Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts).

Description

Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales.

Fractals

Self-similarity is a typical property of fractals.

Scale invariance

Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole.

For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape.

Characteristics

The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales.

Counterexample

As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

See also

• Chaos theory
• Fractal
• Mathematical object
• Mathematics
• Scale invariance
• Self-reference
• Similarity (geometry)

External links

• Self-similarity @ Wikipedia