Tuesday, Sept 25, 2012

We begin by extending the similarity dimension formula to the case of self-similar fractals in which pieces can have different scalings.

Next we introduce a dimension measurement for physical objects.

We end with some examples of dimension computations in science.

This emphasizes that fractal dimension is a measure of roughness. In fact, it is the first quantitative measure of roughness.

First, we review the basics of box-counting dimension.

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.

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

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.

First, we review the basics of box-counting dimension.
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.
The Moran formula extends the similarity dimension fromula to self-similar fractals with different scaling factors.
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.