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"Kissing in Motion""Kissing in Motion" shows the motion of the "shadows" of kissing spheres in a deformation pointed out by J.H. Conway and N.J.A. Sloane, following an observation of H.S.M. Coxeter. The sequence is left-right, right-left, left-right (sometimes called boustrophedon). The image accompanies "Kissing Numbers, Sphere Packings, and Some Unexpected Proofs," by Florian Pfender and Günter M. Ziegler (Notices of the American Mathematical Society, September 2004, p. 873).
--- Bill Casselman
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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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"(1,3,5) Pretzel Knot," by Jarke J. van Wijk (Technische Universiteit Eindhoven). Image courtesy of Jarke J. van Wijk.The yellow tube is a (1, 3, 5) pretzel knot. Such a pretzel knot or link consists of a sequence of angles, where each tangle has a number of twists. The brown surface is a Seifert surface: an orientable surface bounded by the knot. Here the surface is easy to understand; for arbitrary knots such surfaces often have strange and difficult shapes. However, for any knot or link such surfaces can be found, as shown by Herbert Seifert in the 1930's. This image was made with a tool called SeifertView.
--- Jarke J. van Wijk
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"Borromean Rings," by Jarke J. van Wijk (Technische Universiteit Eindhoven). Image courtesy of Jarke J. van Wijk.The Borromean Rings consist of three links. Take one link away and the other links fall apart, but together they are inseparable. Because of this, they are popular as a symbol for strength in unity. Here they are shown from an unusual point of view, and also a Seifert surface is shown. This is an orientable surface, bounded by the links. This image was made with a tool called SeifertView.
--- Jarke J. van Wijk
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"Three Link Chain," by Jarke J. van Wijk (Technische Universiteit Eindhoven). Image courtesy of Jarke J. van Wijk.This knot consists of three similar links, and is threefold-symmetric. The surface shown is a Seifert surface, an orientable surface bounded by the links. Considering only the links, it is hard to imagine that such a surface does exist. However, in the 1930's, the German mathematician Herbert Seifert presented an algorithm to find such surfaces for any knot or link. This image was made with a tool called SeifertView.
--- Jarke J. van Wijk
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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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"Fiddler Crab, opus 446" crease pattern, by Robert J. Lang. Copyright Robert J. Lang (www.langorigami.com).Crease patterns (CPs) provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding, into a single diagram! A CP can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. Mathematical and geometric CPs usually show all the creases, but representational origami rarely shows every crease in the finished form, as it would make the crease pattern impossibly busy. Instead, the crease pattern gives the creases needed to fold the "base," that is, a geometric shape that has the right number and arrangements of flaps. It is still left up to the folder to add thinning and shaping folds. See the final "Fiddler Crab" origami work in this album.
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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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"Night Hunter, opus 469" crease pattern, by Robert J. Lang. Copyright Robert J. Lang (www.langorigami.com).Crease patterns (CPs) provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding, into a single diagram! A CP can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. Mathematical and geometric CPs usually show all the creases, but representational origami rarely shows every crease in the finished form, as it would make the crease pattern impossibly busy. Instead, the crease pattern gives the creases needed to fold the "base," that is, a geometric shape that has the right number and arrangements of flaps. It is still left up to the folder to add thinning and shaping folds. See the final "Night Hunter" origami work in this album.
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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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"African Elephant, opus 322," by Robert J. Lang. Medium: One uncut square of watercolor paper, composed and folded in 1996, 8". 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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"Allosaurus Skeleton, opus 326," by Robert J. Lang. Medium: 16 uncut squares of Wyndstone "Marble" paper, 24". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.This model was inspired by the brilliant Tyrannosaurus Rex of the late Issei Yoshino.
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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