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:
Try some more interesting experiments:
- 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?
- 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...