# Mathematical Imagery

Mathematical artists create strong, stunning works in all media and explore the visualization of mathematics

# Realizations :: Mike Field

An aspect of my art work that I particularly enjoy is that I write the software for all the programs I use and build the computers that run the software. In this sense, I like to feel that theory (mathematics), art (outcome), software (algorithms) and engineering (hardware) are integrated and interdependent and that no part survives without the others.

— *Mike Field*

"InHotPursuit" is a section of a planar repeating pattern of type cm and the pattern was generated using an iterated function system defined on the two-dimensional torus. The resulting pattern on the torus was lifted to the plane to obtain a repeating pattern. The coloring reflects an invariant measure on the attractor of the iterated function system. This image is a bit surprising for an iterated function system as the textures and detail are more suggestive of a deterministic system (the torus maps used to generate the iterated function system are quite discontinuous).

"Thorns" is a bounded symmetric pattern in the plane with symmetry $Z_5$. It is a visual representation of the invariant measure on the attractor of a rational $Z_5$-equivariant planar map. The original image was created in 1996 and was perhaps my first serious attempt to investigate ways one could use methods based on symmetry, dynamics and chaos to achieve artistic ends.

"RedCenter" is a section of a planar repeating "two-color" pattern of type pmm' (or pmm/pm in Coxeter notation). The underlying repeating pattern has reflection symmetries and two-fold rotational symmetries as well as translation symmetries and, less obviously, glide reflection symmetries. Roughly speaking, half the symmetries preserve colors and half interchange colors. (The 46 two-color repeating patterns of the plane were originally classified by H. J. Woods of the Textile Physics Laboratory, University of Leeds, in 1935-36.) The pattern was generated using a determinsitic torus map and the coloring reflects the density of two invariant measures on the torus. The name "RedCenter" is suggested by Uluru (Ayers Rock) in Central Australia.

"RedCenter" is a section of a planar repeating "two-color" pattern of type pmm' (or pmm/pm in Coxeter notation). The underlying repeating pattern has reflection symmetries and two-fold rotational symmetries as well as translation symmetries and, less obviously, glide reflection symmetries. Roughly speaking, half the symmetries preserve colors and half interchange colors. (The 46 two-color repeating patterns of the plane were originally classified by H. J. Woods of the Textile Physics Laboratory, University of Leeds, in 1935-36.) The pattern was generated using a determinsitic torus map and the coloring reflects the density of two invariant measures on the torus. The name "RedCenter" is suggested by Uluru (Ayers Rock) in Central Australia.

"UncertainEnd" is a section of a planar repeating pattern of type $p'_{c}gg$ (or, in Coxeter notation, $cmm/pgg$). Ignoring the colors, the underlying pattern is of type $cmm$ and is the superposition of two colored patterns, each of type $pgg$. The pattern was generated using an iterated function system defined on the two-dimensional torus. The resulting pattern on the torus was lifted to the plane to obtain a repeating pattern. The coloring reflects invariant measures on each of the underlying patterns of type $pgg$ and takes account of overlap, as well as symmetry, using algorithms designed for revealing detail hidden in the dynamics.

"DNAQuilt" is a repeating pattern of type $pgg$. As is the case of the other repeating patterns that have a $pgg$ component, this type of symmetry is particularly dynamic as there are no lines of symmetry in the pattern--only glide-reflection symmetries. Although lines of reflection can be artistically interesting in two-color repeating patterns (for example, in "RedCenter" and "UncertainEnd"), too many lines of symmetry--as in patterns with $p4m$ (square) symmetry--can tend to lead to 'pretty' but ultimately rather dull and static results (at least in patterns without two-color symmetry). Mathematically speaking. the pattern is a visual representation of the invariant measure of a deterministic dynamical system defined on the two-dimensional torus. The pattern is lifted to the plane to obtain a repeating pattern.

"Saw" is a Symmetric Fractal with 11-fold rotational symmetry constructed using methods based on iterated function systems. The image was created many years ago when I was at the University of Sydney, Australia, and appears in Symmetry in Chaos (Mike Field and Marty Golubitsky, OUP, 1992).

"NeuralNet" is is part of the generating tile of a planar repeating pattern of type $pgg$. Repeating patterns of this type have no reflection symmetries but do have many glide reflection symmetries as well as translational symmetries and two-fold centers of rotation. The absence of reflectional symmetries often leads to very fluid and dynamic patterns. The coloring reflects the density of the invariant measure.

"Seasonal Chaos," a Hexagonal Quilt, is a repeating pattern of type $p3m1$ created using methods based on deterministic dynamical systems. The image was created in 1999 and used as the motive for a 'seasonal card.' Hexagonal Quilts are constructed using either a deterministic symmetric dynamical system or a random symmetric dynamical system. In either case, the images shown can be thought of as colored realizations of chaotic symmetric attractors.