Agnes Scott College
Larry Riddle, Agnes Scott College

Golden 120° Symmetric Binary Tree

Taylor observed that there is an equilateral triangle that will contain three copies of a golden 120° symmetric binary tree that are self-contacting. The equilateral triangle has sides of length \(\sqrt{3}(1+\phi)\) when the trunk of each tree is of length 1. Click on the buttons to the left to fit the three copies inside this triangle. Each tree has its trunk positioned at the origin, which is the centroid of the triangle. One tree is rotated 120° clockwise and another is rotated 120° counterclockwise. The trees intersect at three branch tips. [Proofs]
 
 
 
 

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Six copies of the golden 120° tree will fit inside of a regular hexagon with the top edges of the trees along the edges of the hexagon. Each tree is rotated 60° as you move around the middle of the hexagon in a clockwise direction. The sides of the hexagon are of length \(\sqrt{3}\) when the trunks of the trees are of length 1.

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References

  1. Taylor, Tara. Computational Topology and Fractal Trees, Ph.D. Thesis, Dalhousie University (2005).
  2. Taylor, Tara. "Finding Gold in the Forest: Self-contacting Symmetric Binary Fractal Trees and the Golden Ratio".
  3. Taylor, Tara. "Excursions through a Forest of Golden Fractal Trees", in The Beauty of Fractals: 6 Different Views, eds. Denny Gulick and Jon Scott, Mathematical Association of America, 2010. [Excerpts available at Google Books]