"Vedic Geometry & Tactile Mathematics" - March/April 2008

Vedic Geometry & Tactile Mathematics
by Stewart Dickson

The scene opens onto a desert landscape in which a rectangular black monolith stands. The music is the opening fanfare of Richard Strauss' tone poem, "Also Sprach Zarathustra" ("Thus Spake Zarathustra"). Zoroaster Spitama, "Zarathustra" lived from circa 628 to 551 BC in Persia, now known as Iran. [1] Among the Parsi stories and legends about Zoroaster are that he was born laughing, he was imparted wisdom from heaven and that he was assassinated by a priest of the Old Religion, which was threatened by Zoroaster's charisma. [2]

Among Zoroaster's followers were the Magi, three of whom were the "Kings" who visited Jesus Christ after his birth. In 586 BC the First Temple (King Solomon’s Temple) in Jerusalem was destroyed by Nebuchadnezzar, and the Jewish Intelligensia were exiled to Babylon (now Iraq). The Talmudic rabbis remained in Babylon for another 100 years.[3] Pythagoras arrived in Egypt around 547 BC, when he was 23 years old to study mathematics. He stayed in Egypt for 21 years. Cambyses invaded Egypt circa 529 B.C. and Pythagoras was exiled to Babylon. The mathematics of the Zoroastrian magi were orders of magnitude advanced beyond that of the Egyptians. Pythagoras most likely learned his mathematics and numerological mysticism from the Babylonians. It is well documented that Zoroasterian philosophy influenced Judaism (the Kabbalah) and through it Christianity and Islam, but what about the connection between the Middle-East to the Far East?

This question is hard to answer. Possibly one of the most important inventions of early mathematics was the concept of Zero.[4] The numeral Zero first appears in writings on stone columns in India which date from around 250 BC. The Hindu(-Arabic) numerals first appeared in Arabia around 700 AD. Zero was unknown to Europe before Fibonacci introduced it in the 12th Century AD.

The most ancient and sacred Hindu texts are known as the Vedas. They include the sutras (disciplines) of Yoga as well as Tantra (the famous Kama Sutra) and Mathematics.[5] The dates of the Vedic texts are not well documented. Vedic Mathematics is a system of intuitive and visual, mechanical tricks for doing mental arithmetic. ("By 'looking' at the problem.") [6] In "Vedic Mathematics", there is an algebraic proof of the Pythagorean Theorem which possibly predates Pythagoras.[7] Ramesh Raskar and Krishna Kunchithapadam seem to have uncovered evidence that the Hindus could compute Pi to 31 decimal places at the time when the Greeks and Egyptians knew Pi to three places.[8][9] The Vedic Mathematical texts contain numerous expressions of the sacred and mystical nature of numbers. Unfortunately, Vedic Geometry seems to be limited to the Sri Yantra. [Figure 1] (See: The Last Mimzy, 2007)[10]


Figure 1: Great Sri Yantra, source: Mandalas.com

I have been doing mathematical digital sculpture since 1990 and I have heard more than once commentary on the sensuality of pure form, such as that which can be expressed mathematically. Indeed, this has been expressed by no less than a former Fellow at the Princeton University Institute for Advanced Studies -- the same institution where Albert Einstein and John Forbes Nash once worked.[11] There are also the mystical aspect of numbers and Tantric sex. Marcel Duchamp even articulated ideas that the social interaction of men and women takes place in the Fourth Dimension.[12] There must be a way to better synthesize all of this in a contemporary framework -- now that we have the tools to better translate between the purely abstract and the visual, concrete and tactile in 3D computer graphics and animation.

We know Constantin Brancusi and Max Bill, but there is inherent and well-founded mistrust in the ideology of Modernism. I believe that the advent of networked visual computing affords the opportunity to construct micro-communities amid the celebration of diversity on the Internet, which will once again nourish ideas of the mystical attributes of pure form.

How do we appreciate physical form in three dimensions? It is not easy to talk about. It is done through vague notions such as topology and principal curvature.[13] Most concepts from topology are best communicated through pictures and 3D, manipulative models. When a computer-generated mathematical surface is rendered into a physical sculpture, it is not a self-deascribing object! The abstract language which created the sculpture remains inside the computer and I have to use a lot of words to explain what the surfaces mean. The principal curvatures of a surface lie in planes perpendicular to each other and containing the normal vector to the surface -- concepts familiar to 3D computer graphics. Does it make sense to rhapsodize on the beauty of the human physique in the same terms? How about in terms of biomedical computer visualization?

In 2003 I attended a conference on Virtual Reality applications to Medicine, where a working group on Virtual Humans also met.[14][15] At that meeting, Dr. Andries Van Dam of Brown University brought one of his former graduate students, Scott Anderson to present the pre-production R&D work he supervised for Sony Pictures' "Hollow Man". [16][17][18] Art informed the science: The anatomical models in "Hollow Man" became the inspiration piece for Virtual Humans and it represented the vision that the surgeons, physiologists and biomedical engineers had for a 3-D user interface to medical knowledge.

For the next eighteen months, I was on a team who developed a very reduced glimpse of what is possible to achieve in a fully-instrumented, 3D anatomical model. But, the project leader -- chief of surgery at University of Washington Medical Center -- was seeing the full-blown Star Trek Sick Bay in his vision for the operating room of the future.

The DARPA Virtual Soldier was focussed on connecting anatomical knowledge and real-time physiological simulation to a thoracic model in Utah's SCIRun (pronounced "ski-run").[19][20] In principle, however, this is not too different from building anatomically-inspired (e.g., muscled & skinned) full-body, rigged animation character models in a system like 3D Studio max or Maya.[21]

For the DARPA Virtual Soldier, a 3D model of a beating heart was created by linking a static 3D finite-element mesh model to a moving 2D ultrasound echocardiogram slice. The key to accomplishing this is an anatomical Morphometric markup.[22] The markup entails attaching anatomical name markers (e.g., Maya locator nodes) to structures (Nurbs Surfaces) in the model. The surface parameterization is key to obtaining a morph target correlation which is free of mesh topology constraints. I believe this could even be automated by attaching curvature signatures (tip of nose, shoulder, etc.) to the Morphometric markup. Using this technique, it should be much easier to Morph a rigged, generic humanoid animation model to a 3D scan (point-cloud/unorganized mesh) to obtain the rigged digital stand-in for a specific actor. Similarly, the intent with Virtual Humans is to morph a generic, instrumented biomedical surface model to a specific patient's segmented MRI.

CgCharacter's Absolute Character Tools (ACT) are based on a lightweight muscle node (Nurbs Surface) in 3D Studio Max.[23] (In circa 2000, there was a planned version for Maya, which does not seem to have materialized.) If you carefully study cgCharacter's on-line documentation for their biomedical products, you will see glimpses of the idea of morphing a generic 3D anatomical model to slice-wise patient-specific CT or MRI scan data.

In his book, "Anatomy of Hatha Yoga", H. David Coulter, MD -- an anatomist and yoga practitioner -- renders in detail the reflexive, neuro-musculo-skeletal feedback circuits which drive physiological response (like boost in immunity) during the practice of yoga.[24] The Virtual Soldier physiological simulation was limited to the blood chemistry of the cardio-pulmonary system.[25] It is generally known that yoga stimulates the sympathetic nervous and endocrine (glandular) systems, but the scientists are reluctant to make statements like this because these quantities are clinically impossible to measure in real-time. However, if the model is detailed enough, all these quantites are measurable in simulation and the inter-acting connections can be demonstrated.

Many of the pieces have been demonstrated individually and on a limited basis. It has never been done on a large scale and integrated with character animation. I propose to do this. People have told me that the collective market for yoga-related content is huge. Who do we pitch it to? A game company? Has anybody in the Animation Coop ever played Journey to Wild Divine? [26][27]

Surgeons are generally in agreement that to accurately model joint flexure (fingers, hands, shoulders, knees, hips, ...) requires accurate modeling of all tissue layers between the skin and the cartilage and bone, which supports and gives shape to it all. I believe that adding the neuro-muscular feedback circuits to an anatomically-detailed character animation model will allow art to once more inform science: We wil show them a working model of what up to now has been intuitively known, but has never been clinically observable.

The mythology of numerology is fairly well known. In the 1998 film, 'Pi' by Darren Aronofsky, the protagonist is computing the digits 3.14159… At 120 digits, he begins to experience headaches and nosebleeds.[28] He consults Hassidic rabbis of the Khabbalah, who tell him that the unspeakable, true name of G*d is encoded in the digits of Pi. The more digits he computes, the closer he comes to holding the true name of G*d, which cannot be held. What are the metaphysical implications of a Vedic scholar who can mentally compute the digits of Pi as a meditation exercise?

As for demonstrating the mystical aspects of mathematical surfaces in three dimensions and higher? That, I still don't fully know how to do. Here's one idea: British Digital Sculptor Keith Brown finds mystical significance in the Torus -- and rightly so. The Torus is found in many places in physics. See Kenneth Snelson's renderings of atoms: The electron orbitals are conceived as tori.[29] The field of a magnetic dipole in fact has the form of a hypersphere: a nested torus of infinite sheets. [Figure 2] Costa's minimal surface has the form of a torus with three points removed -- literally removed from three-space and flung to infinity.[30] The same operation can be performed by removing three rays (half-lines) from the hypersphere.[31] What shape is the decay of a proton? Given that, in the Inflationary cosmological model, the Big Bang erupted from a quantum point -- what was the shape of the explosion? Does a surface which has shape properties such as these retain any mystical properties?


Figure 2: Fibrations of the Hypersphere atop Laidlaw and Kocak's CGI rendering of the Hypersphere (for Thomas Banchoff)

The ape-people at the opening of Kubrick's "2001: A Space Odyssey" could conceive as little of what they were destined to evolve into as we can conceive of our own next phase of evolution. This was Douglas Trumbull's task in designing the end sequence for the film. Sitting at the forefront of human knowledge -- as we know that the Internet arose from the ARPANet, the NSFNet and the scientific visualization of the National Center for Surpercomputing Applications (authors of NCSA Mosaic -- the precursor to the Netscape Web Browser) -- we find that the more we know about the true structure of the Universe, the more bizarre it appears. It has been said that the mathematics of cosmology are so elegant that mathematics seems almost to be part of the universe, itself.[32]

As for mathematics and eroticism, Marcel Duchamp imagined the woman's uterus as having the form of a Klein bottle. [12][33] In work I have done developing tools for composing tactile captions onto mathematical surfaces, it occurs to me that the action of reading a line of text which describes a mathematical curve, when the caption is printed along the very surface the text is describing, must be a synergetic experience: The reader's hand is tracing the curve as it is reading the abstract statement describing the curve.[34][35][36] I have been fantasizing on the possibility of creating tactile tattoos that make statements about the portion of a person's body on which the tattoos are applied. [37][38][Figure 3] How close is this to John Varley's conception of BodyTalk in "The Persistence of Vision", with its mystical outcome?[39][40]


Figure 3 -- The Principal Curvatures of a Cyberware Figure Scan by Stewart Dickson[37]

Figures:

References:

Stewart Dickson is a Visualization Research Programmer at the Integrated Systems Laboratory of the Beckman Institute, University of Illinois at Urbana-Champaign. (http://www.isl.uiuc.edu/~sdickson)

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