The Golden Ratio

[rocketman]

From posting this information I am trying to get it through my thick skull what exactly the science and math is behind the golden ration and Fibonacci numbers. I dont know how I missed this in my education, maybe I was sick those days or wasted.....dunno. Everyone is welcome to try and beat it through to me in simple terms, well cuz I R simple. If it is already as simple as its going to be, then call me a dumbass and carry on. Math is and always has been my achilles heel. I would love to see god in numbers like Bucky and others have! Help a brother out?


The golden ratio in nature.
So, why do shapes that exhibit the Golden Ratio seem more appealing to the human eye? No one really knows for sure. But we do have evidence that the Golden Ratio seems to be Nature's perfect number. Take, for example, the head of a daisy:


Somebody with a lot of time on their hands discovered that the individual florets of the daisy (and of a sunflower as well) grow in two spirals extending out from the center. The first spiral has 21 arms, while the other has 34. Do these numbers sound familiar? They should - they are Fibonacci numbers! And their ratio, of course, is the Golden Ratio. We can say the same thing about the spirals of a pinecone, where spirals from the center have 5 and 8 arms, respectively (or of 8 and 13, depending on the size)- again, two Fibonacci numbers:



A pineapple has three arms of 5, 8, and 13 - even more evidence that this is not a coincidence. Now is Nature playing some kind of cruel game with us? No one knows for sure, but scientists speculate that plants that grow in spiral formation do so in Fibonacci numbers because this arrangement makes for the perfect spacing for growth. So for some reason, these numbers provide the perfect arrangement for maximum growth potential and survival of the plant.

all text from University of Chicago edu site.

What is the Golden Ratio?

Well, before we answer that question let's examine an interesting sequence (or list) of numbers. We'll start with the numbers 1 and 1. To get the next number we add the previous two numbers together. So now our sequence becomes 1, 1, 2. The next number will be 3. What do you think the next number in the sequence will be? If you said 4, then unfortunately you are incorrect. Remember, we add the previous two numbers to get the next. So the next number should be 2+3, or 5. Here is what our sequence should look like if we continue on in this fashion for a while:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . (I understand this, and have come across this on some tests.) back to the problem at hand.

Now, I know what you might be thinking: "What does this have to do with the Golden Ratio?" The answer is forthcoming. This sequence of numbers was first discovered by a man named Leonardo Fibonacci, and hence is known as Fibonacci's sequence. The relationship of this sequence to the Golden Ratio lies not in the actual numbers of the sequence, but in the ratio of the consecutive numbers. Let's look at some of these ratios:

2/1 = 2.0

3/2 = 1.5

5/3 = 1.67

8/5 = 1.6

13/8 = 1.625

21/13 = 1.615

34/21 = 1.619

55/34 = 1.618

89/55 = 1.618

Aha! Notice that as we continue down the sequence, the ratios seem to be converging upon one number (from both sides of the number)! Notice that I have rounded my ratios to the third decimal place. If we examine 55/34 and 89/55 more closely, we will see that their decimal values are actually not the same. But what do you think will happen if we continue to look at the ratios as the numbers in the sequence get larger and larger? That's right: the ratio will eventually become the same number, and that number is the Golden Ratio! The Golden Ratio is what we call an irrational number: it has an infinite number of decimal places and it never repeats itself! Generally, we round the Golden Ratio to 1.618. We work with another important irrational number in Geometry: pi, which is approximately 3.14. Since we don't want to make the Golden Ratio feel left out, we will give it its own Greek letter: phi. One more interesting thing about Phi is its reciprocal. If you take the ratio of any number in the Fibonacci sequence to the next number (this is the reverse of what we did before), the ratio will approach the approximation 0.618. This is the reciprocal of Phi: 1 / 1.618 = 0.618. It is highly unusual for the decimal integers of a number and its reciprocal to be exactly the same. In fact, I cannot name another number that has this property! This only adds to the mystique of the Golden Ratio and leads us to ask: What makes it so special?

[Rhyno]

http://www.youtube.com/watch?v=wS7CZIJVxFY

Works for me. GLUK!

[zodd]

duhh!!
sorry just demonstrating my utter lack of mathematical ability carry on dude. its al very interesting from my simple pov.
nice link rhyno

[SilvrHairDevil]

See if you can get a copy of Donald Duck in Math Magic Land.

http://www.youtube.com/watch?v=P_ssR7M5Px0

[wildburr]

Originally Posted by SilvrHairDevil

See if you can get a copy of Donald Duck in Math Magic Land.

http://www.youtube.com/watch?v=P_ssR7M5Px0

I'll be damned, I actually do have a copy of that.

[stonedcoder]

very interesting Rocket Man ...does this relate to everything in nature being fractals ...maybe the golden ratio is related to the fractional growth of living things... i r dumb 2 - so we'll both wait until some math heavy hitters chime in eh?

[stonedcoder]

contents pumped to attachment - list of numbers numbers.txt