# Christopher Walsh

Art and mathematics are intertwined processes. When I make art-- drawings, dance, music-- there is always an underlying mathematical pulse of ideas, which helps organize and also catapult the work into different directions. Further, this pulse leads to precise questions, and can become a journey in itself. So began my project to create a geometry of process: what is the shape of musical tones, color harmonies, or number sequences? This question led to the experimental study of rotational symmetry in circle maps and their extensive gauge-planar walks. All the while, as I learn new mathematical ideas I am inspired in unforeseeable ways.

A .3mm technical pen draws a continuous line for over 6 hours with a measured dilution of india ink, which lets pigment aggregate along the envelopes of curves and line-intercept areas. This ink caustic effect visualizes the essential geometric features of a process, letting the primitive elements of points and lines recede from view. The fine, organic quality of line and pigment give an immanence to the mathematical equation; it becomes a material object drawn out in time. The function is a sum of four circular coordinates, computed as a set of interlinked recurrence relations (+,x), with parameters found in a linear search space.

A gauge-planar walk is the set of partial sums of a sequence of vectors. Each of the steps is a mapping from consecutive differences in a numerical sequence to radial angles of direction in the unit circle. The mapping begins with a circle divided into a number of degrees equal to a modulus, in this case it is 49146 degrees. Each point on the circle is a degree of freedom, and so defines a finite field of motion. The sequence used in this map is the set of polygonal numbers for the 49146-gon. Although the polygonal number sequence is discrete, the circle map f(n)→(nk2π)/mod(p) | k∈R is continuous. Here I modulated (k) to extrude a single stack of loxodromic spirals into the chain of eight vortical funnels which exhibit chaotic stability.