| Search results - "icosahedron" |
Dyadic icosahedraThis image exhibits fanciful renderings of the dyadic icosahedra discussed in the article "The p-adic Icosahedron," by Gunther Cornelissen and Fumiharu Kato (Notices of the American Mathematical Society, August 2005, p. 720).
--- Bill Casselman
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"80 circles in an icosahedron pattern," by Bradford Hansen-SmithYou can also see many hexagonal and pentagonal shapes in this pattern. The symbol of the circle is used as metaphor for nothing and for everything, and endless parts in-between. Folding circles appears to have little history: Somewhere in the history of origami lies the circle, unrecognized and discarded in favor of the square; Buckminster Fuller also folded the circle, with informational intent. Fuller is the inspiration for my own exploration into geometry and provided the seed for folding and joining circles-9" paper plates.
-- Bradford Hansen-Smith, Wholemovement
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"20 circles in an icosahedron pattern" by Bradford Hansen-SmithAn icosahedron is a solid with 20 faces. This solid has hexagons on its surface with pentagonal indentations. The symbol of the circle is used as metaphor for nothing and for everything, and endless parts in-between. Folding circles appears to have little history: Somewhere in the history of origami lies the circle, unrecognized and discarded in favor of the square; Buckminster Fuller also folded the circle, with informational intent. Fuller is the inspiration for my own exploration into geometry and provided the seed for folding and joining circles-9" paper plates.
--Bradford Hansen-Smith, Wholemovement
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"Star Corona," by George W. Hart (www.georgehart.com)This 8-inch, diameter, one-of-a-kind, acrylic sculpture consists of an inner red star surrounded by a yellow corona. It is designed to hang and the two components do not touch each other. The star has twelve large 5-sided spikes and twenty smaller 3-sided spikes, all assembled from sixty identical angular components. The corona is assembled from twenty identical curved components, which give the effect of swirling motion. If you look straight down on a spike, you see that arms from five of the yellow parts combine to make a circle around the spike. Both components are based on stellations of the icosahedron. The outer corona is based on the first stellation and the inner star shape is based on number 53 in the list by Coxeter et al. To understand it well, make a paper model from the instructions on my website.
--- George W. Hart (www.georgehart.com)
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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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"Extrapolated Icosahedron," by Bradford Hansen-Smith (2008)52 folded 9” paper plate circles, 13”x13”x13”. "Forty circles have been folded, reformed to an in/out variation of a truncated tetrahedron, then octahedronally joined in pairs, and arranged in an icosahedron pattern. This revealed an interesting form of the icosadodecahedron with open pentagon stars. In this case twelve circles were reformed and added to suggest mouth-like openings found in sea anemones or in opening flower buds. This gives function to the open pentagons. Much of what I explore with folding circles are the structural functions of geometry found in life forms that correlate to the movement forms of the folded circle." --- Bradford Hansen-Smith, Independent consultant, geometer, author, sculptor, Chicago, IL
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"Five Left Tetrahedral Cosets," by Francisco Lara-Dammer, Indiana University, Bloomington (2008)Digital print, 20" x 20". "This is a Klein diagram (named after the nineteenth-century German mathematician Felix Klein) that represents A5, the group of symmetries of the icosahedron. Another way of describing A5 is as the alternating group on five elements, namely, the group of all even permutations of five entities. This diagram emphasizes A5's tetrahedral subgroup A4 (the group of symmetries of the tetrahedron, also the group of even permutations of four entities), which has twelve elements, plus the four left cosets of A4. The general diagram is obtained by centrally projecting an icosahedron onto a sphere (with the center of one face projected onto the north pole) and then making a stereographic projection of the sphere down onto a horizontal plane. Each coset has been identified with one color. The circle contains a hundred and twenty regions from which sixty correspond to the dark blue background, and the other sixty are split with the five left cosets. The reason I have realized Klein diagrams is to understand more clearly the beauty of Group Theory." --- Francisco Lara-Dammer, Research assistant. Center for Research on Concepts and Cognition, Indiana University, Bloomington, IN
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