Visualization and Online Experiments

Mathematical visualization has proven to be an efficient tool for analyzing complex mathematical phenomena, and it has given decisive hints leading to rigorous mathematical proofs of long-standing problems. Visualization is not only a tool to visualize complex objects but in combination with modern numerical methods allows to perform mathematical experiments and simulations in an artificially clean environment. For example, the unveiling of the Costa-Hoffman-Meeks surface [6] (figures Unveiling the Costa Formulas and Costa-Hoffman-Meeks Minimal Surface), or the first numerical examples of compact constant mean curvature surfaces with genus greater than two [5] (figures besides) are among the most prominent results of the fruitful interaction of mathematics with the new toolkit mathematical visualization.

Mathematical experiments require the following goals to be accepted in the mathematical community similar as experiments in physics and chemistry:

1. Validation of experimental data by independent groups
2. Publication and storage of experimental results and data sets
3. Cooperation of researchers on the same experiment while sitting at different places

Up to now visualization has required high-end workstations combined with mainframe computers for numerical computations. Each research lab had developed own software dedicated to specific graphics hardware. The specialization of the software for a specific visualization task or numerical problem as well as the dependence on a specific hardware platform had major drawbacks for the scientific communication. The major reason for the current non-fulfillment of the goals 1. and 2. is a missing world-wide standard and interface for the different software packages, and for goal 3. the missing standard to exchange data sets between experimental software packages and research publications. Currently research publications are paper-based where there is no way back from written publications to digital data sets. Even worse, in order to fit into a restricted page layout and fulfilling the allowed number of pages, publications cannot include all data but restrict to the so-called 'most relevant data'. The presentation of experimental results in publications are incomplete and often moved to the appendix so that it is usually not possible to validate or reproduce them.

In order to allow validation of experiments and of numerical data sets it is essential to allow direct public access to the data in an electronic format in the same way as public access to research publications is given through libraries. This joint publication of experimental and numerical data requires to insist on the similar principles known from the publication of scientific research results:

1. Unambiguous, self-contained digital representation of the data, i.e. no dependence on existing software packages.
2. Reviewing of data sets, e.g. to ensure technical correctness and scientific relevance.
3. Indexing data sets to allow unique references, e.g. when data is used in other experiments.

In the following sections we discuss how online visualization might look like and about the possible difficulties to encounter.