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 bands, 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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"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 FieldMay 21, 2009

"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 FieldMay 21, 2009

"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 FieldMay 21, 2009

"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 FieldMay 21, 2009

"Snowflake Model 11," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 12," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 13," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 1," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 2," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 3," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 4," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 5," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 6," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 7," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009

"Snowflake Model 8," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David GriffeathMay 11, 2009