2. Natural Fractals and Dimensions

Pictured below are the Koch curve and three of its relatives. From top left to bottom right, these pictures become increasingly "fuzzy." Can we find a way to quantify the difference in these pictures? Perhaps such a method could be used to distinguish the coastline of Norway from that of Italy, for example, or the beating of a healthy heart from that of a diseased heart, or the closing prices of a conservative stock from those of a more risky stock, or a text of Shakespeare from one of Bacon. Let us see.

Contents of this page:

A.Ineffective ways to measure problems with trying to measure a shape in the wrong dimension, prelude to noninteger dimensions.

BBox-counting dimension extends the notion of dimension to fractals. Arguing by analogy with Euclidean dimension, we develop an algorithm for determining this dimension.

C. Similarity dimension is a simplified method of computing dimensions for self-similar fractals with all pieces scaled by the same factor. This dimension gives a clear indication of the relation between dimension and complexity.

D. The Moran formula extends the similarity dimension fromula to self-similar fractals with different scaling factors.

E. Other dimensions. Among several variants, we study the mass dimension, a measure of how the mass of an object scales with the size of the object. Objects with hierarchical structures, dust clumps and natural sponges, for example, typically have non-integer mass dimensions.

F. Area-Perimeter Relations. Fractal curves that enclose regions in the plane can reveal their dimensions by a subtle relation between area and perimeter.

G. Some Alegbra of Dimensions. When we build fractals from other fractals, how is the dimension of the whole related to the dimensions of the pieces?

H. Natural Fractals Nature is filled with fractals. We survey a few examples, and of the physical and biological processes forming them.

I. Manufactured fractals. Here we study fractals in industry, and note examples of how dimension directs some specific manufacturing processes.

Here are some calculations of dimensions of variations on the Cantor set.

A.Ineffective ways to measure problems with trying to measure a shape in the wrong dimension, prelude to noninteger dimensions.
BBox-counting dimension extends the notion of dimension to fractals. Arguing by analogy with Euclidean dimension, we develop an algorithm for determining this dimension.
C. Similarity dimension is a simplified method of computing dimensions for self-similar fractals with all pieces scaled by the same factor. This dimension gives a clear indication of the relation between dimension and complexity.
D. The Moran formula extends the similarity dimension fromula to self-similar fractals with different scaling factors.
E. Other dimensions. Among several variants, we study the mass dimension, a measure of how the mass of an object scales with the size of the object. Objects with hierarchical structures, dust clumps and natural sponges, for example, typically have non-integer mass dimensions.
F. Area-Perimeter Relations. Fractal curves that enclose regions in the plane can reveal their dimensions by a subtle relation between area and perimeter.
G. Some Alegbra of Dimensions. When we build fractals from other fractals, how is the dimension of the whole related to the dimensions of the pieces?
H. Natural Fractals Nature is filled with fractals. We survey a few examples, and of the physical and biological processes forming them.
I. Manufactured fractals. Here we study fractals in industry, and note examples of how dimension directs some specific manufacturing processes.
Here are some calculations of dimensions of variations on the Cantor set.