IB MATHEMATICS Topic : Sequences and Series – Von Koch's Snowflake Curve
1. Start with an equilateral triangle with sides of
length 1 (Diagram 1 above).
2. Divide the sides into equal thirds. On the middle
third of each side construct an equilateral triangle with sides of
length one third. Delete the 'base' of each triangle, used to
construct it (Diagram 2 above).
3. On the middle third of each of the twelve sides, build
an equilateral triangle with sides of length 1/9. Erase the base of
each of the twelve new triangles. (Diagram 3).
4. Repeat the process with this 48-sided figure (Diagram
4 onwards).
The "limit curve" defined by repeating this
process indefinitely is called von Koch's SNOWFLAKE CURVE
.
The snowflake curve has some interesting properties that
may seem paradoxical.
The snowflake curve is connected in the sense that
it does not have any breaks or gaps in it.
The snowflake is confined to the unit square.
Building the new little triangles adds more than one unit of length
to the curve. To be precise, the length of the curve is multiplied
by
at
each step so tends to infinity.
Under a magnifying glass, a little piece of the
snowflake looks identical to a larger, unmagnified chunk. Objects
that exhibit this kind of self-similarity are called FRACTALS and
are of great research and applied interest in modern science and
mathematics.
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