Self-Contacting Scaling

A tree is called self-contacting if the tips of some left branches coincide with the tips of some right branches, but no tip coincides with any non-tip point of the tree. For each value of q, exactly one value of r scales the branches to self-contact. This graph shows that r as a function of q. See [FT] for a derivation of this graph.

The angles q = 90 and q = 135 can be viewed as critical angles: here both the tree and the set of branch tips form a filled-in rectangle (q = 90) and filled in triangle (q = 135). These angles alone give the maximum value, 2, for the dimension of the tree. As the angle approaches, achieves, and passes these two special values, the branching geometry undergoes a topological phase transition. These phase transitions are illustrated by the variable tree and variable tip animations.

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