FRACTAL GEOMETRY
IntroductionHarold Brochmann There are many kinds of Geometry:
- Euclidian Geometry. Assumes the existence of objects without thickness (lines and planes) and depicts natural objects as imperfect imitations of polygons. A triangle is the union of three segments.
- Cartesian Geometry. Sets up a one-to-one correspondence between points in space and sets of numbers. A triangle is specified by the coordinates of its vertices.
- Spherical Geometry. Cartesian Geometry on the surface of a sphere.
- Other coordinate systems. You can construct for example triangles on surfaces other than spheres - ellipsoids, cones, etc.
- Turtle Geometry. What we traditionally perceive as 'objects' are seen as procedures or motions of the 'turtle'. A triangle is defined as repeat 3[forward (some distance) rotate 120 degrees]
- Fractal Geometry. Does not concern itself with shapes of objects, but rather, with the an index which quantifies the texture of their surfaces. For reasons that are better not pursued at this time, this index is assumed to be the dimension of the obejct*.
*Traditionally, a line is thought of as 1-dimensional object; a plane as a 2-dimensional object and a prism as a 3-dimensional object. Dimensions are seen as having integer values. The term 'fractal' refers to the notion than some objects have a 'fractional' dimension.
If you are interested in reading what I consider the definitive book on this topic, get a copy of:
A Random Walk Through Fractal Dimensions by Brian H. Kaye
VCH Publishers. 1989 ISBN 0-89573-888-0
Library of Congress Card No.: 89-16494In this series of articles we explore: