JoHN MiLLeR
I've had a decades-long fascination with geometric trees. When the iPad came out, I resolved to employ its multi-touch technology in a tree-drawing app, which I named "Geom-e-Tree". The app enabled me to re-explore the tree form, going into areas I had never ventured. I saw trees snap into grids and shape themselves into polygons and pinwheels. I was shocked to see a three-branched tree form a five-pointed star. I was disoriented because I didn't understand where all these special trees were located in the parameter space, but gradually, patterns emerged. To share this new knowledge, I released a poster of 150 Geom-e-Trees at the Gathering for Gardner last March, and gave a short talk, "What Shape is a Tree?". With this gallery entry, I invite others to explore this wild and beautiful treescape, mathematically and esthetically.
This is a three-branched tree with two 144ยบ obtuse angles between branches at each node throughout the tree. The branches grow in length by the Golden Ratio (phi) on each level away from the trunk in the center. Ternary Tree Star, Version 1.618 is one of an infinity of 5-pointed stars with common ratios ranging from phi (1.618...) up to two (2.0). I found that the Golden Ratio organizes the lines crossing through the interior of the star more highly than other common ratios do, so that a geometric pattern is formed by the white space as well. Ternary Tree Star is also a member of a larger family of polygon-, grid-, and star-shaped patterns formed by geometric trees (Geom-e-Trees) at special angles. [Technical note: The 1800x1800 art catalog image uses 88,574 lines. The web gallery image is downsized to 600x600. The 24 inch digital print has 265,721 lines.]