Dimensions of Fractal Trees

Self-Contacting Trees

With simple trigonometry these sequences are converted into geometric series, giving r as a function of q.

Measuring angles from the dashed line in the figure, the x-coordinate of the tip of the red branch is

rsin(-q) + r³sin(q) + r⁴sin(2q) + r⁵sin(3q) + r⁶sin(2q) + r⁷sin(3q) + ...

with some simplification (summing geometric series) the condition that this x-coordinate is 0 becomes

rsin(-q) + r³sin(q) + (r⁴/(1 - r²))sin(2q) + (r⁵/(1 - r²))sin(3q) = 0

For q = 40 this gives r = 0.577325.

For each value of q, exactly one value of r scales the branches to self-contact.

Note the equations will be different if a different pattern of Left and Right turns is needed before settling down into the LR cycle.

This pattern depends upon the angle, of course. We have considered only one example here.

Return to Self-Contacting Trees.