Francesco De Comité

Associate Professor in Computer Science
Computer Science department University of Sciences of Lille (France)
Lille (France)

Manipulation of digital images, and use of ray-tracing software can help you to concretize mathematical concepts. Either for giving you an idea of how a real object will look like (as with my works on playing cards slide-together presented at Bridges 2010), or to represents imaginary landscapes only computers can handle.

The Apollonian gasket algorithm recursively fills the gaps between tangent circles with new tangent circles. Applying it just once leaves you with a lot of empty circles. Why not applying the algorithm once again to those new circles ?

What happens when you change the number of original tangent circles ? (the basic Apollonian gasket definition uses a set of five tangent circles : one external containing three similar circles, tangent to the first circle, plus one in the middle tangent to the three internal ones; it can be generalized to one external circle, n similar plus on central).

Big Bang
Digital print

There are several ways for drawing tangent circles : Apollonian Gaskets, Steiner's Chains...

Some geometric transformations preserve tangency of circles.

Mixing different ways for generating circles packing and transformations, becomes then a game with unpredictable results.

This image was composed by first using the Apollonian Gasket recursive algorithm, then applying a Mobius transform to distort the original, while maintaining the tangency property.

Empty circles are originally very small, before applying the transformation, hence not filled by the algorithm.

I didn't program the Sierpinski triangle, it is just an unexpected emergent pattern.

Serendipist Alien Angel
Digital Print

Recursively applying the Apollonian gasket algorithm, varying the number of inner circles, creates unexpected shapes...

This image uses only the Apollonian algorithm : adding new circles to fill the gaps between tangent circles, then fill those new circles with an other Apollonian gasket. The number of initial circles in each gasket is determined by the angle of the vector relying the center of the circle to the center of the image, with the horizontal line.

A catalog of Snowflakes
Digital print

Recursivity can help you displaying an infinite number of shape variations in a limited space.

In this image, circles are filled with Steiner chains, the gaps between tangent circles are filled with the Apollonian gasket algorithm, and the whole circle pack is slightly distorted with a Mobius transform, in order to gently break the symmetries.