Francesco De Comité
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).
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.
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.
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.