Cayley Quilt Maker
Create quilt-like patterns from Cayley tables.
Instructions        About


Instructions

Cayley tables show the results of combining elements in a set using various operations. The tables' symmetrical patterns illustrate the sets' mathematical properties. To make a quilt, first paint a collection of small tiles which represent the elements of the set. The set can have 4, 6, or 8 tiles. Set the number of tiles using the Tiles control. Then switch the View control to Tiles, select a paint brush and color, and paint the tiles.

When you have painted a set of tiles, you can view a Cayley table for the set by switching the View control to Table. The Operation control lets you choose the type of Cayley table from the following options :

Addition mod n (shown above)
Multiplication mod n
Dihedral groups D3 and D4
Direct sums Z2+Z2, Z2+Z2+Z2 and Z2+Z3
Quaternion group Q8

Switch the View control to Quilt to see a quilt. Quilts are made using the Cayley tables in various patterns. The Quilt control lets you choose a type of quilt from the following patterns :
  • Reflection: The four quadrants of the quilt are tiled clockwise with vertical and horizontal reflections of the table.
  • Rotation: The four quadrants of the quilt are tiled clockwise with four successive 90 degree clockwise rotations of the table.
  • Fibonacci: The quilt is tiled with 64 tables spaced and sized according to the first numbers of the fibonacci sequence. This option applies to reflection, rotation and color addition quilts.
  • D4 Table: The D4 Table quilt is a Cayley table for the dihedral group D4, whose motions are applied to the image of the table.
  • Golden Rectangle: This is a reflection quilt whose quadrants contain golden rectangles constructed from the original table using recursive subdivision : each golden rectangle is divided into a square and a smaller golden rectangle (which is in turn subdivided).
  • Color Addition: A reflection quilt whose quadrants contain color-shifted copies of the table. The color of each pixel in the image of the table is increased by the sum of its x and y coordinates modulo n, where n is the size of the table's image in pixels.
  • Parallelogram: The table is sheared into a parallelogram which is wrapped around the table's original square. Then it is reflected onto the quilt.

Switch the View control to Save PNG to save a quilt as a PNG image file. Just click on the thumbnail of the quilt image to save it. You will need Adobe Flash Player 10 to use this feature.





About Cayley Quilt Maker

Cayley Quilt Maker - copyright (c) Scott Lopez 2010.

Permission to redistribute unmodified versions of this Flash application for non-commercial use is hereby granted.
Application provided as is, with no warranty.
Use at your own risk.

This program is based on ideas from publications by Forseth & Troutman.
See, for example :
Forseth, S; Troutman, A. 'Designs Exhibiting Mathematical Structure,'
School Science and Mathematics, Volume 74, December 1974.

This program may be run as a stand-alone application in the Adobe Flash Player. To save png images, you will need Adobe Flash Player 10 or better. The program SWF file is available here :
CQM.swf

Windows OS users can get the free user-friendly Windows version of this app here :
cquilter.html
The Windows version saves bmp images, and it has printer support and a help file.