Created 2004-03-09 Modified 2007-01-01
Chelton Evans
Mandelbrot's Peano-snowflake
Peano-snowflake Algorithm from
Picture
The Fractal as a Function of Area
When I saw this fractal picture I wondered how it was generated. The following is an algorithm to generate the fractal.
The triangle is equilateral - all side lengths are equal. Construct points by dividing the lengths in thirds. From the blue lines other points are constructed. Notice the guide lines which pass from one triangle corner/vertex and bisect the opposite side.
The blue arrows indicate the direction of the fractal function applied to length. The black arrow gives the direction. Let the function be defined such that given a starting point B and and end point A the left curve is constructed. Now applying this function to each segment separated by points the right diagram is generated.
From the left diagram you can see the curve which winds itself from B to A. And you can see how applying the function works.
Let a function be defined that converts the bottom large blue triangle into the top complicated one. The triangle has been divided into smaller triangles - 13 with area and 9 without which are holes. Repeating the function on the subsequent 13 triangles with area ... generates the fractal.
So instead of considering the fractal as a curve it can be viewed as a function on area. It is equivalent to the curve in a one-one sense. The area fractal gives a logical explanation for the points by showing their symmetry.
