The connection between mathematics and
art goes back thousands of years. Mathematics has been
used in the design of Gothic cathedrals, Rose windows,
oriental rugs, mosaics and tilings. Geometric forms were
fundamental to the cubists and many abstract expressionists,
and award-winning sculptors have used topology as the
basis for their pieces. Dutch artist M.C. Escher represented
infinity, Möbius ands, tessellations, deformations,
reflections, Platonic solids, spirals, symmetry, and
the hyperbolic plane in his works.

Mathematicians and artists continue to
create stunning works in all media and to explore the
visualization of mathematics--origami, computer-generated
landscapes, tesselations, fractals, anamorphic art, and
more.

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"RedCenter," (detail) by Mike Field (University of Houston)"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. --- Mike Field

"InHotPursuit," by Mike Field (University of Houston)"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). The original image was created in 2003. --- Mike Field

"Thorns," by Mike Field (University of Houston)"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. --- Mike Field

"UncertainEnd," by Mike Field (University of Houston)"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. The original image was created in 2001. --- Mike Field

"DNAQuilt," by Mike Field (University of Houston)"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. --- Mike Field

"Saw," by Mike Field (University of Houston)"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).

--- Mike Field

"RedCenter," by Mike Field (University of Houston)"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.

--- Mike Field

"NeuralNet," by Mike Field (University of Houston)"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. --- Mike Field

"Seasonal Chaos," by Mike Field (University of Houston)"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. --- Mike Field