Interrupt your regular math programming to try this fantastic math doodling investigation!
Anna Weltman wrote a math/art book. It’s great fun for all ages, full of fantastic mathematical explorations — including spirolateral math doodles.
How to Get Started
To make a spirolateral, you first pick a short series of numbers (1, 2, 3 is a traditional first set) and an angle (90° for beginners). On graph paper, draw a straight line the length of your first number. Turn through your chosen angle, and draw the next line. Repeat turning and drawing lines, and when you get to the end of your number series, start again at the first number.
- Download and print Anna’s Loop-de-Loop Lesson Instruction Page and Student Work Page.
- Share with your kids.
- Print some extra graph paper for continued play.
- Check out Anna’s blog post for more ideas.
- Explore what happens when you make spirolaterals on triangle graph paper, too.
Some spirolaterals come back around to the beginning, making a closed loop. Others never close, spiraling out into infinity—or at least, to the edge of your graph paper.
For Further Reading
- Mike Lawler and sons explore Loop-de-Loops: Part 1, and Part 2.
Martin Gardner, “Worm Paths” in Knotted Doughnuts and Other Mathematical Entertainments.
Articles by Robert J. Krawczyk:
- Spirolaterals, Complexity from Simplicity
- The Art of Spirolaterals
- The Art of Spirolateral Reversals
- Curving Spirolaterals
- More Curved Spirolaterals
Anna Weltman appeared on Let’s Play Math blog once before, with the game Snugglenumber. And she’s a regular contributor to the wonderful Math Munch blog.
Denise Gaskins' Let's Play Math 
Thank you so much for sharing these, Denise, they look great. I haven’t stopped to comment here in far too long but I do continue to appreciate all the fab maths ideas you take the time to post. It’s been a few years since we switched to your “Let’s Play Math” approach and we’re all still loving maths. It seems to get more and more fun as the children get older!
Hi, Lula! It’s great to hear from you.
Yes, math gets more fun as the kids get older, because it’s so exciting to see them wrangle with ideas on their own. They never think things through exactly the way I would, and sometimes their ideas are so much better than mine.
For example, with these spirolaterals, my daughter noticed a pattern in the way some loop around while others spiral off into infinity. And after thinking about it for awhile, she came up with a proof for why it happened and predicted what longer series of numbers would do.
Her very own proof! And a conjecture, too. How cool is that?! I love watching her mind work 🙂
Wow! That is very cool!
Yes, I agree that seeing how their minds work is one of the best things about this kind of maths. (Parents who delegate wholesale to workbooks or online programs are missing out on so much!)
I’ve always known my son is a visuospatial learner but it wasn’t until we were looking at some visual patterns together this week that I really saw what that means. His way of deriving the equation for the nth step was completely different from mine and my daughter’s (and probably much more efficient). That was once I’d managed to get him to backwards engineer his answers, of course. 🙂
Backwards-engineering an answer — good phrase! That’s the way professional mathematicians do it, too. I can’t find the quote right now, but someone described it along these lines:
“We notice interesting things and play around with them, and that’s the real work of mathematics. Then later we go back and write a proof, just to make sure our minds haven’t been playing tricks on us.”