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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 tex2html_wrap_inline10 and tex2html_wrap_inline12 in the plane, will calculate points tex2html_wrap_inline14, tex2html_wrap_inline16 and tex2html_wrap_inline18 such that

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?



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Juris Steprans
Thu Mar 27 17:08:47 EST 1997