The mathematical underpinnings

tammybrown · 2024-04-26 02:20:07

Could someone kindly elucidate the fascinating connection between the Fibonacci sequence and the golden ratio, especially considering its frequent appearances in the natural world? I'm eager to understand the mathematical underpinnings of this phenomenon.

Bob · 2024-04-26 04:48:28

hi tammybrown

Welcome to the forum.

Oh bliss! You have asked about one of my favourite topics and I've got more than one post's worth to tell.

Let's start with snowdrops. They grow in woods; coming up usually early in the year; flower; then die back and disappear from the woodland floor. But, next year, up they come again; mostly growing from the bulbs they made the previous year. They can grow from seed; that's how they spread around the wood; but they also spread by budding off a second bulb from the first.

There are biological rules for this. A seed may grow that year then all the nutrients in the leaves are sucked back into a bulb which acts as a store for fresh growth next year. Each bulb produces a single plant stem and flower so it's easy to count the bulbs without having to dig them up. A first year bulb has only enough nutrients to produce a plant stem and flower but a mature bulb (ie. more than one year old) can also bud off a new first year bulb.

So at the end of year 1 the plant makes a first year bulb for next year. At the end of the second year that bulb gets 'promoted' to a mature bulb and so produces a new first year bulb. That's two bulbs for the third year.

At the end of the third year the mature one produces another first year bulb and the first year bulb gets promoted. So we have two matures and one first year.

At the end of the fourth year the two matures produce two first years, and the first year gets promoted. That's three matures and two first years.

I'll put all of this into a table:

end of year firsts matures
1 1 0
2 1 1
3 1 2
4 2 3
5 3 5
6 5 8
7 8 13
..................................................................

It doesn't always work because an animal may eat a bulb or a human may step on a plant and kill it but the majority of clumps have a number of flowers(=bulbs) in the Fibonacci sequence. When a disaster occurs and the clump loses a bulb the rule still applies but maybe not with the familiar numbers eg 1 1 2 3 disaster one lost 2 remain, 4,6,10,16 etc.

Here's some homework for you to try. If you can do it on a spreadsheet that will save you a lot of arithmetic but you can do it on paper with a calculator.

Write out the Fibonacci sequence. Create a new sequence by calculating each number divided by the one before:

eg 1/1 = 1 2/1 = 2 3/2 = 1.5 ..............and so on.

What do you notice about your new sequence? What else about it?

Bob

nycguitarguy · 2024-04-26 10:09:01

Bob wrote:

hi tammybrown
Welcome to the forum.
Oh bliss! You have asked about one of my favourite topics and I've got more than one post's worth to tell.
Let's start with snowdrops. They grow in woods; coming up usually early in the year; flower; then die back and disappear from the woodland floor. But, next year, up they come again; mostly growing from the bulbs they made the previous year. They can grow from seed; that's how they spread around the wood; but they also spread by budding off a second bulb from the first.
There are biological rules for this. A seed may grow that year then all the nutrients in the leaves are sucked back into a bulb which acts as a store for fresh growth next year. Each bulb produces a single plant stem and flower so it's easy to count the bulbs without having to dig them up. A first year bulb has only enough nutrients to produce a plant stem and flower but a mature bulb (ie. more than one year old) can also bud off a new first year bulb.
So at the end of year 1 the plant makes a first year bulb for next year. At the end of the second year that bulb gets 'promoted' to a mature bulb and so produces a new first year bulb. That's two bulbs for the third year.
At the end of the third year the mature one produces another first year bulb and the first year bulb gets promoted. So we have two matures and one first year.
At the end of the fourth year the two matures produce two first years, and the first year gets promoted. That's three matures and two first years.
I'll put all of this into a table:
end of year firsts matures
1 1 0
2 1 1
3 1 2
4 2 3
5 3 5
6 5 8
7 8 13
..................................................................
It doesn't always work because an animal may eat a bulb or a human may step on a plant and kill it but the majority of clumps have a number of flowers(=bulbs) in the Fibonacci sequence. When a disaster occurs and the clump loses a bulb the rule still applies but maybe not with the familiar numbers eg 1 1 2 3 disaster one lost 2 remain, 4,6,10,16 etc.
Here's some homework for you to try. If you can do it on a spreadsheet that will save you a lot of arithmetic but you can do it on paper with a calculator.
Write out the Fibonacci sequence. Create a new sequence by calculating each number divided by the one before:
eg 1/1 = 1 2/1 = 2 3/2 = 1.5 ..............and so on.
What do you notice about your new sequence? What else about it?
Bob

What math course covers Fibonacci and golden ratio ideas?

Bob · 2024-04-26 20:02:26

The 'Maths Is Fun' course, of course.

Thinking back I cannot recall it ever coming up in any maths course. It may feature in biology courses. It might even arise in art and Greek ancient history.

But does it matter where it arises. For me the big thing is how it connects some very different areas of interest: just a few that I know of: art, snail shells, sunflowers, spirals, rationals and irrationals, rabbits, converging sequences, continued fractions, the missing square magic trick, bees.

I will post lesson 2 when someone provides an answer to my homework.

Bob

nycguitarguy · 2024-04-27 05:20:26

Bob wrote:

The 'Maths Is Fun' course, of course.
Thinking back I cannot recall it ever coming up in any maths course. It may feature in biology courses. It might even arise in art and Greek ancient history.
But does it matter where it arises. For me the big thing is how it connects some very different areas of interest: just a few that I know of: art, snail shells, sunflowers, spirals, rationals and irrationals, rabbits, converging sequences, continued fractions, the missing square magic trick, bees.
I will post lesson 2 when someone provides an answer to my homework.
Bob

I decided to let 4 be my starting number.

4/2 = 2

5/2 = 2.5

6/2.5 = 2.4

7/2.4 = 2.91666666667

8/2.91666666667 = 2.74285714286

9/2.74285714286 = 3.28125

I notice that the sequence gets closer and closer to 3.
Is this in any way related to the idea of limits in Calculus 1?

Bob · 2024-04-27 05:52:06

Thanks for trying but this thread is specifically about the Fibonacci Sequence.

https://www.mathsisfun.com/numbers/fibo … uence.html

So you division sequence starts 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...........

Bob

nycguitarguy · 2024-04-27 08:52:51

Bob wrote:

Thanks for trying but this thread is specifically about the Fibonacci Sequence.
https://www.mathsisfun.com/numbers/fibo … uence.html
So you division sequence starts 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...........
Bob

I need to watch videos on YouTube to learn the basics of this sequence if I am to ever understand this material.

Math Is Fun Forum

#1 2024-04-26 02:20:07

The mathematical underpinnings

#2 2024-04-26 04:48:28

Re: The mathematical underpinnings

#3 2024-04-26 10:09:01

Re: The mathematical underpinnings

#4 2024-04-26 20:02:26

Re: The mathematical underpinnings

#5 2024-04-27 05:20:26

Re: The mathematical underpinnings

#6 2024-04-27 05:52:06

Re: The mathematical underpinnings

#7 2024-04-27 08:52:51

Re: The mathematical underpinnings

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