A piece from the gallery
Times-table circles
Every single line in this picture is perfectly straight. So where did that curved heart come from?
Dots, a circle, and the two times table
Put some dots evenly around a circle and number them 0, 1, 2, 3 and so on. Then join each dot to the dot at twice its number: 1 to 2, 2 to 4, 3 to 6.
What happens when you run out of dots? You keep counting around the circle, like a clock. With 10 dots, dot 6 should join dot 12, but there is no dot 12, so you count on past 9 back to 0, 1, 2 and land on dot 2.
Wrapping around: counting that loops back to the start, the way 13 o'clock is 1 o'clock. Mathematicians call it modular arithmetic, and it's everywhere: clocks, calendars, the days of the week.
Now add more dots and watch.
No line ever bends. Each one just touches the curve and moves on, and together the lines trace its outline, like the strings in a string-art picture.
Change the times table, change the shape
The two times table gives a heart. Mathematicians call it a cardioid, from the Greek word for heart. Try the three times table and you get a kidney shape, the nephroid, from the Greek word for kidney.
Spot the pattern? Each times table makes one point fewer than its number. And the numbers in between, like 2.5 or 3.7, are where it gets wild: the shapes shimmer, split and flower as you slide from one to the next.
Try it with paper first
All you need is a circle, a pencil and a ruler. It's a lovely way to start in a classroom, and the moment the heart begins to appear is worth the wait.
- Draw a circle and mark 36 dots around it, one every 10 degrees. Number them 0 to 35.
- Join each dot to the dot at twice its number: 1 to 2, 2 to 4, 3 to 6.
- Past 35, wrap around: subtract 36. So dot 20 joins dot 40 − 36 = 4.
- Keep going and watch the heart appear in your own lines.
Then come back and let the playground draw a few hundred more.
Make your own
Slide the times table and the number of dots, or press play and watch the shapes melt into one another.
Each dot joins the dot at 2 times its number, wrapping around after 200.
The SVG stays sharp at any size: print it, cut it out, or open it in a drawing app. The PNG is 1200 × 630, the shape link previews use, with your circle centered on the background you picked.
Things to try
- Press play from × 2 and stop when something catches your eye. What happens around 2.5?
- Pick a times table, then drag the dots down to 20 and back up to 400. When does the curve appear?
- Try × 10 with 400 dots, then × 11. Can you predict the number of points first?
Where next?
Here, straight lines made a curve. In the Koch snowflake, straight lines make an edge that never ends.