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Home > Chaim GoodmanStrauss :: Symmetries

Chaim GoodmanStrauss :: Symmetries
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"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


"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).


"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).


"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).


"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.


"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).


"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).


"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.


"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).



