Antipodes.
Jim Propp asks whether the two farthest apart points,
as measured by surface distance, on a symmetric convex body
must be opposite each other on the body.
Apparently this is open even for rectangular boxes.
Border
pattern gallery. Oklahoma State U. class project displaying examples
of the seven types of symmetry (frieze groups) possible for
linear patterns in the plane.
Cognitive Engineering
Lab, Java applets for exploring tilings, symmetry, polyhedra, and
four-dimensional polytopes.
Convex
Archimedean polychoremata, 4-dimensional analogues of the
semiregular solids, described by Coxeter-Dynkin diagrams
representing their symmetry groups.
Crystallographic
topology. C. Johnson and M. Burnett of Oak Ridge National Lab use
topological methods to understand and classify the symmetries of the
lattice structures formed by crystals. (Somewhat technical.)
Crystallography
now, tutorial on the seventeen plane symmetry groups by
George Baloglou.
Diamond theory.
Steven Cullinane studies the symmetries of the shapes formed by
splitting each square of a grid into dark and light triangles.
Dreamscope
screen-saver module makes patterns with various Kaleidoscopic symmetries.
Gavrog, a Java package for
visualizing 2d and 3d ornamental patterns with high degrees of symmetry.
Geometry and the Imagination in Minneapolis.
Notes from a workshop led by Conway, Doyle, Gilman, and Thurston.
Includes several sections on polyhedra, knots, and symmetry groups.
Jenn
open-source software for visualizing Cayley graphs of Coxeter groups
as symmetric 4-dimensional polytopes.
Joe's Cafe.
Java applets for creating images of iteration systems
a la Field and Golubitsky's "Symmetry in Chaos".
K12
on G6. Carlo Séquin investigates how to draw a 12-vertex
complete graph as symmetrically as possible on a six-handle surface
(the minimum genus surface on which it can be drawn without crossings).
Kali,
software for making symmetrical drawings based on any of the 17 plane
tiling groups.
Kummer's
surface. Nice ray-traced pictures of a quartic surface with lots of
symmetries.
MagicTile.
Klein's quartic meets the Rubik's cube, by Roice Nelson.
Mirror Curves.
Slavik Jablan investigates patterns formed by crisscrossing a curve around points in a regular grid, and finds examples of these patterns in
art from various cultures.
Moebius
transformations revealed. Video by Douglas N. Arnold and Jonathan
Rogness explaining 2d Moebius transformations in terms of the motions of
a 3d sphere. See also MathTrek.
ProtoZone
interactive shockwave museum exhibits for exploring geometric concepts
such as symmetry, tiling, and wallpaper groups.
Puzzles
with polyhedra and numbers,
J. Rezende.
Some questions about labeling edges of platonic solids with numbers,
and their connections with group theory.
Rational
maps with symmetries.
Buff and Henriksen investigate rational functions invariant under
certain families of Möbius transformations, and use them to
generate symmetric Julia sets.
Rhombic
tilings. Abstract of Serge Elnitsky's thesis, "Rhombic tilings of
polygons and classes of reduced words in Coxeter groups". He also supplied the
picture below of a rhombically tiled 48-gon, available with better color
resolution from his website.
Sighting point.
John McKay asks, given a set of co-planar points, how to find
a point to view them all from in a way that maximizes the
minimum viewing angle between any two points.
Somehow this is related to monodromy groups.
I don't know whether he ever got a useful response.
Soccer
ball pictures,
spherical patterns generated by reflections that form rational angles to each
other.
Spherical
Julia set with dodecahedral symmetry
discovered by McMullen and Doyle in their work on
quintic equations and rendered by
Don Mitchell.
Update 12/14/00: I've lost the big version of this image and can't find
DonM anywhere on the net -- can anyone help?
In the meantime, here's a link to
McMullen's
rendering.
Steve's sprinklers.
An interesting 3d polygon made of copper pipe forms various symmetric 2d shapes
when viewed from different directions.
Wilson
Stothers' Cabri pages.
Geometric animations teaching projective conics,
hyperbolic geometry, and the Klein view of geometry as symmetry.
SymmeToy,
windows shareware for creating paint patterns, symmetry roses,
tessellated art and symmetrically decorated 3D polyhedron models.
Symmetry and Tilings. Charles Radin, Not. AMS, Jan. 1995.
See also his
Symmetry
of Tilings of the Plane, Bull. AMS 29 (1993), which proves that the
pinwheel tiling is ergodic and can be generated by matching rules.
Tilings.
Lecture notes from the Clay Math Institute, by Richard Stanley and
Federico Ardila, discussing polyomino tilings, coloring arguments for
proving the nonexistence of tilings, counting how many tilings a region
has, the arctic circle theorem for domino tilings of diamonds,
tiling the unit square with unit-fraction rectangles, symmetry groups,
penrose tilings, and more. In only 21 pages, including the annotated
bibliography. A nice but necessarily concise introduction to the subject.
(Via Andrei Lopatenko.)
Wallpaper
patterns, R. Morris.
Kaleidoscope-like Java applet for making and transforming symmetric
tilings out of uploaded photos.
A word problem.
Group theoretic mathematics for determining whether a polygon formed out
of hexagons can be dissected into three-hexagon triangles,
or whether a polygon formed out of squares can be dissected
into restricted-orientation triominoes.
