prompter: Why do Fibonacci numbers occur on so many different occasions in both mathematics and the real world?

assistant: Fibonacci numbers are defined by the recursive definition $F_{n+2} = F_{n+1} + F_n$, where $F_0 = 0$ and $F_1 = 1$. One can also start with the values $F_1 = 1$ and $F_2 = 1$, which yields a sequence that is the same (except for the first element). The Fibonacci sequence is one of the most simple linear recursions.

In mathematics, simple concepts are often building blocks in more complex structures, theorems and their proofs. Also, Mathematicians often start exploring the landscape of mathematics by starting with something simple and then exploring the boundaries of the thing they investigate. Consequently, the simply thing often never completely vanishes and even occurs again at some point in some shape or form. Taking this into consideration, it is not surprising that the Fibonacci sequence occurs so often in mathematics.

In nature, the Fibonacci numbers occur, among other things, in concepts like the reproduction of Rabbits or the arrangement of homogeneous structures like the spirals of the Roman cauliflower. These concepts naturally follow (approximately) the Fibonacci numbers due to things like the how Rabbit reproduction is "discretized" into generations and how the Fibonacci numbers naturally describe certain shapes in the real world. In addition to that, nature sometimes finds local evolutionary optima in simplicity. More precisely, evolution might lead to concepts that are simple to formulate, which may correlate to how random mutations may with higher probability lead to concepts that can be encoded into the genome in more simple terms compared to more difficult ones.

In my reply, I focused strongly on the concept of simplicity. But this may indeed be an important factor in both mathematics and nature. Occam's razor empirically plays an important part in science for a good reason.

prompter: This answer is a little complicated for me... can you break it down into more simpler terms?

assistant: Sure! I have simplified the answer for you:
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers. This sequence is found in nature and in math, and is often associated with simplicity. Simple concepts are important in both math and nature, and can lead to evolutionary advantages. The sequence occurs naturally in things like rabbit reproduction and the arrangement of certain structures. Simplicity is an important principle in science, and helps guide scientific investigation.

prompter: What are the importance of fibonacci series?

assistant: Certainly! The Fibonacci sequence is a simple pattern found in mathematics, where each number is the sum of the two numbers before it. This pattern is easy to understand and work with, so it shows up a lot in math problems and studies.

In nature, things like rabbit reproduction and some plant arrangements follow a pattern similar to the Fibonacci sequence. This happens because sometimes the simplest way to do things is the most effective. So, in both math and nature, the Fibonacci numbers show up because they're a simple and efficient pattern that works well in various situations.

prompter: Why would nature follows this specific "simplest way" of Fibonacci sequence? Is there not other sequences that are as simple or simpler?

assistant: Since the Fibonacci numbers follow a simple sequence they can commonly be found in many systems. Fibonacci patterns can be most commonly found amongst plants for efficient packing of similar sized features(e.g. spiky bits on a pineapple, or seeds in a sunflower). They're also found in system related to population growth.

However, while systems may appear to follow these numbers it does not reflect how all things grow in the natural world. And just because a series of numbers can be applied to an astonishing variety of objects that doesn't necessarily imply there's any correlation between figures and reality.

assistant: Scientists have pondered the question for centuries. In some cases, the correlation may just be coincidence. In other situations, the ratio exists because that particular growth pattern evolved as the most effective. In plants, this may mean maximum exposure for light-hungry leaves or maximized seed arrangement.

While experts agree that the Fibonacci sequence is common in nature, there is less agreement about whether the Fibonacci sequence is expressed in certain instances of art and architecture. Although some books say that the Great Pyramid and the Parthenon (as well as some of Leonardo da Vinci's paintings) were designed using the golden ratio, when this is tested, it's found to be false [source: Markowsky].

Mathematician George Markowsky pointed out that both the Parthenon and the Great Pyramid have parts that don't conform to the golden ratio, something left out by people determined to prove that Fibonacci numbers exist in everything. The term "the golden mean" was used in ancient times to denote something that avoided access in either direction, and some people have conflated the golden mean with the golden ratio, which is a more recent term that came into existence in the 19th century.
We celebrate Fibonacci Day Nov. 23rd not just to honor the forgotten mathematical genius Leonardo Fibonacci, but also because when the date is written as 11/23, the four numbers form a Fibonacci sequence. Leonardo Fibonacci is also commonly credited with contributing to the shift from Roman numerals to the Arabic numerals we use today.

prompter: is it possible that the pattern only shows up because we are looking for it? sort of like how we thought the golden ratio was in the pantheon when it was later discovered not to be.

assistant: Scientists have pondered the question for centuries. In some cases, the correlation may just be coincidence. In other situations, like in Nature, the ratio exists because that particular pattern evolved as the most effective. In plants, for example, this may mean maximum exposure for light-hungry leaves or maximized seed arrangement.

prompter: Why is this considered to be so important in scientific studies and observations in Nature?