Physics equations with the mathematical constant Phi?

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Phi, the golden ratio, is often associated with exponential growth and appears in biological contexts, such as the Fibonacci sequence and natural patterns like sunflower florets. However, its direct role in fundamental physics phenomena is less clear compared to Pi. Some discussions suggest that while Phi may not be involved in exponential growth itself, it can replace certain numerical values in mathematical expressions. The conversation highlights the intersection of mathematics and nature, emphasizing Phi's aesthetic and structural significance. Overall, the exploration of Phi in physics remains a topic of interest but lacks definitive applications in fundamental physics equations.
ole cram
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Does the math constant Phi (Φ = 1.618) or its inverse appear in "fundamental" physics formulae?
I know Phi appears often when modelling exponential growth and, probably because of that, also in Biology/Ecology. But does it appear spontaneously in the mathematical description of some fundamental physics phenomenon at all? (As does Pi, the ubiquitous irrational number)
Hope I'm posting on the right forum. Thanks in advance
 
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ole cram said:
I know Phi appears often when modelling exponential growth and, probably because of that, also in Biology/Ecology.
As far as I recall, phi doesn't play a role in exponential growth, but it does play a role in such things as the Fibonacci sequence and the spiral arrangement of the scales on pine cones, the florets on a sunflower, and other examples - https://awkwardbotany.com/2019/12/25/pine-cones-and-the-fibonacci-sequence/.
 
Any time the number 5 comes up you can replace it by (2Φ-1)2.
 
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