Snowflake Curve | Research & Encyclopedia Articles

Snowflake Curve

The snowflake curve is a fractal. It is also known as the Koch snowflake or Koch's island after Helge von Koch who first described it in 1904. To construct it, first draw a line segment AD. Next draw point B and C on AD so that AB, BC and CD all have the same length. Now draw a point E so that the triangle BCE is equilateral. Now erase segment BC. The shape that is left is called the motif of the snowflake curve.

On a separate piece of paper, draw an equilateral triangle with side length equal to the length of AD. Now replace each edge of the triangle with a copy of the motif so that the motif points away from the triangle. Now replace each of edge of what you have with a reduced (by one third) copy of the motif so that the motifs point outwards. Again, replace each edge of what you now have with a reduced (by one ninth) copy of the motif so that the motifs point outward. Continue in this way forever. The result is the Koch curve.

At each new step of the construction, the perimeter of the curve is four thirds times the perimeter of the curve at the previous step. Thus, the perimeter of the Koch curve is infinite.

If s is the number of sides of the island at the nth step then the area of the island at the (n+1)st step is equal to its area at the nth step + s times (1/9) ⁿ⁺¹ times the area of the original equilateral triangle. This is because all the new triangles are lengthwise 1/3 the size of the previous triangles. Thus the area of the new triangles is 1/9 that of the previous. Also there are s new triangles at the n+1st step. The number s is equal to 3*4ⁿ since at each step the number of sides increases by a factor of four and the number of sides of the original triangle is three. These facts imply that the area of the Koch snowflake is equal to the area of the original triangle times one plus one third of the sum as n goes from 0 to infinity of (4/9) ⁿ. The last sum is the sum of a geometric series. Its value is nine fifths. Hence the area of the Koch snowflake is eight fifths of the area of the original triangle.

The Koch curve is nondifferentiable everywhere (see Derivatives and differentials). This means that for any point x on the curve, and any sequence of points y_n on the curve, such that the distance between x and y_n decreases to zero as n increases towards infinity, then the slope of the line segments xy_n does not approach any finite limit. The set of points on the original triangle that are contained in the Koch curve form a Cantor set. Because this set is totally disconnected, the Koch curve contains no line segments.

The snowflake curve can be partitioned into three congruent curves. Each curve is formed from the motif by repeatedly substituting small copies of the motif for each line segment at each step. Each of these curves can then be partitioned into four smaller curves, each a copy of the big curve but one third its size. Then each of these smaller curves can be partitioned into four smaller curves. Of course, this process never ends. This process is similar to another process. A line segment can be divided into two line segments. Then each of the two halves can be divided into two and so on. Also a square can be divided into four squares, each a copy of the original square reduced by one half. A cube can be divided into eight cubes, each a copy of the original cube reduced by one half. Thus, we have that a d- dimensional "cube" can be divided into 2^d "cubes", each a copy of the original reduced by one half. If the reduction factor is s, the number of copies is s^-d. Following this train of thought, the self-similarity dimension of the Koch curve is a number d such that 3^d = 4. This implies that d = log(4)/log(3) which is approximately 1.262. Incidentally, the self-similarity dimension of the Koch curve is equal to its Hausdorff dimension.