Fractals Cayley tree

Scaling relationships and fractal dimensions of the fractal Cayley tree... 

The procedure of fractal set construction can be shown using the Cayley tree so that each fractal set has its own Cayley tree [25,26]. We show a Cayley tree with branch characteristic j = 4. 

The Cayley tree is a pictorial representation of a space that is called ultrametric. Each point of the ultrametric space can be put into correspondence with an element of the fractal set that is, the fractal set and ultrametric space are topologically equivalent sets. We remark that the main feature of an ultrametric space, as well as that of a fractal set, is its hierarchical property. 

This approximate equation means that the ultrametric space has a logarithmic metric. Thus, when constructing a fractal set, each element corresponds to a point of the ultrametric space with geometric image represented by the Cayley tree. 

In many cases, the patterns created via WA-CMB have the characteristics of a Cayley tree structure (a type of fractal structure). The structure of the crystal networks can be then correlated to the rheological properties in terms of fractal dimension. 

Note that Eqns. 33 to 35 of the Cayley tree model, and Eqns. 27 and 28 of the square-channel model are predictions for the same quantity the total current leaving the tree through its "reactive" side-walls. While the square-charmel model is restricted to describe branches of equal width and length, the Cayley the model carries no such limitation. This scaling is quantified by inclusion of the trees two fractal dimensions, Dtree and Damopy, implied by the length and width ratios p and q, respectively. 

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