Last additions 
"Death Mask 2," by Nathan SelikoffThis artwork, from the "Faces of Chaos" series, is a twodimensional plot of the Lyapunov exponent of a chaotic dynamical system. The Lyapunov exponent is a measure of how chaotic the system is, and in this case, the system is a strange attractor with a fourdimensional phase space. Two of the dimensions are static, and the other two vary in the x and y directions of the image. A custom program renders four 16bit grayscale images, which represent the different "components" of the spectrum of Lyapunov exponents. These images are combined in Photoshop using a pseudocolor technique to bring out subtle coloration in the final artwork. See more images from this series at www.nathanselikoff.com/facesofchaos/. Feb 26, 2008


"Prime colorings of the sphere, Euclidean plane and the hyperbolic plane," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)Tilings 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 primefold 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 GoodmanStrauss (AK Peters, 2008).
Jan 31, 2008


"Kaleidospheres," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)There 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.Jan 31, 2008


"Inverse Stereo," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)The pattern on this sphere is not a spherical pattern—that is, its symmetry is not a symmetry of the sphere itself. Symmetry is as much as anything a topological property; the pattern on the sphere is in fact a symmetry of the Euclidean plane, as shown by projecting it down to the plane below. Only seventeen types of symmetrical pattern can cover the Euclidean plane; this one has type 4*2. This image is on the cover of "The Symmetries of Things", by John H. Conway, Heidi Burgiel and Chaim GoodmanStrauss (AK Peters, 2008).
Jan 31, 2008


"Morning Glories 4_2 : 2," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)In addition to the thirtyfive "prime", discrete symmetry types of threedimensional 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 GoodmanStrauss (AK Peters, 2008).Jan 31, 2008


"Dodecafoam I," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)Unlike all of the other images in this collection, the symmetry here is not governed by a group action, but rather by a substitution systema set of replacement rules, based on the stellations of the dodecahedron. Several oddly shaped threedimensional 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 nonperiodic structures; though it may look as if such a structure repeats, in fact it cannot repeat periodically.Jan 31, 2008


"Shells 532," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)There are seven infinite families, and seven more individual types of discrete symmetrical patterns on the sphere. A pattern of type 532 is shown; there are three kinds of gyration points in the patterna 5fold, a 3fold, and a 2fold gyration point are marked. This image is from "The Symmetries of Things", by John H. Conway, Heidi Burgiel and Chaim GoodmanStrauss (AK Peters, 2008).Jan 31, 2008


"Calla Lily 32 infinity," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)The group SL_2(Z) acts on the hyperbolic plane discretely, producing patterns of symmetry type 23 infinity, such as the one shown here. Similarly, the 2fold cover GL_2(Z) acts with symmetry type *23 infinity. This image is from "The Symmetries of Things", by John H. Conway, Heidi Burgiel and Chaim GoodmanStrauss (AK Peters, 2008).Jan 31, 2008


"The Hexacosm," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)This spaceship is flying about in the universal cover of the hexacosm, one of the ten, closed, flat threemanifolds. Equivalently, the pattern is one of the ten discrete cocompact symmetry types of Euclidean space that does not have any fixed points. The type here is (6_1 3_1 2_1) in the ThurstonConway fibrefold notation. This image is from "The Symmetries of Things", by John H. Conway, Heidi Burgiel and Chaim GoodmanStrauss (AK Peters, 2008).Jan 31, 2008


"Tube *X," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)This strange image is just of a distorted but perfectly sensible regular pattern in the Euclidean plane, of type *X. A complicated image like this can be built from simple steps, and can be expressed in just a single formula; the colors of the initial pattern are from the values of
f(x,y) = cos(x  cos(y+a))cos(y  cos(x+a)), with
a = pi/5.  Chaim GoodmanStrauss
Jan 31, 2008


"The Regular Hendecachoron," computer model by Carlo Sequin, University of California, Berkeley.This hendecachoron (a literal translation of "11cell" into Greek) is a regular, selfdual, 4dimensional polytope composed from eleven nonorientable, selfintersecting hemiicosahedra. This object also has 11 vertices (shown as spheres), 55 edges (shown as thin cylindrical beams), and 55 triangular faces (shown as cutout frames). Different colors indicate triangles belonging to different cells. This intriguing object of high combinatorial symmetry was discovered in 1976 by Branko Grünbaum 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 10dimensional 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 10dimensional object might serve as a building block for the 10dimensional universe that some stringtheorists have been postulating.
A more detailed description and visualization of the 11Cell, describing its construction in bottomup 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 Sep 04, 2007


"Fiddler Crab, opus 446," by Robert J. Lang. Medium: One uncut square of Origamido paper, composed and folded in 2004, 4". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.
I'm especially pleased with this model, which involves a combination of symmetry with one distinctly nonsymmetric element. The base is quite irregular, but its asymmetry is mostly concealed. The crease pattern is here.
 Robert J. LangSep 04, 2007


"Allosaurus Skeleton, opus 326," by Robert J. Lang. Medium: 16 uncut squares of Wyndstone "Marble" paper, 24". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.This model was inspired by the brilliant Tyrannosaurus Rex of the late Issei Yoshino.
The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.
 Robert J. LangSep 04, 2007


"African Elephant, opus 322," by Robert J. Lang. Medium: One uncut square of watercolor paper, composed and folded in 1996, 8". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.
 Robert J. LangSep 04, 2007


"Tree Frog, opus 280," by Robert J. Lang. Medium: One uncut square of Origamido paper, composed in 1993, folded in 2005, 5". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.
 Robert J. Lang
Sep 04, 2007


471 files on 32 page(s) 




26  



