Introduction

Visualization has a major impact on the understanding and exploration of complex mathematical phenomena. As in other sciences, images are used in mathematics as helpful illustrations accompanying textual descriptions, as a communication medium to exchange ideas with other cooperating researchers, and to explain deep mathematical results in a comprehensive way to non-experts, just to mention only a few applications.

The process of visualization is a synonym for a broader process than the production of images or video animations. Prior the generation of images, abstract mathematical concepts must be translated into discrete descriptions which are numerical data structures to hold the mathematical information. For example, a smooth surface must be discretized in a set of triangles, and algorithms must be translated from their smooth counterparts acting on smooth geometries to discrete geometries.

In general these conversions are delicate tasks. For example a Riemann surface may have different smooth descriptions requiring non-obvious discrete structures such as triangulated piecewise linear surfaces with specific discrete properties. Continuous Riemann surfaces are identified under conformal maps, and so one would like to perform a similar identification on the corresponding discrete objects. It is of central importance in mathematical visualization to have discrete equivalents of smooth concepts, and discrete algorithms on discrete surfaces which perform the same operation as their counterparts on smooth concepts.

The process of defining useful discrete data structures, finding good discretization algorithms, and deriving methods operating on discrete geometries belongs to the same category of mathematical tasks which have been so important for smooth geometries. Good visualization and numerics rely on perfect discrete definitions which are the basis to assign, measure and transform mathematical properties of discrete geometries. Numerical mathematics has gone a long way in discretizing function spaces but, for example, the understanding of the intrinsic mathematical properties of discrete geometric shapes still leads to challenging questions.

It is important to distinguish between the process of visualizing precomputed numerical data and, in other sciences, of experimentally measured data, and the process of doing numerical experiments and simulations. The first process is an analytic process trying to understand large sets of numbers by the means of finding good visual representations. This includes hiding and emphasizing of data as well as the feature extraction of detail information. Many visualization tools operate in a post-processing step by evaluating large data sets and generating new smaller sets with feature information. For example, one of the current major problems is the visualization of turbulent 3d flows. Because of its complexity the flow might be unaccessible to a direct visual representation but the path of moving vortices may be computed and easier to display. Here the correct definition of vortices in a discrete flow is an essential prerequisite before starting the development of a tracing algorithm. The second process of doing numerical experiments and simulations is a constructive and repetitive process where an experiment is analysized during runtime and information obtained to steer parameters and interact with the numerical process. Here visualization is used to obtain insight into the behavior of a running algorithm rather than into geometric dataset.

Over the past years, visualization has proven to be a successful tool for mathematicians in the investigation of difficult mathematical problems which seem to be unaccessible by standard mathematical tools. It has proven its potential by substantially contributing to the solution of hard mathematical problems.

Mathematicians often have a concrete imagination of abstracts shapes even in higher dimensional spaces, and especially geometers have a deeply visually related thinking. Images are very helpful during their research, they give insight into unknown phenomena and suggest directions for further investigations. Nevertheless there is a remarkable contrast between the visual thinking, and the occurrence of pictures in research publications and inner-mathematical talks. Often images are extremely rare, for example, books on differential geometry, a mathematical subject with one of the largest fundus of easily visualizable complex geometric shapes, contain only a handful figures with most simplest shapes. The lectures on differential geometry of Luigi Bianchi is a comprehensive and very influential book on surface theory, whose German translation was published in 1910, and the classic book on Riemannian Geometry by Gromoll, Klingenberg, and Meyer from 1969, which was the book for a generation of geometers, both books contain no single image of a surface or a geometric shape. Similar examples up to modern days could be listed.

In some sense images have the character of an experiment: they suggest a truth or result but usually do not have the power of a proof. An image can illustrate a mathematical proof but an experiment or image implicitly requests a formal proof of its suggested result. Nevertheless, important experimental results exist which have not been rigorously proven, but they direct theoretical investigations and often give final hints. Experiments have been performed in the whole history of mathematics but only since 1991 there exists a publication media, the journal of Experimental Mathematics.

In contrast to a wide-spread opinion, mathematical history contains a rich set of visual examples. Archimedes was drawing figures into the sand when being bothered by the Romans during the capturing of Syracuse. Euclidean geometry, nowadays referred to as 'elementary geometry', is one of prominent examples where all kinds of drawings always played a central role in the communication and publication of results. Famous non-trivial examples are the copper plate engravings of Hermann Amandus Schwarz shown in the figure besides [10]. He discovered new minimal surfaces which solved long-standing questions in geometry and analysis on the existence of solutions to elliptic boundary value problems. Although his proof was of theoretical nature he found it worthwhile to invest a lot of energy to include visual images of the new surfaces into his research publications. This is remarkable if compared to the much less effort one needs for the generation of computer images today.

One of the most thorough approaches in mathematics to use physical models and experimental instruments in education and research is the famous collection of mathematical models in Göttingen. This model collection already had a long history when Hermann Amandus Schwarz and Felix Klein overtook the direction of the collection. Especially under the direction of Klein the collection was systematically modernized and completed for the education in geometry and geodesy. This collection was considered so important that Klein exhibited the models on the occasion of the World's Columbian Exposition 1893 in Chicago [2]. The models were produced among others by the publisher Martin Schilling in Halle a.S., see his catalog of mathematical models [9]. The price of approximately $250 per model was relatively, and therefore, the large size of the collection of more than 500 plaster models is even more impressive. The collection can still be seen in the mathematical department in Göttingen, and a description including photos of many models is given in Fischer [4]. An example is the plaster model of the Kuen surface in the figure besides.

Shown here is a modern model of the Chen-Gackstatter-Karcher-Thayer minimal surface produced in stereo lithography technique from digital data. The production of models slowed down and finally stopped in the beginning 30s. Following Fischer, the reasons were not purely economical nature but also the appearance of more general and abstract view points in mathematics. It was the time when the books of van der Waerden with only simple illustrations, and of Nicolas Bourbaki with a complete ignorance of images appeared.

It seems that minimal surfaces have been among those geometric shapes which often urged mathematicians and physicists to produce images. In the sequel to Schwarz and the experiments of Joseph Antoine Ferdinand Plateau, it was Richard Courant, Johannes C.C. Nitsche, and Alan H. Schoen who experimented with physical soap films and even produced real models for permanent display and for easier communication. Among the first breakthroughs of mathematical visualization was the proof of embeddedness of another minimal surface. Celsoe Costa discovered the mathematical formulae of a genus 1 minimal surface which was a candidate to solve a 200-year long question: whether there exists a third embedded and complete minimal surfaces with finite total curvature beside the trivial examples, the flat plane and the rotational symmetric catenoid. David Hoffman and William Meeks developed computer programs to visualize Costa's surfaces and watch surface properties which they could later successfully prove after having enough insight into the complex shape of the surface, see figures besides and [7].