by Nina Amenta of the Geometry Center (World Wide Web interface by Paul Burchard via W3Kit)

This special version of Kali for X Mosaic only allows freehand drawing of symmetry patterns! The features of X Mosaic used here are undocumented; however, an improved freehand drawing widget (type=SCRIBBLE) will be included in HTML 3.0 (formerly known as HTML+). We are looking forward to seeing browsers which implement the SCRIBBLE widget!


You can use Kali-Jot to draw Escher-like tilings, infinite knots, and other cool stuff. It lets you draw patterns in all of the 17 planar symmetry groups. 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 crystallographic notation.

How Kali-Jot 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 a set of curves. Each curve 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 lattice point.

(Kali is also available as a standalone program for the Macintosh and for Silicon Graphics Iris computers, from the Geometry Center's software achives)

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The Geometry Center
University of Minnesota
400 Lind Hall 
207 Church Street S.E.
Minneapolis, MN  55455