Symmetry and Group Theory
 The
Aesthetics of Symmetry, essay and design tips by Jeff Chapman.
 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.
 Associating the
symmetry of the Platonic solids with polymorf manipulatives.
 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
fourdimensional polytopes.
 Conceptual proof that
inversion sends circles to circles, G. Kuperberg.
 Convex
Archimedean polychoremata, 4dimensional analogues of the
semiregular solids, described by CoxeterDynkin 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
screensaver module makes patterns with various Kaleidoscopic symmetries.
 Escher's
combinatorial patterns, D. Schattschneider, Elect. J. Combinatorics.
 Michael Field's gallery
of symmetric chaos images.
See his home page
for more links to pages on dynamics, symmetry, and chaos.
 Figure eight knot / horoball diagram.
Research of A. Edmonds into the symmetries of knots,
relating them to something that looks
like a packing of spheres.
The MSRI Computing Group uses
another horoball
diagram as their logo.
 Fractal patterns formed by repeated inversion of circles:
Indra's Pearls
Inversion graphics gallery, Xah Lee.
Inversive circles, W. Gilbert, Waterloo.
 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.
 The Graph of the Truncated Icosahedron and the Last Letter of Galois,
B. Kostant, Not. AMS, Sep. 1995.
Group theoretic mathematics of buckyballs.
See also J. Baez's
review of Kostant's paper.
 Hecatohedra.
John Conway discusses the possible symmetry groups of hundredsided polyhedra.
 The icosahedron, the great icosahedron, graph designs, and Hadamard matrices. Notes by M. Brundage from a talk by M. Rosenfeld.
 Investigating
Patterns: Symmetry and Tessellations.
Companion site to a middle school text by Jill Britton,
with links to many other web sites involving symmetry or tiling.
 Jenn
opensource software for visualizing Cayley graphs of Coxeter groups
as symmetric 4dimensional 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 12vertex
complete graph as symmetrically as possible on a sixhandle surface
(the minimum genus surface on which it can be drawn without crossings).
 Kaleidoscope
geometry, Ephraim Fithian.
 Kali,
software for making symmetrical drawings based on any of the 17 plane
tiling groups.
 Kummer's
surface. Nice raytraced 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.
 Models of Platonic solids
and related symmetric polyhedra.
 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.
 Origami: a study in symmetry. M. Johnson and B. Beug, Capital H.S.
 Parabolic
isometry of an ideal hyperbolic triangulation.
Animation by John Griffin.
 Platonic
solids and quaternion groups, J. Baez.
 Platonic spheres.
Java animation, with a discussion of platonic solid classification,
Euler's formula, and sphere symmetries.
 Platonic
tesselations of Riemann surfaces, Gerard Westendorp.
 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 48gon, available with better color
resolution from his website.
 Sighting point.
John McKay asks, given a set of coplanar 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.
 Symmetries of torusshaped polyhedra
 Symmetry,
tilings, and polyhedra, S. Dutch.
 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.
 Symmetry
in Threshold Design in South India.
 Symmetry web, an exploration of the symmetries of geometric figures.
 Tales of the
dodecahedron, from Pythagoras to Plato to Poincaré. John
Baez, Reese Prosser Memorial Lecture, Dartmouth, 2006.
 Taprats
Java software for generating symmetric Islamicstyle star patterns.
 Temari
dodecahedrally decorated Japanese thread ball.
See also Summer's
temari gallery for many more.
 Morwen Thistlethwait,
sphere packing, computational topology, symmetric knots,
and giant raytraced floating letters.
 Thoughts on the number six.
John Baez contemplates the symmetries of the icosahedron.
 TileDreams
Windows software for creating symmetric patterns with rhombi.
 Tiling the integers with one prototile.
Talk abstract by Ethan Coven on a onedimensional tiling problem on the
boundary between
geometry and number theory, with connections to factorization of finite
cyclic groups.
See also Coven's paper with Aaron Meyerowitz,
Tiling the integers
with translates of one finite set.
 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 unitfraction 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.)
 Tilings and visual symmetry, Xah Lee.
 Transformational geometry.
Leslie Howe illustrates various plane symmetry types with Cabri animations.
 The uniform net
(10,3)a. An interesting crystal structure formed by packing square
and octagonal helices.
 Vision test.
Can you spot the hidden glide reflection symmetry lurking in
(the infinite continuation of) this pattern?
 Wallpaper groups. An illustrated guide to the 17 planar symmetry patterns.
See also Xah Lee's wallpaper group page.
 Wallpaper
patterns, R. Morris.
Kaleidoscopelike 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 threehexagon triangles,
or whether a polygon formed out of squares can be dissected
into restrictedorientation triominoes.
From the Geometry Junkyard,
computational
and recreational geometry pointers.
Send email if you
know of an appropriate page not listed here.
David Eppstein,
Theory Group,
ICS,
UC Irvine.
Semiautomatically
filtered
from a common source file.