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Spiked Rhombic EnneacontahedronThis structure was conceived by taking a 90-sided polyhedron, whose faces are made from two types of rhombi, and placing a pyramid on each face. The construction uses 180 small squares of paper, all folded and interlocked together without glue. See more models on the Origami Gallery.
--- Thomas Hull. Photograph by Nancy Rose Marshall.
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Professor Tom HullTom Hull took his Ph.D. in mathematics at the University of Rhode Island in 1997. His dissertation was on list coloring bipartite graphs, now he mostly studies the mathematics of origami (paper folding).
Tom Hull is an associate professor in the Department of Mathematics at Merrimack College in North Andover, MA.
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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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Nested Hexogonal CollapseThis model is a series of concentric hexagons with "zig-zag" creases coming from the center-most hexagon out to the midpoints of the paper's sides. It can be collapsed in many different ways and twisted into interesting shapes, as done here. See more geometrics and tesselations on the Origami Gallery.
--- Thomas Hull. Photograph by Nancy Rose Marshall.
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Coxeter groupsThis image illustrates two types of infinite Coxeter groups and algorithms involved in computation within those groups: one which generates elements of the group one by one, the "Shortlex automaton," and others, more conjectural, which seem to describe the Kazhdan-Lusztig cells of an arbitrary Coxeter group.
The illustration is described in detail and was created to accompany the article "Cells in Coxeter Groups," by Paul E. Gunnells (Notices of the American Mathematical Society, May 2006, p. 528). The explicit finite state machines required to draw the Kazhdan-Lusztig cells were supplied by Gunnells.
--- Bill Casselman
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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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Hilbert's Square-Filling Curve"Hilbert's Square-Filling Curve" by The
3DXM Consortium
In 1890 David Hilbert published a construction of a continuous curve whose image completely fills a square, which was a significant contribution to the understanding of continuity. Although it might be considered to be a pathological example, today, Hilbert's curve has become well-known for a very different reason---every computer science student learns about it because the algorithm has proved useful in image compression. See more fractal curves on the 3D-XplorMath Gallery.
--- adapted from "About Hilbert's Square Filling Curve" by Hermann Karcher
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Mandelbrot SetA striking aspect of this image is its self-similarity: Parts of the set look very similar to larger parts of the set, or to the entire set itself. The boundary of the Mandelbrot Set is an example of a fractal, a name derived from the fact that the dimensions of such sets need not be integers like two or three, but can be fractions like 4/3. See more at the 3D-XplorMath Fractal Gallery.
--- Richard Palais (Univ. of California at Irvine, Irvine, CA)
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Hopf Fibered Linked Tori"Hopf Fibered Linked Tori," by The
3DXM Consortium
The Hopf map maps the unit sphere in four-dimensional space to the unit sphere in three-dimensional space. The four tori linked in this image are made up of fibers, or pre-images, of the Hopf map. In this visualization, each fiber has a constant color and the color varies with the distance of the fibers. Any two of the four tori are linked, as are any pair of fibers on a given torus. See more surface images on the 3D-XplorMath Gallery.
--- adapted from "Hopf Fibration and Clifford Translation of the 3-Sphere," by Hermann Karcher
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Parametric Breather"Parametric Breather," by The 3DXM Consortium.
This striking object is an example of a surface in 3-space whose intrinsic geometry is the hyperbolic geometry of Bolyai and Lobachevsky. Such surfaces are in one-to-one correspondence with the solutions of a certain non-linear wave-equation (the so-called Sine-Gordon Equation, or SGE) that also arises in high-energy physics. SGE is an equation of soliton type and the Breather surface corresponds to a time-periodic 2-soliton solution. See more pseudospherical surfaces on the 3D-XplorMath Gallery.
--- Richard Palais (Univ. of California at Irvine, Irvine, CA)
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"2 Circles in a bi-pentagon pattern," by Bradford Hansen-SmithThe 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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"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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"36 circles in a dual tetrahedron pattern" by Bradford Hansen-SmithThe tetrahedron has four faces. 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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