Clock arithmetic is similar to normal arithmetic, except on a numberline that wraps around in a circle—just like a clock. For example, 4 hours after 10 o'clock is not 14 o'clock, but 2 o'clock. In clock arithmetic, when the answer is bigger than the largest allowed value, you just take the remainder.

For a clock, the largest value is 12. For angles around a circle, the largest is 360°. In this activity, we start with a maximum of 60, and look at some patterns that arise from clock arithmetic around a circle.



The Heart Curve
Epicycloid patterns from Clock Arithmetic
We build this pattern by drawing a collection of lines, each joining two numbers on the circle. The starting point for each line is just each number in turn. To find the end point, we multiply the number by the selected factor and then take the remainder if necessary using clock arithmetic.

Start with the numbers 1 to 60 and a factor of 2.

        Factor 

About the activity

Here is a worksheet template for this activity from Artful Maths.

Epicycloids are the family of curves generated in this way. A Cardioid is the simplest case, and is named from the Greek word for "heart" (as we also use in cardiac, cardiologist, cardiovascular).

We can also make these curves by rolling a circle around another circle—see the discussion following the Circles on Circles activity.

Thanks to Clarissa Grandi at Artful Maths for this idea.