Post by CIVILISON on Dec 8, 2004 12:17:57 GMT -5
Peace,
Leonardo Fibonacci was an Italian mathematician who came up with the Fibonacci sequence. It is a set of numbers which are obtained by adding the consecutive number i.e. 0,1,1,2,3,5,8,13,21… He came up with this sequence while observing the birth pattern of rabbits.
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. Therefore:
1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. 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.
As a result, we get this:
Or this:
The left side of the graph represents the moths whereas the right represents the amount of pairs of rabbits. Thereby, the Fibonacci sequence is clearly seen.
Another example can be demonstrated with cows.
The English puzzlist, Henry E Dudeney (1857 - 1930, pronounced Dude-knee) wrote several excellent books of puzzles (see after this section). In one of them he adapts Fibonacci's Rabbits to cows, making the problem more realistic in the way we observed above. He gets round the problems by noticing that really, it is only the females that are interesting - er - I mean the number of females!
He changes months into years and rabbits into bulls (male) and cows (females).
“If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?”
Another Fibonacci sequence!
Same rule is prevalent with honeybees.
There are some drone bees who are male and do no work. Males are produced by the queen's unfertilized eggs, so male bees only have a mother but no father!
So female bees have 2 parents, a male and a female whereas male bees have just one parent, a female.
Let's look at the family tree of a male drone bee.
1. He had 1 parent, a female.
2. He has 2 grand-parents, since his mother had two parents, a male and a female.
3. He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one.
4. How many great-great-grand parents did he have?
Here is a graph:
great- great,great gt,gt,gt
grand- grand- grand grand
Number of parents: parents: parents: parents: parents:
of a MALE bee: 1 2 3 5 8
of a FEMALE bee: 2 3 5 8 13
The Fibonacci sequence gives birth to the Fibonacci spiral. Here is a demonstration:
Such spirals are found throughout nature. Here is some examples of some sea shells.
The Fibonacci sequence and spiral corresponds to the physical structure of many plants.
On many plants, the number of petals is a Fibonacci number:
buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.
3 petals: lily, iris. Often lilies have 6 petals formed from two sets of 3.
4 petals Very few plants show 4 petals (or sepals) but some, such as the fuchsia above, do. 4 is not a Fibonacci number! We return to this point near the bottom of this page.
5 petals: buttercup, wild rose, larkspur, columbine (aquilegia), pinks
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria, some daisies
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae family.
Some species are very precise about the number of petals they have - eg buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.
Look at this passion flower.
Back view:
the 3 sepals that protected the bud are outermost,
then 5 outer green petals followed by an inner layer of 5 more paler green petals
Front view:
the two sets of 5 green petals are outermost,
with an array of purple-and-white stamens (how many?);
in the centre are 5 greenish stamens (T-shaped) and
uppermost in the centre are 3 deep brown carpels and style branches)
Fibonacci numbers can also be seen in the arrangement of seeds on flower heads!
Pine Cones:
Cauliflower:
The point of this is to provide the base for the notion that the creation of the world is based upon an intelligence which is manifested in various forms that can be calculated to a precise mathematical detail. It is noteworthy to mention that these sequences and spirals are prevalent with the whole universe. Look at your fingertips, galaxies, the spinning motion of the atom, DNA strains etc. Entertain and deeply contemplate the following quotes:
And.
In Peace,
I-Son
SOURCE: Fibonacci Numbers and Nature
Leonardo Fibonacci was an Italian mathematician who came up with the Fibonacci sequence. It is a set of numbers which are obtained by adding the consecutive number i.e. 0,1,1,2,3,5,8,13,21… He came up with this sequence while observing the birth pattern of rabbits.
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. Therefore:
1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. 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.
As a result, we get this:
Or this:
The left side of the graph represents the moths whereas the right represents the amount of pairs of rabbits. Thereby, the Fibonacci sequence is clearly seen.
Another example can be demonstrated with cows.
The English puzzlist, Henry E Dudeney (1857 - 1930, pronounced Dude-knee) wrote several excellent books of puzzles (see after this section). In one of them he adapts Fibonacci's Rabbits to cows, making the problem more realistic in the way we observed above. He gets round the problems by noticing that really, it is only the females that are interesting - er - I mean the number of females!
He changes months into years and rabbits into bulls (male) and cows (females).
“If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?”
Another Fibonacci sequence!
Same rule is prevalent with honeybees.
There are some drone bees who are male and do no work. Males are produced by the queen's unfertilized eggs, so male bees only have a mother but no father!
So female bees have 2 parents, a male and a female whereas male bees have just one parent, a female.
Let's look at the family tree of a male drone bee.
1. He had 1 parent, a female.
2. He has 2 grand-parents, since his mother had two parents, a male and a female.
3. He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one.
4. How many great-great-grand parents did he have?
Here is a graph:
great- great,great gt,gt,gt
grand- grand- grand grand
Number of parents: parents: parents: parents: parents:
of a MALE bee: 1 2 3 5 8
of a FEMALE bee: 2 3 5 8 13
The Fibonacci sequence gives birth to the Fibonacci spiral. Here is a demonstration:
Such spirals are found throughout nature. Here is some examples of some sea shells.
The Fibonacci sequence and spiral corresponds to the physical structure of many plants.
On many plants, the number of petals is a Fibonacci number:
buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.
3 petals: lily, iris. Often lilies have 6 petals formed from two sets of 3.
4 petals Very few plants show 4 petals (or sepals) but some, such as the fuchsia above, do. 4 is not a Fibonacci number! We return to this point near the bottom of this page.
5 petals: buttercup, wild rose, larkspur, columbine (aquilegia), pinks
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria, some daisies
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae family.
Some species are very precise about the number of petals they have - eg buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.
Look at this passion flower.
Back view:
the 3 sepals that protected the bud are outermost,
then 5 outer green petals followed by an inner layer of 5 more paler green petals
Front view:
the two sets of 5 green petals are outermost,
with an array of purple-and-white stamens (how many?);
in the centre are 5 greenish stamens (T-shaped) and
uppermost in the centre are 3 deep brown carpels and style branches)
Fibonacci numbers can also be seen in the arrangement of seeds on flower heads!
Pine Cones:
Cauliflower:
The point of this is to provide the base for the notion that the creation of the world is based upon an intelligence which is manifested in various forms that can be calculated to a precise mathematical detail. It is noteworthy to mention that these sequences and spirals are prevalent with the whole universe. Look at your fingertips, galaxies, the spinning motion of the atom, DNA strains etc. Entertain and deeply contemplate the following quotes:
“The universe is created by a consciousness hich manifests in physical reality through a geometric blueprint that we call Sacred Geometry which repeats over and over giving the illusion of linear time.”
[/b][/center]And.
“All forces and particles in the world give evidence of being guided by an intelligence that established a universal set of patterns for them to follow. We have already seen that one of these patters – the Fibonacci numbers – arise out of the spiral motion that is the cause of the condensation of the matter that is responsible for the explosion.”
[/b][/center] In Peace,
I-Son
SOURCE: Fibonacci Numbers and Nature
