Editing Scale invariance (section)

==Scale-invariant curves and self-similarity==
In mathematics, one can consider the scaling properties of a [[function (mathematics)|function]] or [[curve]] {{math|''f'' (''x'')}} under rescalings of the variable {{mvar|x}}.  That is, one is interested in the shape of {{math|''f'' (''λx'')}} for some scale factor {{mvar|λ}}, which can be taken to be a length or size rescaling. The requirement for  {{math|''f'' (''x'')}} to be invariant under all rescalings is usually taken to be
:<math>f(\lambda x)=\lambda^\Delta f(x)</math>
for some choice of exponent&nbsp;Δ, and for all dilations {{mvar|λ}}. This is equivalent to {{mvar|f}} &nbsp;  being a [[homogeneous function]] of degree&nbsp;Δ.

Examples of scale-invariant functions are the [[monomial]]s <math>f(x)=x^n</math>, for which {{math|Δ {{=}} ''n''}}, in that clearly

:<math>f(\lambda x) = (\lambda x)^n = \lambda^n f(x)~.</math>

An example of a scale-invariant curve is the [[logarithmic spiral]], a kind of curve that often appears in nature. In [[polar coordinates]] {{math|(''r'', ''θ'')}}, the spiral can be written as

:<math>\theta = \frac{1}{b} \ln(r/a)~.</math>

Allowing for rotations of the curve, it is invariant under all rescalings {{mvar|λ}}; that is, {{math|''θ''(''λr'')}} is identical to a rotated version of {{math|''θ''(''r'')}}.

===Projective geometry===
The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a [[homogeneous polynomial]], and more generally to a [[homogeneous function]]. Homogeneous functions are the natural denizens of [[projective space]], and homogeneous polynomials are studied as [[projective varieties]] in [[projective geometry]].  Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of [[scheme (mathematics)|schemes]], it has connections to various topics in [[string theory]].

===Fractals===
[[File:Kochsim.gif|thumb|right|250px|A [[Koch curve]] is [[self-similar]].]]
It is sometimes said that [[fractal]]s are scale-invariant, although more precisely, one should say that they are [[self-similar]]. A fractal is equal to itself typically for only a discrete set of values {{mvar|λ}}, and even then a translation and rotation may have to be applied to match the fractal up to itself.

Thus, for example, the [[Koch curve]] scales with {{math|∆ {{=}} 1}}, but the scaling holds only for values of {{math|''λ'' {{=}} 1/3<sup>''n''</sup>}} for integer {{mvar|n}}. In addition, the Koch curve scales not only at the origin, but, in a certain sense, "everywhere": miniature copies of itself can be found all along the curve.

Some fractals may have multiple scaling factors at play at once; such scaling is studied with [[multi-fractal analysis]].

Periodic [[External ray|external and internal rays]] are invariant curves .