A Word about Fractals

Robert Rosen expresses profound ideas in his Essays on Life Itself. This is an attempt to summarize his ideas in order to understand.

For the ancient Greeks, the length of a curve was the length of a line when the curve was straightened out. It was presumed that this could always be done without changing the length.¹

Integration (such as a line integral) approximates the curve by line segments of smaller and smaller lengths. When the lengths are infinitely small, the curve is presumably straightened out without distortion, and the resulting length is assigned to the length of original curve.²

However, a fractal curve cannot be straightened out into a line without distortion. If we try, the length of such a line will depend on how we straightened it.³

Robert Rosen writes that it is non-generic for a curve to be straightened out without being distorted. There are, in other words, no algorithms for straightening arbitrary curves, no algorithms for computing their lengths. We can generate (some of) them, but via an infinite process.⁴

Notes:
1. Robert Rosen, Essays on Life Itself, p. 80. This is also related to commensurability. Commensurable means that something is 1 measurable by a common standard, 2) proportionate, or 3) exactly divisible by the same unit an integral number of times (used of two qualities). The Pythagoreans made the assumption that any two line segments are commensurable. (This is not true and is something to explore in another post.)
2. Ibid..
3. Ibid., p. 81.
4. Ibid..

Related posts:
Book Review: Essays of Life Itself
Classical physics is a limiting case of the physics of life
The principle of order from order
Subjectivizing vs. Objectivizing