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“ Though this be madness , yet there is method in it ∗ ” : The Importance Of Mathematical Concepts Beneath Contemporary Visualization

When asked what he thought of Western civilization, Mohandas Gandhi is reported to have replied that he thought it would be a good idea. How would a mathematician respond if asked about the mathematical basis of visualization ... and does the answer matter? Visualization occupies an important role in the pursuit of knowledge as the understanding of real world phenomena. From an external viewpoint, images assist mental processes like perception of spatial relationships, comprehension of complex time-varying behaviours, and detection of patterns. However, underlying the image are mathematical models of the world and its representation. Some of these models are obvious and relatively well-known; these include the calculi of scalar/vector/tensor spaces, the study of manifolds and homotopies. Others are (or at least their applicability to visualization) less well known: multivariate representation, infinite sequences, complex functions, and group theory which have all contributed novel insight into the nature of data and representation, or methods for transforming data into truth-preserving images. The key issues facing the visualization discipline from the “madness” of ever increasing dataset scale, to new challenges of visualizing multi-scale biological systems or huge volumes of streaming real-time data call for an improved understanding and appreciation of its underlying mathematics. Further, at a time when questions about the very core of the discipline, e.g. the relationship between information and scientific visualization, are being raised almost annually within the conference, it seems particular germane to stand back and review what is ultimately what could be the most abstract and neutral picture of visualization itself the mathematics world which ties together the notion of data and transformation.