بسم الله الرحمن الرحيم

May 2, 2010

Symmetries in traditional patterns

There are more than 30 traditional patterns to start a pasang game. All of them exhibit some forms of symmetry.

Perfect symmetry

The following pattern is the most symmetrical. In traditional embroidery, it is used for dithering.

Square symmetry

Most traditional patterns have mirror symmetries with 4 axes, such as these:


Among such patterns, some have finer symmetries in their quadrants. The following have diagonal symmetries in their 5x5 quadrants. The last one also has vertical and horizontal symmetries in its quadrants.


The following pattern is rather unique. Apart from the perfectly symmetrical dithering pattern, it is the only traditional pattern having 6x6 (overlapping) quadrants that are diagonally symmetrical.


4-fold symmetry

Some patterns have 4-fold (90-degree rotational) symmetry, such as these:


A few of such have almost mirror symmetries, but they are broken in the centre.


Diagonal symmetry

One particular pattern has only mirror symmetries diagonally.


Night-and-day symmetry

Two patterns have night-and-day symmetry where the pattern and the background form the same shape of different handedness.


Building unit

A few patterns are made up of units larger than "dots". In particular, the following patterns are made up of, respectively, v's, w's and pluses.

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