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第67行： − + + + − − It is sometimes said that fractals 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 , 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 , but the scaling holds only for values of for integer . 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 and internal rays are invariant curves .
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{{Short description|Features that do not change if length or energy scales are multiplied by a common factor}}
{{Short description|Features that do not change if length or energy scales are multiplied by a common factor}}
[[File:Wiener process animated.gif|thumb|right|500px|The [[Wiener process]] is scale-invariant.|链接=Special:FilePath/Wiener_process_animated.gif]]
[[File:Wiener process animated.gif|thumb|right|500px|The [[Wiener process]] is scale-invariant.
维纳过程具有标度不变性。|链接=Special:FilePath/Wiener_process_animated.gif]]
In [[physics]], [[mathematics]] and [[statistics]], '''scale invariance''' is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
In [[physics]], [[mathematics]] and [[statistics]], '''scale invariance''' is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
===Fractals 分形===
===Fractals 分形===
[[File:Kochsim.gif|thumb|right|250px|A [[Koch curve]] is [[self-similar]].|链接=Special:FilePath/Kochsim.gif]]
[[File:Kochsim.gif|thumb|right|250px|A [[Koch curve]] is [[self-similar]].
科赫雪花具有自相似性。|链接=Special:FilePath/Kochsim.gif]]
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.
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.
有时人们认为分形是标度不变的，尽管更准确地来说，应该说分形是自相似的。分形通常是在某个{{mvar|λ}}值的离散集合内等同于其本身，即使这样，有时也需要通过平移和旋转变换来实现。
有时人们认为分形是标度不变的，尽管更准确地来说，应该说分形是自相似的。分形通常是在某个{{mvar|λ}}值的离散集合内等同于其本身，即使这样，有时也需要通过平移和旋转变换来实现。
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.
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.
因此，以{{math|∆ {{=}} 1}}的'''Koch Curve 科赫雪花'''缩放为例，但是该缩放只适用于{{math|''λ'' {{=}} 1/3<sup>''n''</sup>}}，（{{mvar|n}}为整数）的值。此外，科赫雪花不仅在初始点，而且在某种意义上，在整条曲线上都可以找到其“缩影”。
因此，以{{math|∆ {{=}} 1}}的'''Koch Curve 科赫雪花'''缩放为例，但是该缩放只适用于{{math|''λ'' {{=}} 1/3<sup>''n''</sup>}}，（{{mvar|n}}为整数）的值。此外，科赫雪花不仅在初始点，而且在某种意义上，在整条曲线上都可以找到其“缩影”。
Some fractals may have multiple scaling factors at play at once; such scaling is studied with [[multi-fractal analysis]].
Some fractals may have multiple scaling factors at play at once; such scaling is studied with [[multi-fractal analysis]].
某些分形可能同时具有多个标度因子，可以应用'''Multi-Fractal Analysis 多重分形分析'''进行研究。
某些分形可能同时具有多个标度因子，可以应用'''Multi-Fractal Analysis 多重分形分析'''进行研究。
Periodic [[External ray|external and internal rays]] are invariant curves .
Periodic [[External ray|external and internal rays]] are invariant curves .
周期性外部和内部射线是不变的曲线。
周期性外部和内部射线是不变的曲线。