The Art of Mathematics

June 9, 2016

Vicarage Studio's poignant homage to Messiaen's Quartet for the End of Time is built from the same numerical rules as its inspiration.

 

By Steve Jarratt

 

In 1940, the French composer Olivier Messiaen was captured by German forces and imprisoned in Stalag VIII-A in the east German town of Görlitz. While in captivity, he met with a cellist, a violinist and a clarinettist, for whom he wrote his Quatuor pour la fin du temps, or Quartet for the End of Time. The four of them performed the piece to an audience of some 400 prisoners on a cold and rainy evening on January 15th, 1941.

 

To celebrate its 75th anniversary, Sinfini Music commissioned freelance animation director Simon Russell of Vicarage Studio to produce a video for the piece. The final 3m 46s animation is accompanied by The Crystal Liturgy, one of the eight movements in Quartet for the End of Time.

 

The stunning animation features a single camera move, rendered in one unbroken take, as the viewer is drawn through the mechanical contraptions and polygonal life forms inhabiting an island floating in a monochromatic void. But amid the seeming chaos, the form and movement is all driven by the same numerical laws that Messiaen employed to conceive his music.

 

 

 

 

 

"It's based on a talk by mathematician Marcus du Sautoy,"' explains Russell. "He talked about how Messiaen used mathematics in his music. It's really about the mathematics of nature and how that brings beauty."'

 

Russell suggests that the challenge was in "trying to understand then translate some quite complex mathematical ideas into animation and then form them into a very watchable, interesting format. Once the concept was done it was a matter of dealing with a polygon-heavy scene and making it feel light and synchronised to the music."'

 

The sequence begins with the camera dollying back through a forest of trees and bushes, many of which are generated using the Lindenmayer system or L-System, which employs a set of instructions to describe the growth behavior of plants and trees.

 

"You tell it to 'Go forward two, turn right one',"' says Russell, "and you can also have branching instructions as well. So you can have it like: 'Go forward two steps, and then branch one left and one right.' Then you can say, 'Repeat that'. Or you can do a replacement; you can say, 'Every time I go forward, do the same structure again,' so it branches within itself."'

 

The end result is a complex system that replicates the fractal forms in nature, and in Cinema 4D is built into the MoSpline object's Turtle mode. Having created the spline, it's then a case of using a Sweep object to give it solidity. The system can also insert Clone groups into the structure, which was used to generate the various forms and rotating shapes at the tips of the branches. But while Cinema's L-System was useful, Russell found that he needed more sophistication, which is when he turned to Houdini.

 

Cinema 4D has offered integration with Side Effects Software's app since R16, enabling you to build assets in Houdini Engine and then load them into Cinema. "The setup was straightforward,"' says Russell. "I created a fairly simple L-System node tree in Houdini then wrapped it up as an asset with options to allow me to clone elements onto the system in Cinema."'

 

As the camera moves on we see more geodesic shapes that resemble the growth pattern of sunflower seeds or Romanesco broccoli, both of which follow a spiral based on the Fibonacci series of numbers. To build these, Russell employed the Golden Ratio: "If you put a seed or a point at a certain number of degrees [it's 137.5° – the 'Golden Angle'] and repeat that enough times and keep pushing the pattern out, you start to get these Fibonacci spirals or these really organic structures."' He tried various methods to achieve these shapes, including particles and cloners, but mainly used an XPresso setup to draw a spline and then clone onto it or sweep or extrude it.

 

"MoGraph allowed me to go so far,"' admits Russell, "but when it came to really heavy geometry I built a Houdini asset that allowed me to create even more complex structures. The system took a base piece of geometry – a cube, tetrahedron etc. – and then 'cloned' these shapes onto its own surfaces. Each generation cloned more of the same shapes but smaller in scale. So it created a kind of fractal crystal growing asset. In Cinema 4D you could simply swap in your base piece of geometry and set the number of generations it should grow by. It was interesting to see how the overall shape changed by simply changing the base geometry. Once I was happy with the crystal shape I then baked it out and put it in a Fracture object to animate it."'

 

The overall poly count of the scene (which tops out at around four million) was kept under control by baking each model out to an Alembic file. "With the crystals and the crystal animation, I would have a really, really heavy MoGraph setup, but just in another document,"' explains Russell. "You export that as an Alembic file, render it to disk, and then just pull it back in. You can have a huge amount of geometry. It's the equivalent of doing an edit with loads of really good HD footage, that's all on your drive."'

 

 

 

To read more, click on this link.

 

Steve Jarratt is a long-time CG enthusiast and technology journalist based in the UK.

All images courtesy of Vicarage Studio.

Vicarage Studio Website: http://vicarage.studio

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