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"Crocheted Lorenz manifold, detail," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)Dr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics.
The photograph shows a particularly nice detail of the intriguing geometry of the Lorenz manifold. The wire running through the crocheted work illustrates one of the paths on the surface that end at the origin.
For more information, the crochet pattern and mounting instructions, see: http://www.enm.bris.ac.uk/staff/hinke/crochet/.
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"Crocheted Lorenz manifold, white background," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)Dr. Hinke Osinga and Professor Bernd Krauskopf (Engineering
Mathematics, University of Bristol) have turned the famous Lorenz
equations into a beautiful real-life object, by crocheting
computer-generated instructions of the Lorenz manifold: all crochet
stitches together define the surface of initial conditions that under
influence of the vector field generated by the Lorenz equations end up
at the origin; all other initial conditions go to the butterfly
attractor that has chaotic dynamics.
The white background in the photograph brings out the rotational
symmetry of the Lorenz manifold and gives an idea of the structure of
the mesh.
For more information, the crochet pattern and mounting instructions,
see: http://www.enm.bris.ac.uk/staff/hinke/crochet/.
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"Crocheted Lorenz manifold, white background," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)Dr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics.
The white background in the photograph brings out the rotational symmetry of the Lorenz manifold and gives an idea of the structure of the mesh.
For more information, the crochet pattern and mounting instructions, see: http://www.enm.bris.ac.uk/staff/hinke/crochet/.
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"Crocheted Lorenz manifold, black background," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)Dr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics.
The black background in the photograph brings out the separating properties of the Lorenz manifold: points on one side of the surface can never cross to the other side, even though they will visit both left and right wings of the butterfly attractor in a seemingly unpredictable manner.
For more information, the crochet pattern and mounting instructions, see: http://www.enm.bris.ac.uk/staff/hinke/crochet/.
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"Fiddler Crab, opus 446," by Robert J. Lang. Medium: One uncut square of Origamido paper, composed and folded in 2004, 4". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 35 years, I have developed over 480 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.
I'm especially pleased with this model, which involves a combination of symmetry with one distinctly non-symmetric element. The base is quite irregular, but its asymmetry is mostly concealed. The crease pattern is here.
--- Robert J. Lang
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"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
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"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
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"Night Hunter, opus 469," by Robert J. Lang. Medium: One uncut square of Korean hanji, composed and folded in 2003, 18". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 35 years, I have developed over 480 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.
--- Robert J. Lang
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"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
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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 Field
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"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
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"Bull Moose, opus 413," by Robert J. Lang. Medium: One uncut square of Nepalese lokta, composed and folded in 2002, 6". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 35 years, I have developed over 480 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.
--- Robert J. Lang
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"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
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"Tree Frog, opus 280," by Robert J. Lang. Medium: One uncut square of Origamido paper, composed in 1993, folded in 2005, 5". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 35 years, I have developed over 480 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.
--- Robert J. Lang
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"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
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