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Straight line heart

By , 6 November 2015

Someoen is marking an eight degree angle with a protractor.

Mark every eight degrees.

This simple pattern reveals a heart from a circle and a lot of straight lines!

You will need

  • Pencil
  • Paper
  • Ruler
  • Protractor
  • A compass, or something to draw a big circle with

Marking a circle

A circle with lots of lines coming out of the centre. Someone is erasing them.

Erase the faint lines you drew.

  1. Draw a big circle on your sheet of paper.
  2. Rule a very faint line from the middle of the circle to the edge.
  3. Put your protractor in the middle of the circle on the faint line, and make a mark every 8°. Rule faint lines from the centre, through each mark, to the edge of the circle.
  4. Darken the points where each line crossed the circle, and then erase the faint lines. You now have a circle with 45 evenly spaced marks around the outside.
    Someoen is ruling a line from a point marked 3 to a point marked 6.

    Join 1 to 2, 2 to 4, 3 to 6 and so on.

Joining the dots

  1. Number each of the marks around the circle, from 1 to 45.
  2. Rule a straight line from mark 1 to mark 2.
  3. Rule a straight line from mark 2 to mark 4.
    Someone is ruling a line to a point labelled 1.

    When you get halfway, keep the pattern going.

  4. Rule a straight line from mark 3 to mark 6.
  5. Extend this pattern around the circle, ruling a line from each number to double that number.
  6. The pattern will get a bit difficult when you get to 22, but just keep the pattern going as follows:
  7. Rule a straight line from mark 22 to mark 44.
  8. Rule a straight line from mark 23 to mark 1.
  9. Rule a straight line from mark 24 to mark 3.
    A two lobed heart shape made of straight lines.

    When you’ve finished, you’ll find a heart shape!

  10. When you make it all the way around the circle, you will end up with a heart shape made of straight lines.

What’s happening?

Every line you draw in this activity is a straight line, but the shape you end up with looks curved! If you try again with more marks around the circle, the shape will become even smoother. If you could do an infinite number of marks and lines, your shape would be a very smooth curve.

This shape is known as a cardioid, which is ancient Greek for heart. There are lots of different ways to make this shape. If you take two equal coins, mark a point on the edge of one and then roll it around the other, the marked point will trace a cardioid.

More information

More information about the cardioids, including an animation

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