# Nature, Fibonacci Numbers and the Golden Ratio

*The Fibonacci numbers are Nature’s numbering system. They 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. *— Stan Grist

## Part 1. Golden Ratio & Golden Section, Golden Rectangle, Golden Spiral

*The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form. –Adolf Zeising*

### Golden Ratio & Golden Section

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.

Expressed algebraically:

The golden ratio is often denoted by the Greek letter *phi*.

The figure of a golden section illustrates the geometric relationship that defines this constant. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

### Golden Rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: phi (one-to-phi).

A golden rectangle can be constructed with only straightedge

and compass by this technique:

- Construct a simple square
- Draw a line from the midpoint of one side of the square to an opposite corner
- Use that line as the radius to draw an arc that defines the height of the rectangle
- Complete the golden rectangle

### Golden Spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to j, the golden ratio. Specifically, a golden spiral gets wider (or further from its origin) by a factor of j for every quarter turn it makes.

*Successive points dividing a golden rectangle into squares lie ona logarithmic spiral which is sometimes known as the golden spiral.Image Source: http://mathworld.wolfram.com/GoldenRatio.html*

## Golden Ratio in Nature

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law. Zeising wrote in 1854:

The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.

**Examples:**

*Click on the picture for animation showing more examples of golden ratio.Source: *http://www.xgoldensection.com/xgoldensection.html

*Source:* http://www.goldennumber.net/hand.htm

*A slice through a Nautilus shell revealsgolden spiral construction principle.*

## Golden Ratio in Architecture and Art

Many architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. [*Source:* Wikipedia.org]

**Here are few examples:**

*Parthenon, Acropolis, Athens.This ancient temple fits almost precisely into a golden rectangle.Source: http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm*

*The Vetruvian Man”(The Man in Action)” by Leonardo Da VinciWe can draw many lines of the rectangles into this figure.Then, there are three distinct sets of Golden Rectangles:Each one set for the head area, the torso, and the legs.Image Source >>*

Leonardo’s *Vetruvian Man* is sometimes confused with principles of “golden rectangle”, however that is not the case. The construction of Vetruvian Man is based on drawing a circle with its diameter equal to diagonal of the square, moving it up so it would touch the base of the square and drawing the final circle between the base of the square and the mid-point between square’s center and center of the moved circle:

Detailed explanation about geometrical construction of the Vitruvian Man by Leonardo da Vinci >>

*The Second Pyramid at Giza (Khafre). Length of the slope side (342.5 royal cubits ) divided by half of the side (205.5 royal cubits ) is equal to 1.66666… which is very close to 1.656 (Vitruvian Man ratio). *

Leonardo’s *Vitruvian Man* is sometimes confused with principles of “golden rectangle”, however that is not the case. The construction of Vitruvian Man is based on drawing a circle with its diameter equal to diagonal of the square, moving it up so it would touch the base of the square and drawing the final circle between the base of the square and the mid-point between square’s center and center of the moved circle:

Detailed explanation about geometrical construction of the Vitruvian Man by Leonardo da Vinci >>

# Part 2. Fibonacci Numbers

#### About Fibonacci

Fibonacci was known in his time and is still recognized today as the “greatest European mathematician of the middle ages.” He was born in the 1170’s and died in the 1240’s and there is now a statue commemorating him located at the Leaning Tower end of the cemetery next to the Cathedral in Pisa. Fibonacci’s name is also perpetuated in two streetsthe quayside Lungarno Fibonacci in Pisa and the Via Fibonacci in Florence.

His full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa. He called himself Fibonacci which was short for Filius Bonacci, standing for “son of Bonacci”, which was his father’s name. Leonardo’s father( Guglielmo Bonacci) was a kind of customs officer in the North African town of Bugia, now called Bougie. So Fibonacci grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He then met with many merchants and learned of their systems of doing arithmetic. He soon realized the many advantages of the “Hindu-Arabic” system over all the others. He was one of the first people to introduce the Hindu-Arabic number system into Europe-the system we now use today- based of ten digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9. and 0

His book on how to do arithmetic in the decimal system, calledLiber abbaci(meaning Book of the Abacus or Book of calculating) completed in 1202 persuaded many of the European mathematicians of his day to use his “new” system. The book goes into detail (in Latin) with the rules we all now learn in elementary school for adding, subtracting, multiplying and dividing numbers altogether with many problems to illustrate the methods in detail. ( http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits )

## Fibonacci Numbers

The sequence, in which each number is the sum of the two preceding numbers is known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, … (each number is the sum of the previous two).

The ratio of successive pairs is so-called golden section (GS) – 1.618033989 . . . . .

whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.

The Fibonacci sequence, generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1,

is well known in many different areas of mathematics and science.

#### Pascal’s Triangle and Fibonacci Numbers

The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar KhayyÃ¡m.

Pascal’s Triangle is described by the following formula:

where is a binomial coefficient.

*The “shallow diagonals” of Pascal’s triangle sum to Fibonacci numbers.*

It is quite amazing that the Fibonacci number patterns occur so frequently in nature

( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal “laws of nature”. Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves The Fibonacci numbers are also found in the family tree of honeybees.

## Fibonacci and Nature

Plants do not know about this sequence – they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.

Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space. This is well described in several books listed here >>

So nature isn’t trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. That is why the spirals are imperfect.

The plant is responding to physical constraints, not to a mathematical rule.

The basic idea is that the position of each new growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989 . . . .

Examples of the Fibonacci sequence in nature.

#### Petals on flowers*

Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:

- 3 petals: lily, iris
- 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
- 8 petals: delphiniums
- 13 petals: ragwort, corn marigold, cineraria,
- 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 – e.g. buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.

** Read the entire article here:*

http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

*Related Links:http://britton.disted.camosun.bc.ca/jbfunpatt.htm*

### Flower Patterns and Fibonacci Numbers

Why is it that the number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy has often 34 or 55 petals, etc. Furthermore, when one observes the heads of sunflowers, one notices two series of curves, one winding in one sense and one in another; the number of spirals not being the same in each sense. Why is the number of spirals in general either 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144? The same for pinecones : why do they have either 8 spirals from one side and 13 from the other, or either 5 spirals from one side and 8 from the other? Finally, why is the number of diagonals of a pineapple also 8 in one direction and 13 in the other?

*Passion Fruit . © All rights reserved *Image Source >>

Are these numbers the product of chance? No! They all belong to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (where each number is obtained from the sum of the two preceding). A more abstract way of putting it is that the Fibonacci numbers fn are given by the formula f1 = 1, f2 = 2, f3 = 3, f4 = 5 and generally f n+2 = fn+1 + fn . For a long time, it had been noticed that these numbers were important in nature, but only relatively recently that one understands why. It is a question of efficiency during the growth process of plants.

The explanation is linked to another famous number, the golden mean, itself intimately linked to the spiral form of certain types of shell. Let’s mention also that in the case of the sunflower, the pineapple and of the pinecone, the correspondence with the Fibonacci numbers is very exact, while in the case of the number of flower petals, it is only verified on average (and in certain cases, the number is doubled since the petals are arranged on two levels).

*© All rights reserved.*

Let’s underline also that although Fibonacci historically introduced these numbers in 1202 in attempting to model the growth of populations of rabbits, this does not at all correspond to reality! On the contrary, as we have just seen, his numbers play really a fundamental role in the context of the growth of plants

THE EFFECTIVENESS OF THE GOLDEN MEAN

The explanation which follows is very succinct. For a much more detailed explanation, with very interesting animations, see the web site in the reference.

In many cases, the head of a flower is made up of small seeds which are produced at the centre, and then migrate towards the outside to fill eventually all the space (as for the sunflower but on a much smaller level). Each new seed appears at a certain angle in relation to the preceeding one. For example, if the angle is 90 degrees, that is 1/4 of a turn, the result after several generations is that represented by figure 1.

Of course, this is not the most efficient way of filling space. In fact, if the angle between the appearance of each seed is a portion of a turn which corresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (that is a simple rational number), one always obtains a series of straight lines. If one wants to avoid this rectilinear pattern, it is necessary to choose a portion of the circle which is an irrational number (or a nonsimple fraction). If this latter is well approximated by a simple fraction, one obtains a series of curved lines (spiral arms) which even then do not fill out the space perfectly (figure 2).

In order to optimize the filling, it is necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. This number is exactly the golden mean. The corresponding angle, the golden angle, is 137.5 degrees. (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement). With this angle, one obtains the optimal filling, that is, the same spacing between all the seeds (figure 3).

This angle has to be chosen very precisely: variations of 1/10 of a degree destroy completely the optimization. (In fig 2, the angle is 137.6 degrees!) When the angle is exactly the golden mean, and only this one, two families of spirals (one in each direction) are then visible: their numbers correspond to the numerator and denominator of one of the fractions which approximates the golden mean : 2/3, 3/5, 5/8, 8/13, 13/21, etc.

These numbers are precisely those of the Fibonacci sequence (the bigger the numbers, the better the approximation) and the choice of the fraction depends on the time laps between the appearance of each of the seeds at the center of the flower.

This is why the number of spirals in the centers of sunflowers, and in the centers of flowers in general, correspond to a Fibonacci number. Moreover, generally the petals of flowers are formed at the extremity of one of the families of spiral. This then is also why the number of petals corresponds on average to a Fibonacci number.

#### REFERENCES:

- An excellent Internet site of Ron Knot’s at the University of Surrey on this and related topics.
- S. Douady et Y. Couder, La physique des spirales vÃ©gÃ©tales, La Recherche, janvier 1993, p. 26 (In French).

*Source of the above segment:http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html© Mathematics and Knots, U.C.N.W.,Bangor, 1996 – 2002*

### Fibonacci numbers in vegetables and fruit

Romanesque Brocolli/Cauliflower (or Romanesco) looks and tastes like a cross between brocolli and cauliflower. Each floret is peaked and is an identical but smaller version of the whole thing and this makes the spirals easy to see.

*Brocolli/Cauliflower© All rights reserved Image Source >>*

* * *

### Human Hand

Every human has two hands, each one of these has five fingers, each finger has three parts which are separated by two knuckles. All of these numbers fit into the sequence. However keep in mind, this could simply be coincidence.

To view more examples of Fibonacci numbers in Nature explore our selection of related links>>.

### Human Face

Knowledge of the golden section, ratio and rectangle goes back to the Greeks, who based their most famous work of art on them: the Parthenon is full of golden rectangles. The Greek followers of the mathematician and mystic Pythagoras even thought of the golden ratio as divine.

Later, Leonardo da Vinci painted Mona Lisa’s face to fit perfectly into a golden rectangle, and structured the rest of the painting around similar rectangles.

Mozart divided a striking number of his sonatas into two parts whose lengths reflect the golden ratio, though there is much debate about whether he was conscious of this. In more modern times, Hungarian composer Bela Bartok and French architect Le Corbusier purposefully incorporated the golden ratio into their work.

Even today, the golden ratio is in human-made objects all around us. Look at almost any Christian cross; the ratio of the vertical part to the horizontal is the golden ratio. To find a golden rectangle, you need to look no further than the credit cards in your wallet.

Despite these numerous appearances in works of art throughout the ages, there is an ongoing debate among psychologists about whether people really do perceive the golden shapes, particularly the golden rectangle, as more beautiful than other shapes. In a 1995 article in the journal Perception, professor Christopher Green,

of York University in Toronto, discusses several experiments over the years that have shown no measurable preference for the golden rectangle, but notes that several others have provided evidence suggesting such a preference exists.

Regardless of the science, the golden ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside a nautilus shell is remarkably close to the golden section, and the ratio of the lengths of the thorax and abdomen in most bees is nearly the golden ratio. Even a cross section of the most common form of human DNA fits nicely into a golden decagon. The golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to spark interest in the mathematical community.

Dr. Stephen Marquardt, a former plastic surgeon, has used the golden section, that enigmatic number that has long stood for beauty, and some of its relatives to make a mask that he claims is the most beautiful shape a human face can have.

The Mask of a perfect human face

Egyptian Queen Nefertiti (1400 B.C.)

An artist’s impression of the face of Jesus

based on the Shroud of Turin and corrected

to match Dr. Stephen Marquardt’s mask.

Click here for more detailed analysis.

“Averaged” (morphed) face of few celebrities.

Related website: http://www.faceresearch.org/tech/demos/average

You can overlay the Repose Frontal Mask (also called the RF Mask or Repose Expression – Frontal View Mask) over a photograph of your own face to help you apply makeup, to aid in evaluating your face for face lift surgery, or simply to see how much your face conforms to the measurements of the Golden Ratio.

Visit Dr. Marquardt’s Web site for more information on the beauty mask.

Source of the above article (with exception of few added photos):

http://tlc.discovery.com/convergence/humanface/articles/mask.html