by Nina Amenta of the
(World Wide Web interface by
Paul Burchard via
CLICK HERE TO START.
For more about tilings, go to
You can use Kali to draw Escher-like tilings, infinite knots, and
other cool stuff. It lets you draw patterns in all of the 17 planar
To learn more about symmetry groups, take a look at the
Geometry and the Imagination
course notes (with pictures) now available on this Web server!
Choosing a Symmetry Group
Select the symmetry group you wish to work in by clicking on one of
the icons on the symmetry group selection panel below the picture.
Each icon shows some of the symmetries which your pattern will contain.
Each group is also labeled in either Conway notation or the traditional
How Kali Works
Every symmetry group is defined by a lattice and a set of two
generators, which are either a rotation, reflections, or glide
reflections. A pattern is just a list of lines. Each line is first
reflected (if there are any reflection or glide reflection generators)
and then all reflections are rotated (if there is a rotation
generator). Finally the resulting figure is redrawn around every
(Kali is also available as a standalone program for
the Macintosh and for Silicon Graphics Iris computers, from the
Geometry Center's software achives)
The Geometry Center
University of Minnesota
400 Lind Hall
207 Church Street S.E.
Minneapolis, MN 55455