Fracta Sketch Manual

FractaSketch: a tool for exploring fractal geometry by Peter Van Roy FractaSketch is a Macintosh program that allows students, mathematicians, graphic artists, and curious people to rapidly design and draw shapes of great intricacy. The figures range from symmetric and regular structures to patterns that seem completely random. The figures are actually fractals, shapes described by a dimension that need not be a whole number. This sets them apart from shapes in traditional Euclidean geometry such as points, lines, and planes, which have dimensions of 0, 1, and 2 respectively. A fractal with a dimension between 1 and 2 is intermediate between a line and a plane. It is more than a line (it has infinite length) and less than a plane (it has zero area). When its dimension gets closer to 2 the shape tends to fill the plane more and more. All fractals created by FractaSketch are made of many small replicas of themselves, i.e. they are self-similar in mathematicians' jargon. An example of this is the Koch snowflake, which has a dimension of 1.26: It is made of four smaller copies of itself: copy 2 copy 3 copy 1 copy 4 The program calculates the dimension of a fractal exactly. A fractal can be saved to a file in a compact form, or printed in high resolution. Since fractals are complex figures, care has been taken to ensure that no detail is lost when transferring a picture through the clipboard or when printing. The drawing algorithms have been optimized for the greatest drawing speed. The mathematics underlying fractal geometry was developed near the turn of the century, and has remained relatively unknown until recently. The recent blossoming of the field is fueled partly by the ability to make pictures of these complex shapes, and partly by the realization that the world around us is filled with irregular shapes that are actually fractals. The first views of these shapes were provided by Benoît Mandelbrot in his book “Fractals: Form, Chance and Dimension”. Since then many isolated views have been made, mostly by solitary explorers armed with massive computing power. FractaSketch was developed to allow everyone to be an explorer. It runs well on a Macintosh with 512K of memory. Creating a fractal in FractaSketch consists of two parts: First you create a template, which is a shape that is the heart of the fractal. Then you choose the drawing commands that draw this shape in the desired manner. All of this is done graphically and quickly. A template consists of a series of line segments that is entered with the mouse. A complex shape can be the result of a simple template. For example, the template below: is the basis of the following intricate drawing (of dimension 1.75): Many objects in the world around us are fract…