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Se non è vero, è ben trovato! One of the two talks of Werner Benger at the Second High-End Visualization Workshop from April 2005 is quite originally entitled “Visualizing Curved Space in the Human Brain”. The tensor visualization techniques treat scientifically both the brain and the universe on equal footing. However it seems to us that, while Benger’s title is being “ben trovato” it is also “vero” in a more precise and way deeper sense: our brain contains information about the whole universe! This should be actually the essential quality of the man as a homo cosmicus. It is this vision the one which created the religious conscience in the first place!

It seems that we have indeed to visualize tensors everywhere, inside as well as outside us. Especially the light, inasmuch as gauge phenomenon seems to be best described, at least in a linear approximation, as a tensor phenomenon associated with the deformation of space. However, the most profound meaning of the Benger’s presentation title pops up when one notices that there is no reference for any motion in space: one cannot say that a trajectory or an orbit is what it appears to be locally. Take for instance the Moon: it appears as revolving around Earth, which appears as revolving around Sun, which appears as revolving around galactic center, which appears as revolving… The precise space shape of a trajectory is a matter of scale. If one asks about the true motion of the Earth for instance, one certainly cannot answer. However, in a relational natural philosophy, we can think of the structure of the Earth as determined by its relation with the whole universe: a kind of general Mach principle. Now, if we are a little more liberal in our natural philosophy, and accept, for instance, the religious argument on equal footing with the scientific argument, a way suggests itself, helping us to see the ultimate motion of the Earth, and its relation with the evolution of its structure. This way should be one recorded in the human brain by a kind of tensor glyphs, in spite of the fact that we are not yet completely aware of it. In other words, the evolution of brain reproduces the evolution of Earth, and perhaps the brain carries somehow in it the very fate of the Earth.

It is only lately that a school of thought initiated by Professor Nir Sochen of Tel Aviv University realized that the visualization of tensors is basically related to a kind of Dirichlet principle characterizing the harmonic mappings, so important in general relativity and even when the Kepler motion is regarded as a measure of the inertial field. One can follow the essential traits of this school of thought, within the specific problem of regularization of tensor field rendering, in the doctoral dissertation of Yaniv Gur, a pupil of Professor Sochen (Gur, 2008). The correlation of this fundamental theoretical physics fact with the acute problem of tensor visualization, and implicitly with the problem of glyph representation of tensors in general, should be considered as a lucky strike of the science in general.

Reference

Gur, Y. (2008): Tensor-Valued Image Regularization via Geometric Flows, PhD Dissertation, Tel Aviv University, Israel