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## 'Smaller and Smaller' printed from http://nrich.maths.org/

Here is an equilateral triangle which has sides of length 1
unit:

We could take out the middle third of each side and
replace it with a spike. If the spike is made of two lines which
are the same length as the missing section, it will look like
this:

We could do the same again. Taking out the middle
third of each line and replacing it with a spike in the same way
gives this:

Remembering that the length of each side of our
original equilateral triangle is 1 unit, what is the perimeter of
the first shape we drew above? And the second? Can you see a
pattern? Perhaps you can explain why this pattern comes about.

Can you predict, without drawing, what the
perimeter of the next shape will be if we continue drawing them in
the same way?

Could you predict the perimeter of any shape in the
series?

The shapes above are called
fractals. A fractal is made up of lots of copies of itself, each
one smaller than the last. See also the problems
Von Koch Curve,
The Cantor Set and
How Long is The Cantor Set? To find
out more about fractals, look at the Motivate
site.