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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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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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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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"A Piece of Hyperspace," by Sarah Mylchreest and Mark NewboldThe quilt depicts a polyhedron known as the Great Triambic Icosidodecahedron. It was paper-pieced by Sarah Mylchreest from a design generated by Mark Newbold using his "Hyperspace Star Polytope Slicer" Java applet. It won a ribbon in the 2002 Vermont Quilt Festival. The Dogfeathers.com site has a description of the quilt pattern.
--- Photograph and image copyright 2005 by Mark Newbold, dogfeathers.com.
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"The Hexacosm," by Chaim Goodman-Strauss, University of ArkansasThis spaceship is flying about in the universal cover of the hexacosm, one of the ten, closed, flat three-manifolds. Equivalently, the pattern is one of the ten discrete co-compact symmetry types of Euclidean space that does not have any fixed points. The type here is (6_1 3_1 2_1) in the Thurston-Conway fibrefold notation. 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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"Dodecafoam I," by Chaim Goodman-Strauss, University of ArkansasUnlike all of the other images in this collection, the symmetry here is not governed by a group action, but rather by a substitution system--a set of replacement rules, based on the stellations of the dodecahedron. Several oddly shaped three-dimensional cells based on the stellations of the dodecahedron are used; a rule then gives a method for dividing each cell into small copies of the others. Such techniques are commonly used to produce highly ordered non-periodic structures; though it may look as if such a structure repeats, in fact it cannot repeat periodically.
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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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"Sierpinski Tetrahedron (View I)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadinfinitum.com)
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"Sierpinski Tetrahedron (View II)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadinfinitum.com)
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"Octahedral Cluster" in white opalite glass, seed beads, Nymo nylon thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo, and beAd Infinitum. Copyright 2005 by Gwen L. Fisher.The regular octahedron has 8 triangular faces, 6 vertices each of valence 4, and 12 edges. These 12 edges correspond to the 12 largest (white) beads in the Octahedral Cluster beaded bead. The 6 vertices of the octahedron appear as 6 stars with 4 points each. The 8 triangular faces correspond to where the points of the stars meet. This beaded bead is hollow, yet structurally stable. The stability comes from the way the small beads fit snugly into the spaces between the larger beads. The beaded bead shows virtually no thread; only beads are visible. It is springy between the fingertips and reforms its shape remarkably well. When free from compression, it is round from every angle.
-- Gwen L. Fisher (www.beadinfinitum.com)
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"Mobius Frame with 2 Holes (View II)" in seed beads, Nymo nylon thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo, and beAd InfinitumThis Mobius Frame is woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3-space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. This Mobius Frame represents two distinct mathematical objects. First, one can view this object as assembled from cube (or cube-like) shapes. We might be tempted to try to build such an object from wood using three long beams and two short beams. However, like the Impossible Triangle, this Mobius Frame cannot be built in 3D using all straight lines and right angles. In connecting the beams at their ends, the sides of the beams need to twist. The flexibility of the thread connecting the beads allows the beaded frame to twist to accommodate the necessary turns to build this object in 3D. The second way to view this object is to see it as a patch of an infinite surface with no thickness and two holes. Since the surface has no thickness, ignore the layer of purple beads in the middle. The blue and green coloring of the largest faces shows that this surface has two distinct faces. --- Gwen L. Fisher (www.beadinfinitum.com)
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