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February 27th, 2012 by Math Tricks | No Comments | Filed in Math Art, Spirolaterals

Spirolaterals – A Perfect Match for Doodlers!

Have you ever had to wait in an office for a long time for an appointment without a book?  Sure, they may have magazines at the dentist’s office or what have you, but chances are that they are months old.  What do you do to entertain yourself while you wait?  One solution is to draw spirolaterals – a great way to draw interesting math art with a few simple rules – plus a notepad, pen, and patience!

Spirolaterals are spiral structures generated by setting up simple rules for drawing lines, and performing reiterated iterations of these rules.  Using complex algorithms, the number of spirolaterals that can be generated may indeed be infinite.  These structures were first investigated in 1968 by Hal Abelson and his colleagues.  Spirolaterals are open or closed, and can range from simple squares and rectangles to complex curves.

A very simple spiral is a square.  The rules for drawing a square are:

Draw a line of length one, rotate your paper by 90 degrees and repeat.  Repeat procedure until a closed design is generated.

I have outlined these steps using the excellent spirolateral program at http://math.fau.edu/MLogan/Pattern_Exploration/Spirolaterals/SL.html:

Iteration 1

spirolateral square 1








Iteration 2

spirolateral square 2








Iteration 3

spirolateral square 3








Iteration 4

spirolateral square 4








So now what do you get if you use the above algorithm, but change the angle to 60 degrees?  You get a geometric figure which has
an angle of 60 degrees between all of its line segments:

hexagon spirolateral







That’s right – a hexagon!

If you vary the line length as you progress in your iterations, much more interesting shapes can be made.  Of course, you can do all of this roughly with pen and ink, but here I will generate the designs with the program mentioned above.

Using variable lengths 1 and 2 (here it is set to letter codes MZ, which codes for 13 and 26, which is the same as 1 and 2, but the program draws the design bigger this way – LOL!), and an angle of 120 degrees, we get this design:

1 2 120 spirolateral








This one was drawn using 1 5 9 lengths, and 35 degrees.  It reminds me of the designs i used to make with the Spirograph toy when I was a kid:

1 5 9 35 spirolateral







Length 1 (actually, it is set to Z, or 26) and 30 degrees:

1 30 spirolateral







1 2 3 4 5 6 7 and 35 degrees:

1 2 3 4 5 6 7 35 spirolateral







Give it a try, either with pen and paper or by using the program.  It really fun and interesting, and you get a sense of how patterns should develop before you actually draw them.

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