Fractals 1

Fractals 1
Introduce a simple idea of a fractal – something split up into infinitely small
sections in a recursive way. Eg start with an equilateral triangle. The recursion
is: take every straight line, divide it into 3 equal parts, construct a new
equilateral triangle based just on the central section, erase the original central
section. It goes something like this:
It is worth noting that the perimeter of this shape becomes infinitely large
although the shape itself is ‘contained’ – you could say it is infinitely wiggly!
Students may like to try their own designs.
Fractals were first studied by geographers trying to determine how long the
coast of Great Britain was – as they used more and more detailed maps the
coastline got longer and longer!
Lots of interesting and curious results have been found,
Eg (1): in each of these shapes, the length of the staircase diagonal is 2. As you
keep going, the line gets closer and closer to the diagonal of the square.
Therefore the diagonal of the square has length 2??
Eg (2) Again start with an equilateral triangle, this time divide it into 4
equilateral triangles and shade out the central one. Repeat…..
Compare the result with Pascal’s Triangle, where all even numbers are shaded
(continues down infinitely…)