| Search results - "five" |
Five Intersecting TetrahedraThis is a version of the Ow-Hull "Five Intersecting Tetrahedra." The visually stunning object should be a familiar sight to those who frequent the landscapes of M.C. Escher or like to thumb through geometry textbooks. Read about the object and how it is constructed on the Origami Gallery.
--- Thomas Hull. Photograph by Nancy Rose Marshall.
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"The Susurrus of the Sea," by George W. Hart (www.georgehart.com)Soft waves, suggestive of both sky and water, travel around the globe along six different criss-crossing equators. The susurrus (murmur) of the sea is suggested as a sense of harmony in this sphere. Technically difficult, the 60 transparent blue acrylic plastic components had to be made very precisely to fit together. Heat-formed, the components were formed and assembled on special jigs which imparted the proper dimensions and angles. Mathematically, the blue spirals are helixes that follow the edges of an icosidodecahedron. This is a polyhedron that was known to the ancient Greeks, but the oldest known drawing of it is by Leonardo da Vinci. Formally constructed of triangles and pentagons (which show up here as the openings) it can also be seen as an arrangement of six equatorial regular decagons. Each equator makes ten twists in a complete path, crossing the other five equators at two opposite points. If one "walks along" a dark blue edge, making right-angle turns where edges meet, one traces a large five-pointed star before returning to oneā€™s starting point.
--- George W. Hart (www.georgehart.com)
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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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"Eights," by George W. Hart (www.georgehart.com)This six-inch diameter paper sculpture is made of sixty identically shaped parts. Parts of any one color form a type of tetrahedron, and there are five such, deeply interlocked. No glue is used; they parts just hook into each other. I call this type of design "modular kirigami". It took me about four hours to assemble after several hours of false starts and figuring out how to do it. I generated a computer-rendered view down a five-fold axis. The "8"-shaped parts each link with many others. So they could not be made as single pieces of paper unless they were glued or taped together after being linked. But I wanted to be a purist and use no glue or tape, so I designed the parts as two overlapping "3"-shaped pieces.
--- George W. Hart (www.georgehart.com)
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"Morning Glories 4_2 : 2," by Chaim Goodman-Strauss, University of ArkansasIn addition to the thirty-five "prime"¯ discrete symmetry types of three-dimensional Euclidean space, there are 184 "composite"¯ types; these each can be projected down an axis to produce one of the 17 discrete symmetry types of the plane. This pattern in space, for example, with type 4_2 : 2, is a kind of attenuated planar pattern with type 4 * 2. This image is from "The Symmetries of Things"¯ by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss (AK Peters, 2008).
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"Kaleidospheres," by Chaim Goodman-Strauss, University of ArkansasThere are five types of kaleidoscopic symmetry on the sphere (two of which are infinite families). Four are shown here: *532, *432, *332, and *22N. It is quite amusing to make real, physical kaleidoscopes that produce images like these.
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"Prime colorings of the sphere, Euclidean plane and the hyperbolic plane," by Chaim Goodman-Strauss, University of ArkansasTilings of the sphere, the Euclidean plane and the hyperbolic plane are shown. In each case, we have triangular faces, but on the sphere, the triangles meet in fives; in the Euclidean plane, the triangles meet in sixes, and in the hyperbolic plane, they meet in sevens. To a great degree, this is forced. It is impossible, for example, to have a tiling of the sphere with triangles meeting only in sevens (try it!).
In each case, a prime-fold coloring of the pattern is shown. It is helpful to realize that there are more similarities than differences among the three geometries.
The symmetry of the hyperbolic plane pictured above was known to Felix Klein by 1878, and has a tremendous number of interesting topological, geometric and algebraic properties.
This image is from "The Symmetries of Things"¯ by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss (AK Peters, 2008).
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"Starfruit," by David MakinDavid Makin is a British computer programmer born in North Wales, who loves fractal geometry and science fiction. The majority of his work comes from his investigations into the use of coloring algorithms. In this case he employed three algorithms applied to a Julia set. The first of his algorithms, named "MMF3-Turning Points," generated the starred forms that characterize the image and suggested the title of the shape immediately to him (the starfruit is a tropical fruit whose cross section produces a five-pointed star). With the second algorithm, "MMF3-Orbital Waves," he used the idea of complementing the first layer with the handsome curved lines that accentuate the set. At this point he proceeded to include the third algorithm, "MMF3-Alternative fBm II," which provides a more organic texture. Finally, David Makin took considerable time in combining the three layers with color palettes and the algorithms described that produced the final result.
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"Infinite Curl 7," by Matjuska Teja Krasek in collaboration with Dr. Clifford Pickover (2006)Digital print, 9.9" x 10.1". Kraskek's interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon. Her artworks also illustrate properties such as golden mean relations, self similarity, ten- and fivefold symmetry, Fibonacci sequence, inward infinity and perceptual ambiguity. She employs contemporary computer technology as well as classical painting techniques.
"The image represents the behavior of mathematical feedback loops, and more particularly the iteration of a complex function. The figure is our rendition of a visually interesting quartic variant of a Ushiki Phoenix Julia set. As with other fractals, the image exhibits a wealth of detail upon successive magnifications. The image ‘Infinite Curl 7’ has been made in collaboration with Dr. Clifford Pickover, the author of more than thirty books about mathematics, art, and science." --- Matjuska Teja Krasek, Freelance artist, Ljubljana, Slovenia, EU
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"Interlaced Hearts," by Matjuska Teja Krasek (2001)Digital print, 12.6" x 11.9". Kraskek's interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon. Her artworks also illustrate properties such as golden mean relations, self similarity, ten- and fivefold symmetry, Fibonacci sequence, inward infinity and perceptual ambiguity. She employs contemporary computer technology as well as classical painting techniques.
"The implicit decagon constituted of five smaller decagons expresses tenfold and fivefold rotational symmetry. The image where golden heart-like shapes are exposed shows self-similarity, the richness of relations between the decagons, pentagonal stars, Penrose rhombs, kites and darts with the golden ratio used several times as a scale factor." --- Matjuska Teja Krasek, Freelance artist, Ljubljana, Slovenia, EU
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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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