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Fibonacci Numbers and The Golden Section

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**1. **Fibonacci Numbers and The Golden Section

**2. **Who was Fibonacci? The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa
The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa

**3. **The four works from this period which have come down to us are:
Around 1200, Fibonacci returned to Pisa where, for at least the next twenty-five years, he worked on his own mathematical compositions.Around 1200, Fibonacci returned to Pisa where, for at least the next twenty-five years, he worked on his own mathematical compositions.

**4. **Fibonacci's Mathematical Contributions 1 2 3 4 5 6 7 8 9 and 0
Roman Numerals
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000
For instance, 13 would be written as XIII or perhaps IIIX.
2003 would be MMIII or IIIMM. 99 would be LXXXXVIIII and 1998 is MDCCCCLXXXXVIII
For example, XI means 10+1=1 but IX means 1 less than 10 or 9. 8 is still written as VIII (not IIX)
He was one of the first people to introduce the Hindu-Arabic number system into Europe - the positional system we use today - based on ten digits with its decimal point and a symbol for zero:
His book on how to do arithmetic in the decimal system, called Liber abbaci (meaning Book of the Abacus or Book of Calculating) completed in 1202 persuaded many European mathematicians of his day to use this "new" system.
The method in use in Europe until then used the Roman numerals:
You can still see them used on foundation stones of old buildings and on some clocks.
Later, an abbreviation became popular where the order of letters did matter and, if a single smaller value came before the next larger one, it was subtracted and if it came after, it was added as usual.He was one of the first people to introduce the Hindu-Arabic number system into Europe - the positional system we use today - based on ten digits with its decimal point and a symbol for zero:
His book on how to do arithmetic in the decimal system, called Liber abbaci (meaning Book of the Abacus or Book of Calculating) completed in 1202 persuaded many European mathematicians of his day to use this "new" system.
The method in use in Europe until then used the Roman numerals:
You can still see them used on foundation stones of old buildings and on some clocks.
Later, an abbreviation became popular where the order of letters did matter and, if a single smaller value came before the next larger one, it was subtracted and if it came after, it was added as usual.

**5. **The Fibonacci Series
Hindu-Arabic Number System The Fibonacci series
is formed by starting with 0 and 1 and then adding the latest two numbers to get the next one: 0 1 --the series starts like this. 0+1=1 so the series is now 0 1 1 1+1=2 so the series continues... 0 1 1 2 and the next term is 1+2=3 so we now have 0 1 1 2 3 and it continues as follows ...
On these pages, we shall say the first Fibonacci numbers is 1, as is the second, the third being 2, the fourth 3 and so on. We will often include Fib(0)=0 too:
The Fibonacci series
is formed by starting with 0 and 1 and then adding the latest two numbers to get the next one: 0 1 --the series starts like this. 0+1=1 so the series is now 0 1 1 1+1=2 so the series continues... 0 1 1 2 and the next term is 1+2=3 so we now have 0 1 1 2 3 and it continues as follows ...
On these pages, we shall say the first Fibonacci numbers is 1, as is the second, the third being 2, the fourth 3 and so on. We will often include Fib(0)=0 too:

**6. **Patterns in the Fibonacci Numbers & Cycles in the Fibonacci Numbers Here are some patterns people have already noticed:
Pattern Number 1
0,1,1,2,3,5,8,13,21,34,55,...
Pattern Number 2
00, 01, 01, 02, 03, 05, 08, 13, ...
For the last three digits, the cycle length is 1,500
For the last four digits,the cycle length is 15,000
For the last five digits the cycle length is 150,000
and so on...
There is a cycle in the units column - the cycle of units digits (0,1,1,2,3,5,8,13,21,34,55,...) repeats from n=60 and again every 60 values.
There is also a cycle in the last two digits, repeating (00, 01, 01, 02, 03, 05, 08, 13, ...) from n=300 with a cycle of length 300.
There is a cycle in the units column - the cycle of units digits (0,1,1,2,3,5,8,13,21,34,55,...) repeats from n=60 and again every 60 values.
There is also a cycle in the last two digits, repeating (00, 01, 01, 02, 03, 05, 08, 13, ...) from n=300 with a cycle of length 300.

**7. **Fibonacci's Rabbits
The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances. Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...
How many pairs will there be in one year?
At the end of the first month, they mate, but there is still one only 1 pair.
At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...
How many pairs will there be in one year?
At the end of the first month, they mate, but there is still one only 1 pair.
At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

**8. **Fibonacci Puzzles Making a bee-line with Fibonacci numbers
Here is a picture of a bee starting at the end of some cells in its hive. It can start at either cell 1 or cell 2 and moves only to the right (that is, only to a cell with a higher number in it). There is only one path to cell 1, but two ways to reach cell 2: directly or via cell 1. For cell 3, it can go 123, 13, or 23, that is, there are three different paths. How many paths are there from the start to cell number n? Here is a picture of a bee starting at the end of some cells in its hive. It can start at either cell 1 or cell 2 and moves only to the right (that is, only to a cell with a higher number in it). There is only one path to cell 1, but two ways to reach cell 2: directly or via cell 1. For cell 3, it can go 123, 13, or 23, that is, there are three different paths. How many paths are there from the start to cell number n?

**11. **What is the Golden Section (or Phi)? (Also called The Divine Proportion)

**12. **The Golden Section in Architecture The Parthenon and Greek Architecture
Even from the time of the Greeks, a rectangle whose sides are in the "golden proportion"
(1 : 1.618 which is the same as 0.618 : 1) has been known since it occurs naturally in some of the proportions of the Five Platonic Solids (as we have already seen). This rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in Athens, Greece. (There is a replica of the original building (accurate to one-eighth of an inch!) at Nashville which calls itself "The Athens of South USA".) The Acropolis, in the centre of Athens, is an outcrop of rock that dominates this ancient city. Its most famous monument, now largely ruined, is the Parthenon, a temple to the goddess "Athena" built around 430 or 440 BC. Though no original plans of the temple exist, it appears that the temple was built on a square-root-of-5 rectangle, that is, it is 5 times as long as it is wide. These are also the dimensions of the longest side view of the temple. Also, the front elevation is built on a Golden Rectangle, that is, it is Phi times as wide as it is tallhas been known since it occurs naturally in some of the proportions of the Five Platonic Solids (as we have already seen). This rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in Athens, Greece. (There is a replica of the original building (accurate to one-eighth of an inch!) at Nashville which calls itself "The Athens of South USA".) The Acropolis, in the centre of Athens, is an outcrop of rock that dominates this ancient city. Its most famous monument, now largely ruined, is the Parthenon, a temple to the goddess "Athena" built around 430 or 440 BC.Though no original plans of the temple exist, it appears that the temple was built on a square-root-of-5 rectangle, that is, it is 5 times as long as it is wide. These are also the dimensions of the longest side view of the temple. Also, the front elevation is built on a Golden Rectangle, that is, it is Phi times as wide as it is tall

**13. **Golden Section in Art is a picture that looks like it is in a frame of 1:sqrt(5) shape (a root-5 rectangle). Print it and measure it - is it a root-5 rectangle? Divide it into a square on the left and another on the right. (If it is a root-5 rectangle, these lines mark out two golden-section rectangles as the parts remaining after a square has been removed). Also mark in the lines across the picture which are 0·618 of the way up and 0·618 of the way down it. Also mark in the vertical lines which are 0·618 of the way along from both ends. You will see that these lines mark out significant parts of the picture or go through important objects. You can then try marking lines that divide these parts into their golden sections too.
is a picture that looks like it is in a frame of 1:sqrt(5) shape (a root-5 rectangle). Print it and measure it - is it a root-5 rectangle? Divide it into a square on the left and another on the right. (If it is a root-5 rectangle, these lines mark out two golden-section rectangles as the parts remaining after a square has been removed). Also mark in the lines across the picture which are 0·618 of the way up and 0·618 of the way down it. Also mark in the vertical lines which are 0·618 of the way along from both ends. You will see that these lines mark out significant parts of the picture or go through important objects. You can then try marking lines that divide these parts into their golden sections too.

**14. **Golden Section In Nature

**15. **Nature Continued

**16. **BIBLIOGRAPHY http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html