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Using several structural elements with associated probabilities of occurrence produces a family of recognizably related but individual forms (Figure 10b). It requires very few rules, which can be com- pactly encoded genetically, to distinguish maple trees from oaks, for example. Figure 11a shows tree-like branching with characteristic angles defined by atomic structure in growth of dendritic crystals. Fractal dendritic crystal growth is a prob- lem that results in shorting of rechargeable batter- ies, but it also produces the infinite variations of snow crystals. Figure 11 also shows the branching patterns


in neurons and in the airways in the human lung. Many anatomical structures, such as the human vascular system, are fractal, and differences in the dimensions oſten distinguish healthy from abnormal conditions [9]. A fractional dimen- sion for these branching patterns relates the total measured length to the resolution of ever-smaller branches or tributaries. A cumulative log-log plot of total length as a function of the length of included skeleton branches provides a measure of the fractal dimension. When Mandelbrot [10] introduced the term


“fractal” to describe structures that have more and more detail present at ever finer scales, one exam- ple was the length of the coast of Britain. Plotting the measured length as a function of the ruler used for measurement [11] produces a straight line plot on log-log scales, whose slope gives the fractional dimension. Tere are more convenient ways to determine the dimension, one of which is shown


Figure 9: Creation of a Sierpinski gasket as described in the text.


Figure 10: Iterating a branching shape forms a tree: (a) varying the rules alters the resulting tree shape; (b) using probabilities for different branch patterns produces a family of different but recognizably related forms.


Figure 11: Natural branching shapes: (a) growth of copper dendrites; (b) neurons in the brain; (c) 3D reconstruction of airways in the human lung. 2020 March • www.microscopy-today.com 15


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