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		1.2.1— Two Types of Self-Similarity

		1— Geometrical Self-Similarity

		In geometrical self-similarity little pieces of an object are exact smaller copies of the whole object.

		The little pieces are geometrically similar to the whole object.

		This is usually true only for mathematically defined objects.

		2— Statistical Self-Similarity

		The little pieces of real biological specimens are usually not exact smaller copies of the whole object. Thus, they cannot be geometrically self-similar. However, the little pieces of real biological specimens can be kind-of-like their larger pieces. The statistical properties of the little pieces can be geometrically similar to the statistical properties of the big pieces. This is called statistical self-similarity.

		For example, the statistical property could be the length of the perimeter of an organ. Statistical self-similarity means that the length measured at one resolution is geometrically similar, that is, proportional to the length measured at other resolutions. If is the length measured at resolution r, and is the length measured at resolution ar, then where k is the constant of proportionality.

		The statistical properties of an object are described by the number of pieces of each size that make up the object. The function that tells how many pieces of each size that make up the object is called the probability density function (pdf). The formal mathematical definition of statistical self-similarity is that the probability density function (pdf) measured at resolution r is geometrically similar, that is, has the same shape, as the probability density function (pdf) measured at resolution ar.

		We use the shorter expression ''self-similarity" to denote the statistical self-similarity of biological objects in space or processes in time. This means that the smaller pieces are like the larger pieces but they are not exact copies of the larger pieces.

 

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