Cosmologic philosophy: fibonacci sequence (Introduction)

by David Turell , Saturday, January 20, 2024, 20:02 (308 days ago) @ David Turell

It is everywhere:

https://bigthink.com/starts-with-a-bang/what-explains-fibonacci-sequence/?utm_campaign=...

"One of the most fascinating facts about the natural world is that so many entities within it — both biologically and purely physically — obey a specific set of patterns and ratios. Many galaxies exhibit spiral shapes and structures, as do a wide variety of plant structures: pinecones, pineapples, and sunflower heads among them. Ammonites, shelled animals that went extinct more than 60 million years ago, also show that spiral pattern, where one of the key features of spirals is that the next “wind” around outside the prior one displays a specific length ratio to the size of the prior, interior winding.

"That ratio, in any such structure, is often extremely close to the ratio of two adjacent numbers found in the Fibonacci sequence. This mathematical sequence, often taught to children, simply starts with the numbers “0” and “1” and then gets the next term in the sequence by adding the two prior terms together. It’s arguably the most famous mathematical sequence of all, but what explains the sequence’s pattern, and is it truly, inextricably linked to nature?

***

"The few spirals that do show that Fibonacci-like pattern are a part of a class of spirals known as Grand Design Spiral Galaxies, and these represent only about 1-in-10 spiral galaxies, as opposed to the most common types with multi-arm spirals (including the Milky Way) and the second most common type with subtle, many-laned spiral structure known as flocculent spiral galaxies. These “grand design” spirals are almost exclusively galaxies that have recently undergone or are currently undergoing a gravitational interaction with a nearby companion galaxy, and it’s only that external gravitational influence that pulls the outermost arms and features into shapes that are more consistent with ratios found within the Fibonacci sequence.

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"However, the Fibonacci-like patterns and ratios found in many biological organisms, including in plants, truly are related to the Fibonacci sequence: both in a mathematically rigorous fashion and also for an evolutionary reason that makes perfect sense. Let’s tackle the biological properties first, and return to the mathematics.

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"It turns out there’s nothing special about the starting point of the Fibonacci sequence, either. You can start with any two non-negative numbers that you like where at least one of them is non-zero: they need not be “0” and “1,” they need not be whole numbers, they need not be close together. All you need to do is follow the same formula, where you take the first two numbers and add them together to make the next (third) number, and then add that number with the previous to make the next subsequent number, and so on. No matter which numbers you start with, the ratio of any two successive numbers will quickly approach φ, the golden ratio. (my bold)

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"...the original sequence, i.e., the sum of the numbers in the Fibonacci sequence sorted by decimal places, equals 0.1/8.9, or 1/89. And that’s why the Fibonacci sequence isn’t inherent to nature, but rather, to pure mathematics instead. It appears in nature because the golden ratio has a biological utility, but wherever it appears in the physical sciences, including in some spiral galaxies, it’s only by pure coincidence!"

Comment: only a small number of galactic spirals are Fibonacci. But it turns up in many organisms. The golden ratio is mentioned and is an ancient discovery:

"Golden ratio, in mathematics, the irrational number (1 + Square root of√5)/2, often denoted by the Greek letter ϕ or τ, which is approximately equal to 1.618. It is the ratio of a line segment cut into two pieces of different lengths such that the ratio of the whole segment to that of the longer segment is equal to the ratio of the longer segment to the shorter segment. The origin of this number can be traced back to Euclid, who mentions it as the “extreme and mean ratio” in the Elements."

https://www.britannica.com/science/golden-ratio