Editing Similarity (geometry) (section)

==In general metric spaces==
[[File:Sierpinski deep.svg|thumb|300px|[[Sierpiński triangle]]. A space having self-similarity dimension <math>\tfrac{\log 3}{\log 2} = \log_2 3,</math> which is approximately 1.58. (From [[Hausdorff dimension]].)]]

In a general [[metric space]] {{math|(''X'', ''d'')}}, an exact '''similitude''' is a [[function (mathematics)|function]] {{mvar|f}} from the metric space {{mvar|X}} into itself that multiplies all distances by the same positive [[scalar (mathematics)|scalar]] {{mvar|r}}, called {{mvar|f}} 's contraction factor, so that for any two points {{mvar|x}} and {{mvar|y}} we have

<math display=block>d(f(x),f(y)) = r d(x,y).</math>

Weaker versions of similarity would for instance have {{mvar|f}} be a bi-[[Lipschitz continuity|Lipschitz]] function and the scalar {{mvar|r}} a limit

<math display=block>\lim \frac{d(f(x),f(y))}{d(x,y)} = r. </math>

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 {{math|(''X'', ''d'')}} is a set {{mvar|K}} for which there exists a finite set of similitudes {{math|{ ''f{{sub|s}}''}{{sub|''s''∈''S''}}}} with contraction factors {{math|0 ≤ ''r{{sub|s}}'' < 1}} such that {{mvar|K}} is the unique compact subset of {{mvar|X}} for which

:[[File:Epi17.png|thumb|A self-similar set constructed with two similitudes: 
<math>\begin{align} 
z' &= 0.1[(4+i)z+4] \\
z' &= 0.1[(4+7i)z^* + 5 - 2i]
\end{align}</math>]]

<math display=block>\bigcup_{s\in S} f_s(K)=K.</math>

These self-similar sets have a self-similar [[measure (mathematics)|measure]] {{math|''μ{{sup|D}}''}} with dimension {{mvar|D}} given by the formula

<math display=block>\sum_{s\in S} (r_s)^D=1 </math>

which is often (but not always) equal to the set's [[Hausdorff dimension]] and [[packing dimension]]. If the overlaps between the {{math|''f{{sub|s}}''(''K'')}} are "small", we have the following simple formula for the measure:

<math display=block>\mu^D(f_{s_1}\circ f_{s_2} \circ \cdots \circ f_{s_n}(K)) = (r_{s_1}\cdot r_{s_2}\cdots r_{s_n})^D.\,</math>