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Kleinian PearlsPeople have long been fascinated with repeated patterns that display a rich collection of symmetries. The discovery of hyperbolic geometries in the nineteenth century revealed a far greater wealth of patterns, some popularized by Dutch artist M. C. Escher in his Circle Limit series of works.
This cover illustration portrays a pattern which is symmetric under a group generated by two Möbius transformations. These are not distance-preserving, but they do preserve angles between curves and they map circles to circles. The image accompanies "Double Cusp Group," by David J. Wright (Notices of the American Mathematical Society, December 2004, p. 1322).
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Hilbert's Square-Filling Curve"Hilbert's Square-Filling Curve" by The
3DXM Consortium
In 1890 David Hilbert published a construction of a continuous curve whose image completely fills a square, which was a significant contribution to the understanding of continuity. Although it might be considered to be a pathological example, today, Hilbert's curve has become well-known for a very different reason---every computer science student learns about it because the algorithm has proved useful in image compression. See more fractal curves on the 3D-XplorMath Gallery.
--- adapted from "About Hilbert's Square Filling Curve" by Hermann Karcher
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"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 Griffeath
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"Fractal Effervescence," by David AprilThis image comes from the fusion of the three separate image files generated with the software Apophysis, designed by Mark Townsend. Each one of the files contains different types of transformations-linear, polar, and spherical-that produce a curious dialog between the vertical lines, the sinuous curves with the appearance of smoke, and the bubbling circular shapes. In this type of fractal there is only one method of coloring. Sometimes a tiny change to the color gradient can distort the image dramatically. Fractal artists, hunting for treasure, can tease out unexplored forms, but a slight difference in this or that parameter can make them pass by that secret treasure without seeing its hidden beauty. David April lives in Illinois (USA). His interest in fractals is relatively recent, but he compensates for that with an enormous enthusiasm and fascination for finding new forms.
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"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 Griffeath
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"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 Griffeath
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"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 Griffeath
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"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 Griffeath
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"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 Griffeath
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"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 Griffeath
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"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 Griffeath
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"Snowflake Model 9," 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 Griffeath
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"Snowflake Model 10," 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 Griffeath
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"Indra Family," by Jos LeysJos Leys is a Belgian mechanical engineer who has always shown a special interest for mathematics in general and fractal art in particular since he programmed his first fractal image 25 years ago. "Indra Family" is a tribute to the professors David Mumford, Caroline Series and David Wright, the authors of the book "Indra’s Pearls: The Vision of Felix Klein." The iterative calculation techniques of the Kleinian Groups described in this book reveal new fractal images that until then had remained unexplored. The name "Indra's Pearls" is a Hindu and Buddhist concept that represents a network of silk strings that extend to infinity in all directions, and contains at each intersection a very bright and luminous pearl that reflects each of the pearls of the network, that then reflect the others and so on, without end, like mirrors reflecting to infinity.
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"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 Griffeath
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