Mark Thompson
 Math Education
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 Tartans

The tartan on the left border is one of the traditional MacTavish Thompson dress tartans.  I was delighted when I realized how quick and easy it is to construct such patterns using a simple drawing program such as Microsoft Paint, which is included with the Windows operating system.  For example, the one below right, another MacTavish Thompson tartan, took about 10 minutes from start to finish.  This might be a good lesson for a computer-art class.  Here are a couple examples, with instructions below for drawing your own.  At the bottom are some other geometric designs suitable for tiling on your desktop -- all drawn by me, so copy and distribute them freely:

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Left:  a tartan I made up.

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A tartan is a plaid pattern composed of colored stripes in a certain sequence, crossing each other at right angles.  The same sequence is used vertically and horizontally, and the sequence is always symmetrical.  In the Thompson tartan that I use as a border on this page, the sequence is yellow-black-white-black-blue-red, followed by the same colors in reverse order.  The widths of the different stripes are variable.  To draw a tartan in MS Paint, Zoom to view individual pixels, and draw the color pattern you want, diagonally.

Then (making sure that “Draw Opaque” is not checked, so that white pixels are “clear”) copy the line and Rotate it 180 degrees, and carefully align it with the first line.  Now you have a symmetrical line of colors.

From here on, everything is select - copy - paste - reposition.  To begin with, make a second copy of your line and reposition it as shown, one step below and to the right of the first copy.  There should be white space between the two lines.  It’s critical that the repositioning be done to exactly the right place.

Then copy the pair of lines, paste and reposition so that you now have four rows, exactly positioned, and continue.  Soon, as shown below, you will have a long swath of colored stripes going northwest-to-southeast.

Finally, copy the entire swath, flip it horizontally, and carefully position it so that the colors of one swath show through the white space of the other, as in the bottom illustration.  The diamond shape where the swaths cross is now your tartan!  By continuing to copy it across the page you can make the pattern as large as you like.

By the way -- personally I like my tartans a little “grainy,” so in my initial row of pixels shown in the first illustration, I would actually draw each “pixel” as a square of four pixels.

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 Below left:  two lattices based on an Islamic design I saw once in a book (I think it may be in Hermann Weyl’s Symmetry).  Below right:  two images that, when “tiled,” form the Cairo tiling.

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Questions, corrections, comments:  Send me e-mail at  markthom@flash.net

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