocks. Akin to the scientists’ search for the right discretization of continuum, this film itself is composed of fragments – individual characters of different ages, temperaments and scientific approaches – which form a single continuous melody. The question of where the boundaries lie between mathematics and the lives of those who are involved in it and how much they are willing to sacrifice is as important as the search for precise scientific answers. A unique and unprecedented dive into the unknown world of mathematicians.




How can you make good flat maps of the round earth?' Our story begins with Mercator's world map of 1569, the first angle-preserving (or 'conformal') world map. His idea fell on fruitful soil, from which a new branch of mathematics has developed. The movie shows some of the highlights of this development, yielding a series of elegant visual forms which arise as 'conformal maps' on a variety of surfaces in 2- and 3-D. Featuring non-technical language, a simple aesthetic, compelling animation, and an original score, the movie builds an accessible bridge from everyday experience to a beautiful but little-known mathematical theory that continues to bear technological fruit today in fields such as computer graphics and architecture.


Schrödinger's Smoke (SIGGRAPH 2016)

s2016_Technical-Papers_0032_Schrodingers-Smoke-1024x683.png© 2016 California Institute of Technology, Technische Universität Berlin, and Google

We describe a new approach for the purely Eulerian simulation of incompressible fluids. In it, the fluid state is represented by a ℂ²-valued wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamical system is Hamiltonian and governed by the kinetic energy of the fluid together with an energy of Landau-Lifshitz type. The latter ensures that dynamics due to thin vortical structures, all important for visual simulation, are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids. Our implementation uses a simple splitting method for time integration, employing the FFT for Schrödinger evolution as well as constraint projection. Using a standard penalty method we also allow arbitrary obstacles. The resulting algorithm is simple, unconditionally stable, and efficient. In particular it does not require any Lagrangian techniques for advection or to counteract the loss of vorticity. We demonstrate its use in a variety of scenarios, compare it with experiments, and evaluate it against benchmark tests. A full implementation is included in the ancillary materials.

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Inside Fluids: Clebsch Maps for Visualization and Processing (SIGGRAPH 2017)


© 2017 California Institute of Technology, Technische Universität Berlin (Photo: Business Wire)

Clebsch maps encode velocity fields through functions. These functions contain valuable information about the velocity field. For example, closed integral curves of the associated vorticity field are level lines of the vorticity Clebsch map. This makes Clebsch maps useful for visualization and fluid dynamics analysis. Additionally they can be used in the context of simulations to enhance flows through the introduction of subgrid vorticity. In this paper we study spherical Clebsch maps, which are particularly attractive. Elucidating their geometric structure, we show that such maps can be found as minimizers of a non-linear Dirichlet energy. To illustrate our approach we use a number of benchmark problems and apply it to numerically given flow fields.

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