Imagine you are in a new city you’ve never been in before, and you want to get lost in a new way. You write down three numbers, say, 3, 4, 5, and then use them to go for a walk. Here’s what you do:
Walk 3 blocks North, then turn right.
Walk 4 blocks East, then turn right.
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Walk 5 blocks South, then turn right. You’ve used the 3, 4, and 5, so now you repeat.
Walk 3 blocks West, then turn right.
Walk 4 blocks North, then turn right.
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Walk 5 blocks East, then turn right.
Walk 3 blocks South, then turn right… and so on.
You keep walking like this until you get back to where you started. And surprisingly, you do get back to where you started! It takes 12 turns in all, and in that time you’ll have walked a total of 60 blocks.
The shape of your path is known as a spirolateral . Since I used the numbers 3, 4, 5 to make this one, I’ll call it a (3,4,5) spirolateral. It’s quite a pretty object, and fun to make. Get yourself some graph paper and try drawing your own! All you need to do is pick some numbers and repeat the process of walk, turn, walk, turn, and so on. To make a (1,2,3) spirolateral, for example, you’d simply draw 1 unit North, 2 units East, 3 units South, 1 unit West, and so on. The finished shape would look like this.
Here are some puzzles to ponder while you play with spirolaterals.
Puzzle 1.
Pick any three numbers, a, b, c , and use them to build a spirolateral. Will you always come back to where you started? Why or why not?
Puzzle 2.
What happens when you choose four numbers, a, b, c, d , and use them to build a spirolateral? Will you return to where you started?
Solutions
Puzzle 1.
After you play with enough examples of spirolaterals, you’ll probably become convinced that any three numbers you pick will generate a spirolateral that loops back to where it started. But how can you know for sure?
Keeping track of moves more systematically can help. Let’s make a table of the moves we make as we walk our (3,4,5) spirolateral.
Walk 3 units | Walk 4 units | Walk 5 units |
North | East | South |
West | North | East |
South | West | North |
East | South | West |
Take a look at the chart, and you’ll notice that by the time you’ve walked these twelve lengths, you’ll have walked 3 units in the North, West, South, and East directions. Put these together, and they all cancel out! The same thing happens for the distances of 4 units and of 5 units: they cycle through the directions and cancel once they’ve gone through all four.
This argument works for any three distances a,b,c you might choose. So every spirolateral based on three distances will loop back to its starting point after 12 moves.
Puzzle 2.
What happens if we use four distances instead? Let’s imagine we were building a (3,4,5,6) spirolateral, and keep track of the moves.
Since there are four lengths, and four directions, there’s no cancellation: you just keep walking the same loop over and over again!
In general, this will happen no matter what four distances you choose. The only exception is where the North and South directions and the same, and the East and West directions are the same, as in a (3, 4, 3, 4) spirolateral. But that’s really just a (3, 4) spirolateral, so it doesn’t feel like a genuine counterexample to me personally.
As you can tell, the exploration can keep going from here! What happens if you use even more different lengths? What happens if you change the angles? My interest in spirolaterals came recently from an art exhibit I helped curate, where I saw the work of John Critchett. These photos of his works Six Spirals and Nine Triangle Hexagon are built from spirolaterals made by using different turn angles and number sequences. The possibilities are truly beautiful.
Happy puzzling!