Similarity in General Metric Spaces
In a general metric space (X, d), an exact similitude is a function f from the metric space X into itself that multiplies all distances by the same positive scalar r, called f's contraction factor, so that for any two points x and y we have
Weaker versions of similarity would for instance have f be a bi-Lipschitz function and the scalar r a limit
This weaker version applies when the metric is an effective resistance on a topologically self-similar set.
A self-similar subset of a metric space (X, d) is a set K for which there exists a finite set of similitudes with contraction factors such that K is the unique compact subset of X for which
These self-similar sets have a self-similar measure with dimension D given by the formula
which is often (but not always) equal to the set's Hausdorff dimension and packing dimension. If the overlaps between the are "small", we have the following simple formula for the measure:
Read more about this topic: Similarity (geometry)
Famous quotes containing the words similarity, general and/or spaces:
“Incompatibility. In matrimony a similarity of tastes, particularly the taste for domination.”
—Ambrose Bierce (1842–1914)
“To judge from a single conversation, he made the impression of a narrow and very English mind; of one who paid for his rare elevation by general tameness and conformity. Off his own beat, his opinions were of no value.”
—Ralph Waldo Emerson (1803–1882)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)