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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