- #1
Ventrella
- 29
- 4
Does there exist a binary fractal tree…
(reference: http://ecademy.agnesscott.edu/~lriddle/ifs/pythagorean/symbinarytree.htm )
…whose leaves (endpoints) lie on a circle and are equidistant?
Consider a binary fractal tree with branches decreasing in length by a scaling factor r (0 < r < 1) for each consecutive generation of branching. My first question is: do there exist two numbers representing the left and right branching angles θ1 and θ2 - being constant for every generation of branchings - such that the smallest branches converge on a circle at the limit? My initial hunch is that the answer is no, given the various shapes found by adjusting these angles, and some intuitions about fractals not being very good at imitating Euclidian geometry.
As a thought experiment: what if the lengths and angles could be anything? What if the tree could be drawn in a free-form manner (as if you were doodling on paper), with the only constraint being that it must be topologically equal to a binary fractal tree? In this case, it seems that the answer would be yes. But I want to preserve some of the classic tree’s beautiful self-similar nature by way of some elegant branching and length relationships, and to essentially come up with an expression that has as few factors as possible.
Specifically: can such a fractal tree exist if the scaling factor r is kept constant, and if the branch angles θ1 and θ2 change by some delta d per generation of branching? If so, I suspect that θ1 and θ2 would have to change in a nonlinear way in order to enable convergence to a circle. If the algorithm is sufficiently elegant and compact, then it could be used as a thought-provoking (if impractical) expression of a circle.
(reference: http://ecademy.agnesscott.edu/~lriddle/ifs/pythagorean/symbinarytree.htm )
…whose leaves (endpoints) lie on a circle and are equidistant?
Consider a binary fractal tree with branches decreasing in length by a scaling factor r (0 < r < 1) for each consecutive generation of branching. My first question is: do there exist two numbers representing the left and right branching angles θ1 and θ2 - being constant for every generation of branchings - such that the smallest branches converge on a circle at the limit? My initial hunch is that the answer is no, given the various shapes found by adjusting these angles, and some intuitions about fractals not being very good at imitating Euclidian geometry.
As a thought experiment: what if the lengths and angles could be anything? What if the tree could be drawn in a free-form manner (as if you were doodling on paper), with the only constraint being that it must be topologically equal to a binary fractal tree? In this case, it seems that the answer would be yes. But I want to preserve some of the classic tree’s beautiful self-similar nature by way of some elegant branching and length relationships, and to essentially come up with an expression that has as few factors as possible.
Specifically: can such a fractal tree exist if the scaling factor r is kept constant, and if the branch angles θ1 and θ2 change by some delta d per generation of branching? If so, I suspect that θ1 and θ2 would have to change in a nonlinear way in order to enable convergence to a circle. If the algorithm is sufficiently elegant and compact, then it could be used as a thought-provoking (if impractical) expression of a circle.
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