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第93行：
第93行： − 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。一个分形通常是等于它自己的一个{{mvar|λ}}值离散集合，即使这样，一个平移和旋转可能不得不应用到匹配分形到它自己。+ 第99行：
第99行： − 因此，举例来说，Koch 曲线的尺度为，但是该尺度只适用于整数的值。此外，科氏曲线不仅在起源处，而且在某种意义上说，“到处”都是科氏曲线的缩影: 曲线上到处都可以找到科氏曲线本身的缩影。+ 第105行：
第105行： − 某些分形可能同时具有多个标度因子，这种标度是用多重分形分析研究的。+ 第111行：
第111行： − 周期性的外部和内部射线是不变的曲线。+
→Fractals 分形
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.
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.
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.
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.
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.
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]].
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。
Periodic [[External ray|external and internal rays]] are invariant curves .
Periodic [[External ray|external and internal rays]] are invariant curves .
Periodic external and internal rays are invariant curves .
Periodic external and internal rays are invariant curves .
周期性外部和内部射线是不变的曲线。
==Scale invariance in stochastic processes==
==Scale invariance in stochastic processes==