Phyllotaxis Packing on Surfaces of Revolution

Tools: Rhinoceros, Grasshopper, Python

There’s something magical about stepping back in time, especially during the holiday season. This year, why not add a touch of nostalgia to your Christmas tree by creating your own vintage-inspired toys ?

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When I think of my childhood Christmas tree, a few unique ornaments stand out in my memory—ones shaped like corn, fruits, or pinecones.

In this special edition, we’re revisiting those organic designs by recreating them using simple mathematical formulas that mimic the natural patterns found in plants, known as phyllotaxis. For instance, in pinecones, corn, and pineapples, the seeds or kernels are often arranged in spiral formations that follow distinct Fibonacci sequences. Plants “came up with” the this strategy as a result of millions of years of evolution and natural selection. If you’re curious about the math behind these patterns, check out my earlier post on phyllotaxis.

This time, we’ll use the same principles to craft decorations with a unique twist: we want these patterns to wrap around any surface of revolution.

There are several ways to achieve this goal. Before, we used trigonometry to create polar coordinates for a flat model and cylindrical coordinates to add height. The number of points was set as an input, and we enlarged the surface area by increasing this number. This time, we’ll take a different approach: integration

I stumbled upon this approach in a video by Entagma, which itself references the formula from the paper “The Use of Positional Information in the Modeling of Plants.” The video demonstrates how to generate phyllotactic patterns on surfaces of revolution using Houdini with VEX code. We’ll adapt this code with couple of changes:

1. We rewrite th code for Grasshopper in python.
2. We’ll introduce a feature to adjust the size of the “organs”.

This method determines the precise position of each successive organ on the revolved surface. For a detailed explanation, please refer to the paper or the video, but here’s a brief summary:

First, we need a curve that, when rotated around an axis, generates a surface of revolution. We also define the size of the organs we want to place. Starting at the initial point of the curve, t=0, we move step by step toward the end point, t=1. The distance we travel along the curve is denoted as S, and each step corresponds to an increment, ΔS. As S increases, whenever it grows enough to accommodate an organ, we place one, rotate by a fixed angle, and then move forward.

As a result, the pattern will cover the entire surface. The number of points is an output primarily determined by the ‘size’ of the organ and the total area of the surface.

The “classic” phyllotactic angle is approximately 137.5°, a value derived from the golden ratio and commonly observed in nature. I left it as an input for a more general approach.

Here’s the code I used:

import Rhino.Geometry as rg
import math

a = 0.0 
s = 0.0 
i = 0   

points = []
curve_length = curve.GetLength() 

while s < curve_length:
    while a < 1 and s < curve_length:
        t = curve.LengthParameter(s) 
        t = t[1]
        pos = curve.PointAt(t)
        x = pos.X
        a += (2 * x / (size ** 2)) * deltas
        s += deltas
        
    a -= 1
    new_pos = rg.Point3d(pos.X,0,pos.Z)
    
    rotation = rg.Transform.Rotation(math.radians(angle)*i, rg.Vector3d.ZAxis, rg.Point3d.Origin)  
    new_pos.Transform(rotation)
    
    points.append(new_pos)
    i += 1

a represents the area.
s corresponds to S.
i is a counter — not strictly necessary, but it’s generally a good practice

Curve is planar, drawn in the XZ plane. I used global Z-axis as the center of rotation.

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As you might notice, the points here are evenly distributed, maintaining the same distance from each other. This is because the variable area is fixed. In nature, however, plant organs typically grow larger as they develop further from the center. To create a more natural appearance, we need to introduce a variable that allows the area to increase proportionally to its position.

Here is an example of the same surface covered with spheres (a simplified representation of the organ). In the image on the left, with 561 points, the areas are the same size. In contrast, the image on the right, with 765 points, shows areas scaled according to their distance from the center.

Here, I used a variable called ratio to adjust the organs size. To calculate it, I first determine the circumference C of a circle centered on the Z-axis in the XY plane, positioned at t₀ of our curve, with the radius defined as the distance from the axis to t₀ ( X coordinate in my case). For each S in the loop, I calculate the C for tₛ and then find the ratio between C(tₛ) and C(t₀). This results in a list of ratios for each point, which can then be used to scale the spheres accordingly

import Rhino.Geometry as rg
import math

a = 0.0
s = 0.0
i = 0

C = 2*math.pi * curve.PointAtStart.X 

points = []
ratios  = []
curve_length = curve.GetLength()

while s < curve_length:
    while a < 1 and s < curve_length:
        t = curve.LengthParameter(s) 
        t = t[1]
        pos = curve.PointAt(t)
        x = pos.X
        current_C = 2 * math.pi * x
        ratio = current_C/C
        a += (2 * x / (size ** 2 * ratio)) * deltas
        s += deltas
        
    a -= 1
    new_pos = rg.Point3d(pos.X,0,pos.Z)
    
    rotation = rg.Transform.Rotation(math.radians(angle)*i, rg.Vector3d.ZAxis, rg.Point3d.Origin)
    new_pos.Transform(rotation)
    
    points.append(new_pos)
    ratios.append(ratio)
    i += 1

This ratio is not fixed and will vary depending on the desired outcome. You can reduce it (for example, by using its square root) for a smaller difference, or increase it for a more noticeable effect.

You can alternate the organs using a call pattern component. Below are a few examples with colors:

You can also experiment with angles other than the golden ratio:

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Now all you need is access to a 3D printer, and voilà!

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The principles of phyllotaxis aren’t limited to plants—they’re part of broader physical and mathematical laws that govern growth and packing. Similar patterns appear in systems like galaxy spirals and crystal formations, showing that phyllotaxis is a fundamental principle of nature.

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Resources:

1. Prusinkiewicz, Przemyslaw, Lars Mündermann, Radoslaw Karwowski, and Brendan Lane. ‘The Use of Positional Information in the Modeling of Plants’. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, 289–300. ACM, 2001. https://doi.org/10.1145/383259.383291.
2. McCormack, Jon. ‘A Method for Generating Phyllotaxis over Surfaces of Revolution’, n.d.
3. https://www.youtube.com/watch?v=yGwhnt7mZ50&ab_channel=Entagma
4. https://blog.wolfram.com/2011/07/28/how-i-made-wine-glasses-from-sunflowers/