Math for Art’s Sake

“He who does not understand the supreme certainty of mathematics is wallowing in confusion.”  Leonardo da Vinci.

Artists of the Italian Renaissance were driven to understand the perfection of math and the natural laws of geometry (notably linear perspective) in order to create beauty in representational art.

There are some art movements where math is an aesthetic end in itself – such as fractal art.  But for Renaissance artists, geometry was not an explicit effect, but utilised in order to create naturalistic environments and proportions.  In other words, the math behind the art was a practical tool, not art itself.

Vitruvian Man, by Leonardo da Vinci, ca 1490, Gallerie dell'Accademia, Venice

In this pursuit of naturalistic perfection through math, the world of contemporary 3D rendering and animation and the Italian Renaissance are linked.

I recently came upon the excellent BBC4 programme Vitruvian Man the first programme in the Beauty of Diagrams series and was struck by the common aspirations of Leonardo and 3D animators.

Leonardo in his diagram Vitruvian Man, seeks to discover within the body of man, a set of universally perfect proportions which he could then use to create perfectly proportioned architecture (which later evolved into Palladian Architecture). 1

Why try and find these perfect proportions in a man’s body?  At the time, it was believed that all of God’s creation was geometrically perfect, and man’s body was the supreme creation of God. The challenge for artists and philosophers was to find the nature of the perfection in the form of equations and laws, and apply these to their own creations.  His diagram not only documents a set of harmonious proportions but also predicts the position of man’s body when animated according to these calculations.

Harmonic Co-ordinates Explained, Pixar Video 2007

Fast forward five hundred years and mathematicians at Pixar studio seek to discover mathematical computations which will move 3D characters naturalistically through space.  This problem has been tackled rather successfully already as we have now enjoyed decades of quite realistic character movement in 3D movies and games.  However, the challenge is always to do it more efficiently (using up less processing power per frame) and more accurately (without need to intervene when computational output looks ‘wrong’). The really exciting thing is when an entire new kind of equation is developed for the purposes of visual art, that may not have otherwise come to light in the scientific community. Computer scientist Tony DeRose of Pixar Animation Studios describes how his research group developed a new application of Harmonic Coordinates in order to reduce time and effort in character animation.

“It’s new mathematics inspired by the film industry.”2

For those who can understand the math  the entire paper Harmonic Coordinates for Character Articulation is freely available.

An excellent film on YouTube, Harmonic Co-ordinates Explained also created by Pixar is also available, and is presents the main concepts that non-mathematicians can understand.

In Leonardo’s and Pixar’s quest for naturalistic representation, they have uncovered math for art’s sake.  For Leonardo, Palladian Architecture was the end result of math-art fusion while for Pixar it is character animation – and for both, the beneficiaries were not an elite art world, but a very wide and appreciative public.

See also The Volumetric Eye

References

1. Vitruvian Man, Programme 1/6 in BBC4′s series The Beauty of Diagrams (Available to view via iPlayer until 31 Dec 2010).  Mathematician Marcus du Sautoy explores Leonardo’s passions for anatomy, for the mechanics of the human body and for geometry.

2. Pixar’s Tony DeRose Illuminates the Mathematics of Animation, Oct 15, 2009,

Resources

• Harmonic Co-ordinates for Character Animation, a paper by Pushkar Joshi, Mark Meyer, Tony DeRose, Brian Green, Tom Sanocki, May 2007
• Harmonic Co-ordinates Explained, Pixar Video 2007