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Spirolaterals ala Carte

About thirty years ago, someone introduced me to the idea of spirolaterals. I shared the idea with my students and found that it sparked their interest, while a lot of geometry was being done at the same time...even more when LOGO came on the scene....and then fractals.

Consider the number sequence [2,3,5]. Start at a grid point in the center of a sheet of graph paper, pretend you are facing to the right, draw a line 2 units forward, turn 90 degrees clockwise, draw a line 3 units forward (i.e. down), turn 90 degrees clockwise, draw a line 5 units forward (now to the left), turn 90 degrees clockwise, repeat the process....until you either find you are retracing a pattern or it is obvious that no repeating pattern will occur. This is called the spirolateral for the sequence 2,3,5.

Make the spirolaterals for these sequences: [1,2,5], [1,3,2], [1,2,3,4,5], [1,2,3,4,5,6], [1,1,2,2]...or make up your own sequences. What observations can you make about the resulting spirolateral patterns for the different sequences:

  • How is the spirolateral impacted by the number of digits in the number sequence?
  • How is the spirolateral impacted by changing the order of digits in the number sequence?
  • Given a number sequence, can you predict whether the spirolateral will repeat or "spiral off" into infinity?
  • If a spirolateral repeats for a number sequence, can you predict its length?
  • Could a number sequence produce a spirolateral that does not "spiral off" to infinity, yet it does not end up in a repeating pattern?
Try some more interesting experiments:
  • Build the spirolateral for the digit sequence in your address, your phone, or your social security number. Note: How will you handle the 0?
  • Build the spirolateral for your name (or any word) using this code to assign a digit to a letter...

    123456789
    ABCDEFGHI
    JKLMNOPQR
    STUVWXYZ

Now, I hear you saying there must computer programa that do all of this, as the drawing becomes tiresome by hand (but it has great educational value)? First, students can write programs for drawing spirolaterals on graphing calculators (or consider the PDF on page 17 of this site). Second, the NSF project Pattern Exploration offers a very powerful spirolateral-creation program (with directions). Use it...play with it...share it...learn from it.