Tuesday, May 30, 2017

Spiral Phyllotaxis patterns in OpenSCAD

Spiral Phyllotaxis

In just uploaded a YouTube video about Spiral Phyllotaxis patterns in OpenSCAD. This time it’s a script that I made to demonstrate Spiral Phyllotaxis. Phyllotaxis is a term used for patterns that emerge during the growth of plants. Spiral Phyllotaxis is observed in the heads of sunflowers, in pine-cones and pineapples, and in a variety of other plants.

The script that I wrote uses a mathematical description of Spiral Phyllotaxis called Vogel’s formula. Vogel’s formula actually exists of two equations, one for an angle theta, and one for a radius, describing the a pattern of seeds in a sunflower head. I’ll put a link in the description if you want to know more about Vogel’s formula. These simple equations can generate beautiful patterns that have some interesting mathematical properties.

The number of spirals derived from Vogel’s formula has a close relationship with the Fibonacci sequence. Exactly 55 spirals go counterclockwise, 34 smaller spirals go clockwise and 21 even smaller spirals go counterclockwise and so on. These numbers of spirals are all integers in the Fibonacci sequence.

The Golden Angle or Fibonacci Angle of 137.5 degrees is key in these Spiral Patterns. This angle results in the best distribution of the seeds. A slightly smaller or larger angle leads to a less optimized distribution.

Not only can the Spiral Pattern be examined in OpenSCAD. A big bonus of the program is that the user can create an stl file that can be printed. The physical model can then be studies further.

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