Before I tell you about the Fibonacci Ratio, I have to tell you first about the Fibonacci Series.
It's named for the guy who invented it ... Leonardo Pisano Bigollo Fibonacci, an Italian mathematician born in 1170 in Pisano, Italy.
The "Fibonacci Series" is a very special series of numbers. We find things that proceed according to the Fibonacci Series all around us in Nature, and it's used in Art and Architecture.
To make the Fibonacci Series of numbers, first write down " 0, 1 ". After that, add as many more numbers to the series as you want. Get EACH new number by ADDING the last two numbers that you already have. So the first 12 numbers of the series are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 . . . . .
Each number is the SUM of the 2 numbers just before it. Pretty easy. But it has at least a thousand interesting things about it. There have been whole books written about the series, and about the 'Fibonacci Ratio' that comes from the series.
Now for the Fibonacci Ratio:
A ratio is what you get when you divide two numbers. Every fraction is a ratio. Like, the ratio of 1/2 is 1÷2 = 0.5 . The ratio of 2/5 is 2÷5 = 0.4 . The ratio of a person's arms to legs is 2/2 = 1 .
To get the Fibonacci Ratio, take any pair of neighboring numbers in the Fibonacci Series and divide them.
The first pair is (0, 1). Their ratio is (0/1) = 0. Not too interesting.
The next pair is (1, 1). Their ratio is (1/1) = 1. Still not very interesting.
The next pair is (1, 2). Their ratio is (1/2) = 0.5 . Weird.
The next pair is (2, 3). Their ratio is (2/3) = 0.66.. Is something happening ?
The next pair is (3, 5). Their ratio is (3/5) = 0.60 . something is definitely happening.
The next pair is (5, 8). their ratio is (5/8) = 0.6250 . Now they're sticking around 0.6 ?
The next pair is (8, 13). Their ratio is (8/13) = 0.6154 . wow
The next pair is (13, 21). Their ratio is (13/21) = 0.6190 not changed much
The next pair is (21, 34). Their ratio is (21/34) = 0.6176 changed even less
The 11th pair is (55, 89). their ratio is (55/89) = 0.6180 smaller change
As you go along and keep taking the ratio of two neighbors in the Fibonacci Series, it almost looks like they're getting closer and closer to some special number where they want to settle down. And that is true. We've only looked at 11 pairs in the Fibonacci series, and their ratio is already settling down and changing very little from one pair to the next.
The number they're heading for is (1/2) · (1 + √5) .
This is the "Golden Ratio". It's the first ratio you get when you play around with the Fibonacci Series, like I just started showing you.
You can never write it down exactly with only numbers, because √5 is an "irrational number" that can never be completely written down as a decimal. But you can get as close to it as you want to be, by using a longer piece of the Fibonacci Series, and carrying out the division to more decimal places.
A few weird things about it:
-- Take the number. Divide 1 by it. Add 1 to the answer. You've got the number again. [ F = 1/F + 1 ]
-- Since the time of the ancient Greeks, people have always felt that this number is the way to draw and shape things that are most pleasing to us.
-- The (length/width) of several ancient Greek buildings is the Fibonacci Ratio.
-- The dimensions of the face of the "Mona Lisa" painting by Leonardo DaVinci have the Fibonacci Ratio in them.
-- If a snow plow starts out at 12:00 in a steadily-falling snow, and plows equal volumes of snow in equal amounts of time, and is able to make 2 miles of progress in the first hour and 1 mile of progress in the second hour, then the snow started falling (the Fibonacci Ratio of an hour) before 12:00 .
-- The leaves sprout out of a willow-tree twig in directions that are distributed according to the Fibonacci Series.
-- The patterns of markings on the back of a turtle's shell have the Fibonacci Ratio in them.
. . . and thousands of other Fibonacci things in nature. You should go on line and read about some of them.
= = = = = = = = = =
The 20th pair in the Fibonacci Series: (4181, 6765).
Some approximate numbers:
Their ratio is (4181, 6765) = 0.61803399...
The reciprocal is (6765/4181) = 1.61803396...
(1/2) (1 + √5) = 1.61803398...
1 / 1.61803399 = 0.6180398...
= initial launch speed = ?
= horizontal component of initial launch speed = 60 m/s
= vertical component of initial launch speed = 80 m/s