I have always been fascinated by the connections between mathematics, art and nature. I began my studies as an art major, but later switched to mathematics. After I bought my first computer in the late eighties I spent all my spare time programming, trying to use mathematical ideas to imitate art and nature.
An Appolonian Gasket is constructed as follows: starting with any three mutually tangent circles the two circles that are tangent to the original three are added.. At the next stage 6 new circles are added, each one tangent to three of the circles from the previous stage.. Continuing in this pattern; at each stage, for every triple of circles, new circles tangent to each of the three are constructed.
In my 2011 Bridges paper I described a method of creating interesting designs by attaching vectors to curves in the plane. In this picture a vector f(z), for a function f of a complex number is attached to the points on the graph of an eight-leaved rose. The vectors are semi-transparent and rendered over a background of computer generated vines. The color and value of a vector is a function of its slope.
The clouds are the result of a midpoint algorithm. The flowers and leaves are generated using string rewriting sysems. The trees are fractals with controlled random lengths and angles designed to avoid regularity. A blade of grass is a sequence of very short lines whose slopes and lengths vary randomly.