| Search results - "knot," |
ashley.jpg"Ashley Knot," by by Rob Scharein (Centre for Experimental and Constructive
Mathematics, Simon Fraser University, B.C., Canada)
This example illustrates the SE rendering mode in KnotPlot, which visualizes the symmetric energy distribution. KnotPlot is a program to visualize and manipulate mathematical knots in three and four dimensions, and the website includes a wealth of resources and pictures. This picture is a direct screen capture from KnotPlot, rendered entirely in OpenGL, an environment for portable, interactive graphics applications.
--- Rob Scharein
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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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"Recursive Figure-8 Knot" by Carlo Sequin, University of California, BerkeleyIn some depictions of a mathematical knot, some of the meshes formed between the criss-crossing strands resemble the overall outline shape of the whole knot. It is then possible to fit a reduced copy of the knot into every one of these meshes and reconnect the strands so as to obtain again a mathematical knot consisting of a single closed strand. Then this process can be continued recursively resulting in a self-similar pattern. This general process was applied to the 4-crossing Figure-8 knot. But rather than performing this process in a drawing plane as outlined above, subsequent generations of reduced knot instances were placed in planes that are roughly perpendicular to one another, resulting in a truly 3-dimensional sculpture. --- Carlo Sequin
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"Arabic Icosahedron" by Carlo Sequin, University of California, BerkeleyMoorish patterns found in the Alhambra often depict lattices of interlocking knots. Here such a pattern composed of interlocking trefoil knots has been wrapped around an icosahedron. Each of the 20 faces is replaced with a trefoil knot, which interlocks along the triangle edges with three adjacent trefoils. The exact nature of the linking between adjacent trefoils leaves some freedom to the designer: In the simplest case two adjacent trefoils interlock with just one lobe each. In the "Arabic Icosahedron" they are linked with two lobes each, resulting in a much tighter meshing. --- Carlo Sequin
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"Knot divided" (snow sculpture), by Carlo Sequin (University of California, Bekeley), Stan Wagon (Team Captain), John Sullivan, Dan Schwalbe, and Rich SeeleyCan a DIVIDED KNOT be NOT DIVIDED? When carving this sculpture out of a 10x10x12 foot block of hard compacted snow, we started with the simplest possible knot: the overhand knot, also known as the trefoil knot. We then split lengthwise the whole ribbon forming the three big loops. But there is a twist that may lead to surprises: The original knotted strand was actually a triply twisted Moebius band! Thus the question: Does our cut separate the structure into two pieces, or does it form a single, highly knotted twisted strand? Read more about this snow sculpture. --- Carlo Sequin
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"Twice Iterated Knot No. 1," by Robert Fathauer, Tessellations Company (2008)Third Prize, 2009 Mathematical Art Exhibition. Digital print, 19" x 12". Fathauer makes limited-edition prints inspired by tiling, fractals, and knots. He employs mathematics in his art to express his fascination with certain aspects of our world, such as symmetry, complexity, chaos, and infinity.
"The starting point for this iterated knot is a nine-crossing knot that has been carefully arranged to allow seamless iteration. Four regions of this starting knot are replaced with a scaled-down copy of the full starting knot, incorporated in such a way that the iterated knot is still unicursal. These same four regions are then replaced with a scaled-down copy of the iterated knot, resulting in a complex knot possessing self similarity." --- Robert Fathauer, Small business owner, puzzle designer and artist, Tessellations Company, Phoenix, AZ
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"Figure-8 Knot," by Carlo H. Séquin, University of California, Berkeley (2007)Second Prize, 2009 Mathematical Art Exhibit. Bronze, 9" tall. "The Figure-8 Knot is the second simplest knot, which can be drawn in the plane with as few as four crossings. When embedded in 3D space it makes a nice constructivist sculpture. This particular realization has been modeled as a B-spline along which a crescent-shaped cross section has been swept. The orientation of the cross section has been chosen to form a continuous surface of negative Gaussian curvature." --- Carlo H. Séquin, Professor of Computer Science, EECS Computer Science Division, University of California, Berkeley
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"Chinese Button Knot," by Carlo H. Séquin, University of California, Berkeley (2007)Bronze, 8" tall. "The Chinese Button Knot is a nine-crossing knot, number 9-40 in the knot table. It actually has more symmetries than one would infer from the usual depiction in these tables. This has been brought out in this 3D sculpture, which has one 3-fold and three 2-fold rotational symmetry axes. It has been implemented as an alternating over-under path on the surface of a sphere, realized by a ribbon of continuous negative Gaussian curvature." --- Carlo H. Séquin, Professor of Computer Science, EECS Computer Science Division, University of California, Berkeley
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