- 3123 - FRACTAL MATH - Fibonacci’s fractals? Could fractals be the link to the very small world of quantum mechanics and the very large world of galaxies and galaxy clusters? Maybe the whole Universe was formed just like breeding rabbits.
- ----------------------- 3123 - FRACTAL MATH - Fibonacci’s fractals?
- In 1202 Leonardo Pisano, whose pen name was Ribonacci, wrote down this problem: How many pairs of rabbits will be produced from a single pair if each pair gives birth to a new pair ( one male one female) each month starting the second month after birth? Each rabbit pair takes 2 months to mature before starting to spawn more rabbits.
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- Start a chart with months on one axis and rabbit pairs on the other. The monthly totals are 1,1,2,3,5,8,13,21 …….. out to the 8th month. This series is known as the Fibonacci sequence after the inventor Leonardo Pisano.
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- Note carefully that each number is the sum of the two previous numbers. There are 21 pairs of rabbits in 8 months. So, to continue the series: 1,1,2,3,5,8,13,21,34,55,89,144 for the 12th month. You have 144 pairs, or 288 rabbits at the end of the year.
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- If each pair of rabbits matured and reproduced each month, after one month instead of two months how many rabbit pairs would you have at the end of the year?
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- Use the same method to get a sequence of 1,2,4,8,16,……. and you get 4,096 pairs of rabbits in 12 months. 8,192 rabbits total when the number doubles each month for 12 months (2^12). You need a bigger farm.
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- If it took 3 months for rabbits to mature and reproduce you would have 41 rabbits at the end of the year. The monthly sequence would work like this: 1,1,1,2,3,4,6,9,
Each number in the sequence is equal to the sum of the previous number and the number three places back in the sequence: 1+3=4, 2+4=6, 3+6=9, …….
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- The Fibonacci sequence of numbers is F(n+1) = F(n) + F(n-1). If you calculate the slope, or growth rate, of Fibonacci’s numbers you get F(n+1) / F(n) which converges to the Golden Ratio.
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- The Golden Ratio = 1.61803399.…………. the further you go in the sequence the more accurate the approximation to this irrational number it becomes.
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- The Golden Ratio appears in the Parthenon, Leonardo da Vinci’s paintings, Frank Lloyd Wright’s architecture. Volumes of books have been written about the Golden Ratio. It is an irrational number that artists claim creates a great aesthetic appeal.
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- The seeds of sunflowers spiral outward according to Fibonacci’s numbers. Same with pinecones, rings of palm trees, snail shells, plant leaves, pineapples. The growth rate of these numbers, called the Golden Ratio is the only number that is equal to its own reciprocal +1. GR = 1/GR+1
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------------------------------- GR^2 = GR + 1.
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- Therefore from the quadratic equation: GR^2 - GR -1 = 0 of which the Golden Ratio is the solution, = 1.6180339887.………
The Golden Ratio is represented by the continuous fraction: 1+1/(1+1/1+1/1+1/………..
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------------------------------- GR = a/b = b/(a-b)
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------------------------------- “a” is the larger of two numbers “a” and “b”
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---------------------- a/b = 1/(a/b-1) = GR = its reciprocal minus 1: a = 1.618 and b = 1
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- The spread of matter throughout space follows a pattern laid out at the beginning of time and scaled up to incredible proportions by nearly 14 billion years of expansion. Matter is spread more or less evenly throughout space.
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- The fractal falls into the same patterns in large and small versions at whatever scale you look, galaxies, clusters of galaxies, or super clusters of galaxies. In other words the Universe is a fractal.
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- A fractal is a geometric shape that when subdivided at any scale into parts that are reduced-size copies of the whole. Fractals were first defined in 1975. Today astronomers are thinking the Universe is an irregular distribution of matter that never evens out into a smooth structure but repeats itself at ever grander scales.
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- Fractals are also everyday things. We see then in the branches of trees, the curves of coastlines, lungs, turbulence, clouds. Movie sets use simple computer programs that generate fractals creating clouds, mountains, coastlines, and trees in the backgrounds.
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- Think of broccoli, a tiny branch looks much the same as the whole vegetable. Fractals can be beautiful to look at.
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- The galaxy and galaxy cluster data to date suggests that the galaxies are fractal regardless of whether someone can explain why. It even suggests that space-time itself is a fractal.
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- Einstein’s theory of relativity is based on the laws of physics remaining the same regardless of the motion of the observer. An extension of that theory being proposed is that the laws of physics remain the same regardless of the scale at which the Universe is being observed. If that is the case the underlying space-time of such a theory would have to be a fractal.
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- Could fractals be the link to the very small world of quantum mechanics and the very large world of galaxies and galaxy clusters? Yes the whole Universe was formed just like breeding rabbits.
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- Astronomers are looking for this pattern to see if they can figure it out. The fact that mathematics can so well represent the realty of the world we live in is amazing. Does man invent math to fit reality, or, is it already there and does man discover it?
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- Fibonacci (1175-1250) was the pen name for Leonardo Pisano of Pisa, Italy. He was the greatest mathematician of the Middle Ages. He solved problems that took mathematicians 300 years later to repeat. In 1202 he introduced Europe to the Hindu-Arabic numerals and the decimal system that we use today. He wrote the book “Book of the Abacus” ( Liber Aback).
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- April 13, 2021 FRACTAL MATH - Fibonacci’s fractals? 752 3123
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