In mathematics, the Fibonacci sequence is a sequence where each number is the sum of the previous two, and where the first two terms, by definition are 1 and 1.
F1 = 1
F2 = 1
Fn = Fn - 1 + Fn - 2
The first numbers of the Fibonacci sequence are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
The sequence is named after the mathematician Leonardo Fibonacci. (Figure 1 below)
The Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci. Fibonacci considers the growth of an idealized rabbit population (Figure 2 below), assuming that: 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; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on.
Fibonacci numbers are closely related to golden ratio.
Fn / Fn - 1 = Φ
where
Φ = 1.6180339887...
the relation between a Fibonacci number and its following is the mutual of the golden ratio
1 / Φ = 0.6180339887...
The Fibonacci sequence appear everywhere in nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.
Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem as the Phyllotaxis (Figure 3 below). The leaves on this plant are staggered in a spiral pattern so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
In the case of pineapples (Figure 4 below) or pinecones (Figure 5 below), we see a double set of spirals, one going in a clockwise direction and one in the opposite direction. When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers.
Similarly, sunflowers (Figure 6 below) have a golden spiral seed arrangement. This provides a biological advantage because it maximizes the number of seeds that can be packed into a seed head.
As well, many flowers have a Fibonacci number of petals (Figure 7 below):
Humans exhibit Fibonacci characteristics, too. The golden ratio is seen in the proportions in the sections of a finger (Figure 10 below). We have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand.r
No one has discover yet the reason why the Fibonacci sequence appears so frequently in nature, but I think it is really interesting how many natural things follow this sequence.
F1 = 1
F2 = 1
Fn = Fn - 1 + Fn - 2
The first numbers of the Fibonacci sequence are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
The sequence is named after the mathematician Leonardo Fibonacci. (Figure 1 below)
The Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci. Fibonacci considers the growth of an idealized rabbit population (Figure 2 below), assuming that: 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; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on.
- At the end of the first month, they mate, but there is still 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 produces yet another new pair, the female born two months ago produces her first pair also, making 5.
Fibonacci numbers are closely related to golden ratio.
Fn / Fn - 1 = Φ
where
Φ = 1.6180339887...
the relation between a Fibonacci number and its following is the mutual of the golden ratio
1 / Φ = 0.6180339887...
The Fibonacci sequence appear everywhere in nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.
Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem as the Phyllotaxis (Figure 3 below). The leaves on this plant are staggered in a spiral pattern so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
In the case of pineapples (Figure 4 below) or pinecones (Figure 5 below), we see a double set of spirals, one going in a clockwise direction and one in the opposite direction. When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers.
Similarly, sunflowers (Figure 6 below) have a golden spiral seed arrangement. This provides a biological advantage because it maximizes the number of seeds that can be packed into a seed head.
As well, many flowers have a Fibonacci number of petals (Figure 7 below):
- 3 petals: lily and iris
- 5 petals: buttercup, wild rose, larkspur, columbine and pinks
- 8 petals: delphiniums
- 13 petals: ragwort, corn marigold, cineraria and some daisies
- 21 petals: aster, black-eyed susan and chicory
- 34 petals: plantain and pytethrum
- 55, 89 petals: michelmas daisies and the asteraceae family
Humans exhibit Fibonacci characteristics, too. The golden ratio is seen in the proportions in the sections of a finger (Figure 10 below). We have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand.r
No one has discover yet the reason why the Fibonacci sequence appears so frequently in nature, but I think it is really interesting how many natural things follow this sequence.
Figure 1 Figure 2
Figure 3 Figure 4
Figure 5 Figure 6
Figure 7 Figure 8
Figure 9 Figure 10