Phyllotactic Patterns are a good start to using math for fun. There are a number of sites describing this pattern, including The Algorithmic Beauty of Plants, specifically Chapter 4 where I originally implemented this formula.
Deborah R. Fowler
Phyllotactic Pattern
Mathematical Foundation
Reduced to optimal packing on a disc, these equations stem from H Vogel 1979 in Mathematical Biosciences.
With adj (x coordinate) and opp (y coordinate), we have (x, y). When r and θ vary, we substitute the following formulas:
Implementation Resources
You can use just about any graphics interface/programming language. In class (VSFX350/721), we use Houdini. Additional Notes.
- Scratch: View projects
- Python: Turtle Sunflowers
- Mel: Mel Phyllotaxis
- Houdini (Python): Examples
- Houdini (VEX): Point Wrangle snippets
- Houdini (VOPs): VOPs comparison
- L-Systems: HIP file (Rule:
+(137.508)[F(i^0.5,1,1,1)]A(i+1)) - 3D Printing: OpenSCAD/BlocksCAD
Note on Wrangle Precision: Using
137.5 causes patterns to break at high counts.
An approximation of 137.508 is better. For extreme cases (90k+ points), enable 64-bit precision in the Bindings tab.
Additional Examples
Check out the Zoetrope Phyllotactic Pattern for more details.
