12/12/2003

### von Koch Fractal Image Sets

##### (new 7/31/02)

The images available here are of fractals similar to the von Koch Triadic Island.

Each series of fractal images is drawn using a minimum edge length appropriate for each series. The minimum edge lengths are usually 4, 6, or 8 pixels. The edge length determines the number of steps used to construct that particular portion of the fractal, and thus the fineness of detail. The minimum edge length corresponds roughly to the minimum step size that can be used in measuring the boundary fractal dimension, Db. The upper limit of the step size for valid Db measurements seems to be on the order of a tenth of the diameter of the fractal (see series4 plot).

I made the images using Lispix. Lispix also measures boundary fractal dimension.

The images are for testing image processing software that measures the boundary fractal dimension, Db, of particles or other objects.

##### von Koch Triadic Island

Db for this fractal is 1.262...

a. Island outline as drawn with edge length of 4 pixels.

b. Image of outline. (192x200 pixels)

c. Filled outline.

#### Construction of a von Koch Fractal

Construction of a von Koch fractal starts with a base polygon – here an equilateral triangle. At each stage of construction, each straight line is replaced by four shorter ones, two each of length a and b. For the triadic Koch island,

a = b = 1/3, and
Db = 1.2619...

For other values3 of a and b, the value for Db is the solution of:

(Stoyan & Stoyan '94)

(Lispix solves this equation for Db numerically.)

Any two of the four parameters a, b, alpha, and Db, determine the other two parameters, and thus the fractal. Here, Db and a are the independent parameters, where Db is the fractal dimension, and a is the shape parameter.

 Fractal Shape vs. Db and a. This chart shows the ranges of Db and a that result in drawable fractals. Colors - show minimum edge length (in pixels) where image is faithful to the fractal *. Small numbers in white boxes show parameters for a single image in a given series. Most series of images are confined to the regions of edge length 2 - 8 pixels (the blue - pink - yellow - white regions). The gray area represents non-physical particles because drawing overlaps itself, except for very coarse renderings with very large minimum edge lengths -- these can hardly be called fractals.   * Tested on fractals of 300 - 1000 pixels diameter. Minimum step length for which the perimeter of the fractal image is within 5% of the length of the drawn fractal.