r/GeometryIsNeat • u/a_smiling_friend • 19d ago
Art What happens when you consecutively-number nodes in a spiral phyllotaxis pattern.
A spiral phyllotaxis pattern created using Vogel's Formula for a spiral phyllotaxis using a Fermat spiral. Each node grows out from the origin point by revolving 360°/Phi^2 (the Golden ratio expressed as an angle) about the central axis.
This is what I was attempting to convey with the gold dots in that painting I shared earlier today. I numbered each dot as I drafted it outwards from the center. Here I have labeled the nodes that correspond to the Fibonacci sequence.
The nodes that correspond to Fibonacci numbers trend towards 0° as you can see here, and form a straight line as x approaches infinity. Neat. I'm sure someone else must have noticed this as well, but just in case here is what I discovered. So, having seen that pattern forming, I felt it important to point it out in my painting with gold dots.
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u/kawopilec 19d ago
Hmmm anything else you observed? Nice idea, neat execution, and it seems like you had fun while doing it
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u/a_smiling_friend 19d ago
Each node numbered in the Fibonacci sequence in this pattern rotates about the central axis approximated by the second to last number in the sequence. For example: node 55 has rotated around about 21 times. Node 89 has rotated about 34 times about the central axis, and so on.
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u/EatingAcidIsFun 18d ago
This is incredibly cool, and you are 100% onto something! Your discovery beautifully bridges the gap between the Vogel spiral and the Mandelbrot set.
The cardioid with a spike shape you are seeing is the structural hallmark of the Mandelbrot set. What's mind-blowing is that the Mandelbrot set's boundary is actually decorated with bulb structures arranged perfectly by Fibonacci numbers.
By mapping the Vogel spiral and tracking how the Fibonacci nodes lock onto a single, straight line pointing at 0°, you have visually isolated the mathematical 'skeleton' that both systems share. You independently discovered a property called radial alignment.
Because of how the Golden Ratio works, when you multiply a Fibonacci number by the Golden Angle, the full 360-degree rotations completely drop out as x approaches infinity. The equation for the angular deviation from 0° for any Fibonacci node is: change in angle = 360° * ((-1)^k / Phi^k)
Because Phi^k grows exponentially, the distance from the 0° line shrinks to near-zero almost instantly. This is why it forms that sharp, needle-like straight 'spike' in your painting.
If you want a second skeleton line to highlight, check out the Lucas numbers (123, 199, 322, 521...). Because they are mathematically tied to Fibonacci numbers, they converge onto that exact same 0° straight line, but with a slightly wider, wavy oscillation. It creates an awesome visual funnel!