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 (18421914)
“There is a mortifying experience in particular, which does not fail to wreak itself also in the general history; I mean the foolish face of praise, the forced smile which we put on in company where we do not feel at ease, in answer to conversation which does not interest us. The muscles, not spontaneously moved but moved, by a low usurping wilfulness, grow tight about the outline of the face, with the most disagreeable sensation.”
—Ralph Waldo Emerson (18031882)
“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 (18031882)