The largest number of confusing articles we can find in nature regarding the origin of spirals in nature. No one knows why does it happen, and starts “sacred”, magical, hidden secret of the universe. Well, abundance of the spirals does make us amazed, but not enough to feel that it is god’s choice. “We find spirals in the DNA double helix, sunflowers, the path of draining water, weather patterns (including hurricanes), vine tendrils, phyllotaxis (the arrangement of leaves on a plant stem), galaxies, the horns of various animals, mollusc shells, the nautilus shell, snail shells, whirlpools, ferns and algae”. The reason for the spiral is simple.

There are four numbers e, phi and pi, they are also connected by i, e^2 + phi^2 = pi^2. Therefore if a term increases by unity fraction it is phi, if a number increases as production of fraction it is e, when both of their square increases it twists the angle to a difference plane, try an orthogonal triangle, you would see how 60 degree and 30 degree angles between the triangle would change, when it completes to circularly covered path we got to complete the pi. This is the reason for forming a spiral. Not the equation but the physical significance of e and phi suggests the most simplest form of growth of a material, one is additive and the other is multiplicative. Most of us are unaware of the physical significance of e and phi, a little google search would show that there are plenty of articles which suggest why and how these two series are the simplest route to account for growth.

The identity requires a 3D term to accommodate mismatch, e^2 + phi^2 + z^2 = pi^2. Then we get z= i/e this is also another series with a phase that changes the term. There is a orthogonal phase editing that helps creating a vortex from spiral.

Origin of spiral or vortex is hidden in the physical significance of e and phi, its continued fraction expressions, that brings them into a half circle.

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Hi, Anirban! I follow your great work. The phi equation you wrote is not exact, as you know, but there is an imaginary golden mean that can be find in reciprocal addition of parallel resistors, based on the equality Φ𝑖=𝑒±𝜋𝑖3

https://math.stackexchange.com/questions/1851698/imaginary-golden-ratio

https://mathforums.com/threads/imaginary-golden-ratio.17605/

The yin-yang symbol, that is a circle, a wave and an spiral all in one has it clearly embeded, but it was not discovered until recently: https://www.cut-the-knot.org/do_you_know/GoldenRatioInYinYang.shtml

Best, Miguel

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