Fractals: Session 3

von Koch curve

Like the Sierpinski gasket, the von Koch curve is a fractal curve which is formed by successive iterations of simple rules. Again starting with an equilateral triangle:

• remove the central third of each line segment
• in its place, add the other two sides of an equilateral triangle of length equal to the segment you have just removed

The first three stages are shown below:

In this session, we are going to use a geometric and trigonometric approach to find the area of the von Koch curve as well as using geometric series.

Task 1: constructing the von Koch curve

For the first task, you will need plain paper, compasses and ruler, and a sharp pencil - you are going to make an accurate construction of the first stage of the von Koch curve (more if you want!). Start by constructing an equilateral triangle inscribed in a circle (make both reasonably large). Next, you need to trisect one side of the triangle, AB:

1. Draw an arbitrary ray from A in the general direction of B, but not through B (outside the triangle is probably a good idea to avoid cluttering your drawing).
2. Mark C where your ray cuts the circle again.
3. Mark D on the ray beyond C, such that CD = AC (use compasses to measure the distance rather than a ruler).
4. Draw a ray from D through B and extend it.
5. Mark E on this ray beyond B, such that BE = DB.
6. Join C and E. Where this segment cuts the line AB, mark F.
7. The distance BF should be precisely one third of the distance AB.
8. Copy this distance (using compasses) from each vertex of the triangle along each side, to trisect each side of the triangle.
9. Construct equilateral triangles on each inner third of the sides of the original triangle.

Now prove that this construction does in fact work.

If you have done your construction accurately, you should see that the small triangles each have a vertex on the circle. Can you prove this?

Task 2: limits on the area contained within the von Koch curve

It would appear from this construction that the von Koch curve is contained within the circle which circumscribes the original equilateral triangle, and it can be shown that this is true. Use this fact to find the lower and upper limits of the area of the von Koch curve at all stages? (You should find your answers as exact values, using fractions, pi and surds, rather than decimals).

Task 3: area of the von Koch curve

Find the area of the von Koch curve at the first few stages, and hence at the nth stage. If n tends to infinity, what will the area be?