# Deborah R. Fowler

## Phyllotactic Pattern

Updated on June 22 2013

Updated with wrangles/lsystems on Sept 10 2016

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.

Other sites include:

http://www.math.smith.edu/phyllo//EXPO/ExpoIntro.html

http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.htm

http://demonstrations.wolfram.com/topic.html?topic=Plant+Biology&limit=20

Here is a brief description. Recently I have implemented this pattern in Houdini, C++/OpenGL, Python and Scratch as well as in the 90's with GL/C on Silicon Graphics Workstations (for those of you who remember what those are).

Now we have adj, which is our x coordinate and opp, which is our y coordinate, thus we have ( x, y )

Suppose now that r, θ also are allowed to vary.

Where do these equations come from? Well from a paper by H Vogel 1979 in Mathematical Biosciences.

However, what this has done is reduced the problem to optimal packing on a disc.

When we substitute these equations in for r and θ it results in the phyllotactic pattern.

### Implementing this in Houdini or Python
or ...

So now you have the formulas, now what? You can use just about any graphics interface/programming language as a playground.

Exercise 1 VSFX350/721 will be implementing this in Houdini. Click here for some additional notes.

If you would like to see an implementation in

- Scratch, click here
- Python, click here
- Mel, click here
- Python in Houdini click
here

- Point Wrangle node in Houdini, click here
- Point Vop in Houdini, click
here

- Lsystems in Houdini, 15.5.565 hip file