So it all started with a challenge that was posted on the Xenodream pix list to work with the Pythagorean Tree. As illustrated below, The Pythagoras tree is a plane fractal constructed from squares. It is named after Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. Garth, the creator of Xenodream had posted the initial .xep file with the simple repeating squares. From the left each image is an iteration of this function growing exponentially more complex. The top is a 45,45,90 triangle and the bottom is a 30, 60, 90 triangle resulting in a lopsided tree.

Since squares are 2 dimensional I decided to add a third dimension by making the squares into cubes and exploring the geometry in 3 dimensional space. The tree itself was interesting but since I’m reading Samuel Colmans “Harmonic Proportion and Form” I decided to test some of the ideas he’s pushing in the book; I decided to play with angles derived from the geometries of platonic solids and lo and behold I got a Dragon Curve. The cubes were rendered 95% transparent so you can see the geometry inside of it.

(above) Pythagoras Dragon Curve .xep
(below) Same image with outlines instead of cubes and not viewed from an angle. .xep

 

I continued to play with the angles of rotation I found an interesting setting: if you rotate the left square 135 degrees and the right 225 degrees you end up with perfectly nestled fractal geometry of the Pythagorean Triangle, which ends up extending and filling a space proportionate to the original triangle. The squares themselves are subdivided by four of the triangles illustrating their geometric relation. I suspect the white lines are due to the manner we are drawing lines in Xenodream and that perhaps the geometry would fit together perfectly if I were to model it more precisely, which I will probably do with Processing at some later date. Click the image to explore the 4000+ pixel render of this image!

Pythagoras Triangle Fractal (above) and close up of detail (below) .xep

 

Below are a few other images I made playing around with this shape

geomeTree

Below is a side by side comparison of the by now familiar tree and below that the same geometry with a little variation; instead of a square I created a shape that was very very vaguely trunk like. I apologise for the mid ’90s 3D quality of this image I wasn’t setting out to create a realistic tree, Just to see how convincing the branching patterns were. And for the little amount of work that went into it I’m impressed.

very 90’s looking 3D tree .xep

While researching for this article I found an artist named Koos Verhoeff who has made a bronze sculpture using this base geometry.

All the xep files which are for Xenodream are released under a WTFPL public license and you can do whatever you want with them.
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This article was posted on Wednesday, January 9th, 2008 in the category: Fractal.
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