| Search results - "intersecting" |
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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"72 Pencils," by George W. Hart (www.georgehart.com)"72 Pencils" is a geometric construction of 72 pencils, assembled into a work of art. The form is an arrangement of four intersecting hexagonal tubes that penetrate each other in a fascinating three-dimensional lattice. For some viewers, part of the interest lies in the form of the interior. The four hexagonal tubes are hollow, so the sculpture as a whole is hollow. But, what shape is its cavity? What would someone on the inside see? To the mathematician, the answer is "the rhombic dodecahedron," a geometric solid bounded by twelve rhombuses. See two other views, showing how it looks along various axes of symmetry, at www.georgehart.comwww.georgehart.com. --- George W. Hart (www.georgehart.com)
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"Recursive Construction for Sliding Disks," Adrian Dumitrescu, University of Wisconsin, Milwaukee (2008)Digital print, 11” x 5”. "Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in the plane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk slides in the plane without intersecting any other disk, so that its center moves along an arbitrary (open) continuous curve. One can easily show that 2n moves always suffice, while the above construction shows pairs of configurations that require 2n-o(n) moves for this task, for every sufficiently large n. Disks in the start configuration are white, and disks in the target configuration are shaded. " --- Adrian Dumitrescu, University of Wisconsin, Milwaukee
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"The Regular Hendecachoron," computer model by Carlo Sequin, University of California, Berkeley.This hendecachoron (a literal translation of "11-cell" into Greek) is a regular, self-dual, 4-dimensional polytope composed from eleven non-orientable, self-intersecting hemi-icosahedra. This object also has 11 vertices (shown as spheres), 55 edges (shown as thin cylindrical beams), and 55 triangular faces (shown as cut-out frames). Different colors indicate triangles belonging to different cells. This intriguing object of high combinatorial symmetry was discovered in 1976 by Branko GrĂĽnbaum and later rediscovered and analyzed from a group theoretic point of view by geometer H.S.M. Coxeter. Freeman Dyson, the renowned physicist, was also much intrigued by this shape and remarked in an essay: "Plato would have been delighted to know about it." The hendecachoron has 660 combinatorial automorphisms, but these can only show themselves as observable geometric symmetries in 10-dimensional space or higher. In this image, the model of the hendecachoron is shown with a background of a deep space photo of our universe, to raise the capricious question, whether this 10-dimensional object might serve as a building block for the 10-dimensional universe that some string-theorists have been postulating.
A more detailed description and visualization of the 11-Cell, describing its construction in bottom-up as well as in top down ways, can be found in a paper by Sequin and Lanier: “Hyperseeing the Regular Hendecachoron”. There are additional images and VRML models for interactive inspection here. --- Carlo Sequin
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"Flow 1," by Elizabeth Whiteley (2008)Laminated canvas and acrylic paint, 7.5" x 14" x 7.5". "'Flow 1' is created by intersecting two Golden Triangles (base angles of 72° and vertex angle 36°). The plane of each triangle is partially bisected and then curved to create an aesthetically pleasing form. One triangle is smooth; the other has a textured surface. The sculpture changes our perception of a static and planar geometric shape and makes for a dynamic visual experience. The curves move the eye around the form and suggest multiple points of view." --- Elizabeth Whiteley, Studio artist, Washington, DC
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