Spirolaterals!
By Jasmine, 4 November 2014 Activity
You will need graph paper, a coloured pencil and a ruler.
In this activity, you’ll draw one long line that spirals and snakes around the page, known as a spirolateral. Start with a simple one, and then go on to invent your own spirolaterals!
You will need
 Graph paper with 1 cm x 1 cm squares. (You can download and print some as a PDF or a Word document.)
 A coloured pencil
 Ruler
What to do

Start near the centre of your graph paper and draw a line one square long, going straight up.
Starting at a corner near the centre of your graph paper, draw a line one square long, going straight up.
 At the end of that line turn 90 degrees right, and draw a line two squares long.
 Continuing the line, turn right again and draw a line three squares long. This time you will be drawing straight down the page.
 Again, drawing from the end of the line, turn right and draw a line four squares long. This time your pencil should travel from the right to the left of the page.
 From the end of your line, turn right and draw a line five squares long. This line will go straight up, in the same direction as the first one.
 Repeat the whole process again starting from the end of the length five line.
Turn right and draw a line two squares long, then turn right for three, four and five squares.
As before, turn to the right and draw a line one square long. Be careful! This time, the one square line goes from the left to the right of the page, instead of going straight up. Continue the pattern you made above, drawing lines that are two, three, four and five squares long, turning right between each segment.
 Repeat step 6 two more times, drawing lines one to five squares long rotating right each time. This should return you to the initial point you chose to start your spirolateral!
Bonus spirolaterals to try
Instead of drawing sides of length one through to five, see what happens when you use any one of the procedures below.
 One, two and then repeat.
Repeat the whole process until you end up back where you started.
 One, two, three, four and then repeat.
 One, two, three, four, five, six … Go as high as you dare, be careful you don’t run out of space on your page!
 For more advanced work, see what happens to your spirolaterals if you change the angle you turn. This might be easier with triangular graph paper.
 Make up your own instructions and then get a friend to draw it!
What’s happening?
Now that you know how to draw spirolaterals, you can describe new ones by listing the side lengths and naming the angle to turn. Small changes in the instructions can create a huge variety of different spirolaterals!
Spirolaterals are grouped into two main types: open and closed. The first spirolateral in this activity is closed, because it eventually joins up into a closed loop. If you want to find out what an open spirolateral is, try drawing the second bonus spirolateral. Instead of joining back to the start, open spirolaterals keep going forever. If you do try drawing it, please do not keep drawing your spirolateral forever, as you still need to eat, drink and learn!
Written by Dennis Price
More information
Spirolateral generator
Maths investigations: spirolaterals
Spirolaterals, complexity from simplicity (more advanced)
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