ABSTRACT
The seven general ways in which chaos theory uses the word "dimension" are: The Euclidean dimension (the number of measurable coordinates or variables needed to describe a Euclidean shape) For example, one measured coordinate or dimension (namely length) describes a straight line. Two dimensions (length and width) describe a basketball court (area), and three (length, width, and depth) describe a cabinet drawer. Also, as mentioned earlier, a point has a dimension of zero. Euclidean dimensions usually are envisioned as directions or axes at right angles to one another (Fig. 20.1 a). Therefore, specifying the pertinent values in one, two, or three dimensions locates any point within a line, area, or volume, respectively. The topological dimension The topological dimension of a body is I + the Euclidean dimension of the simplest geometric object that can subdivide that body. For instance, think of a line as a series of connected points. Removing any point from the chain (except for the two endpoints) subdivides that line into two smaller lines. So, a point (dimension zero) is the simplest geometric object that subdivides a line. According to the formula, a line's topological dimension therefore is 1 +0 = 1 (Fig. 20.1 b, left side). Now, what about a surface? To subdivide a surface into two smaller surfaces, removing a point from that surface won't do the job (it just creates a small hole in the surface). Instead, we need a line. A line has a Euclidean dimension ofl, so a surface's topological dimension is 1 + 1 = 2 (Fig. 20.1 b, right side). Similarly, a line won't subdivide
Three-dimensional system of weights
(d) Three-dimensional vector
Figure 20.1 Continued.
