## *
Approximating the Koch snowflake
*

There are various ways of constructing continuous curves which have
not tangent at *any * point. One such construction, due to Helga
Koch, will be illustrated with the help of Maple.
The starting point is that given a straight line segment,it is
possible to remove the middle third of the segment and replace it by
two sides of an equilateral triangle whose sides have the same length
as one third of the original segment. The resulting object is a
crooked curve consisting of four straight line segment -- all of
which have the same length as one third of the original segment.

Starting with an equilateral triangle, this procedure can be applied
to all three of it sides. The resulting object is a six-pointed star
with twelve sides. The basic procedure can be applied to each of these
sides and the result is a more complicated curve with 48 sides. The
basic procedure can be applied to each of these sides. This process
can be carried out infinitely often and the limiting curve is known as
the *Koch Snowflake*. It is a continuous curve which has has no
tangent at any point.

Write a Maple procedure which will draw the intermediate stages along
the way to constructing the Koch Snowflake. Begin by writing a
procedure which, given points and in the plane, will
calculate points , and such that

- is on the line segment joining and and the
distance from to is one third of the length of the segment
from to
- is on the line segment joining and and the
distance from to is one third of the length of the segment
from to
- the points , and form the vertices of an
equilateral triangle whose side have length one third of the length of
the segment from to
- travelling in a train from to you would see by
looking out the left window

Using this procedure write another procedure which takes any sequence of
points in the plane (which can be imagined as forming the vertices of
some polygon)
and applies the first procedure to each pair of successive points to
produce a new sequence of points in the plane.
Apply this new
procedure to the sequence of points forming the vertices of an
equilateral triangle several times and use plot to draw the resulting
polygon.

Use Maple to calculate the length of the Koch Snowflake. What is the
area of the Koch Snowflake?

## Instructor

Juris Steprans

email address: steprans@mathstat.yorku.ca

Department of Mathematics and Statistics.

Ross 536 North, ext. 33921

York University

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