Nature, Fibonacci Numbers and the Golden Ratio

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 (? or ?).
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: j (one-to-phi),
that is, 1 :  or approximately 1:1.618.

A golden rectangle can be constructed with only straightedge
and compass by this technique:

  1. Construct a simple square
  2. Draw a line from the midpoint of one side of the square to an opposite corner

  3. Use that line as the radius to draw an arc that defines the height of the rectangle

  4. 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 on
a 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 reveals
golden spiral construction principle.

These spiral shapes are called Equiangular or Logarithmic spirals. The links from these terms contain much more information on these curves and pictures of computer-generated shells.

Here is a curve which crosses the X-axis at the Fibonacci numbers

The spiral part crosses at 1 2 5 13 etc on the positive axis, and 0 1 3 8 etc on the negative axis. The oscillatory part crosses at 0 1 1 2 3 5 8 13 etc on the positive axis. The curve is strangely reminiscent of the shells of Nautilus and snails. This is not surprising, as the curve tends to a logarithmic spiral as it expands.

Nautilus shell (cut) . © All rights reserved. Image source >>


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 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).  Click to Enlarge.


The Vitruvian Man” by Leonardo Da Vinci
We 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 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, called Liber 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 Khayyám.

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.

One-petalled …

white calla lily

Two-petalled flowers are not common.

euphorbia

Three petals are more common.

trillium

Five petals – there are hundreds of species, both wild and cultivated, with five petals.

 

 

Eight-petalled flowers are not so common as five-petalled, but there are quite a number of well-known species with eight.

bloodroot

Thirteen, …

black-eyed susan

 

Twenty-one and thirty-four petals are also quite common. The outer ring of ray florets in the daisy family illustrate the Fibonacci sequence extremely well.  Daisies with 13, 21, 34, 55 or 89 petals are quite common.

shasta daisy with 21 petals

Ordinary field daisies have 34 petals… a fact to be taken in consideration when playing “she loves me, she loves me not”. In saying that daisies have 34 petals, one is generalizing about the species – but any individual member of the species may deviate from this general pattern. There is more likelihood of a possible under development than over-development, so that 33 is more common than 35.

* 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:

  1. An excellent Internet site of  Ron Knot’s at the University of Surrey on this and related topics.

  2. S. Douady et Y. Couder, La physique des spirales végétales, 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.

Mona Lisa

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

The Golden Mean or Phi occurs frequently in nature and it may be that humans are genetically programmed to recognize the ratio as being pleasing. Studies of top fashion models revealed that their faces have an abundance of the 1.618 ratio.  

?Worth a Look

Comments

  1. says

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  2. says

    1_ x 12 = 12_____________1 + 2 = 3
    2_ x 12 = 24_____________2 + 4 = 6
    3_ x 12 = 36_____________3 + 6 = 9
    4_ x 12 = 48___4 + 8 = 12 = 1+2 = 3
    5_ x 12 = 60______________6+0 = 6
    6_ x 12 = 72______________7+2 = 9
    7_ x 12 = 84____8+4 = 12 = 1+2 = 3
    8_ x 12 = 96____9+6 = 15 = 1+5 = 6
    9_ x 12 = 108___________1+0+8 = 9
    10 x 12 = 120___________1+2+0 = 3
    11 x 12 = 132___________1+3+2 = 6
    12 x 12 = 144___________1+4+4 = 9

    13 x 12 = 156__1+5+6 = 12 = 1+2 = 3
    14 x 12 = 168__1+6+8 = 15 = 1+5 = 6
    15 x 12 = 180____________1+8+0 = 9
    16 x 12 = 192__1+9+2 = 12 = 1+2 = 3
    17 x 12 = 204____________2+0+4 = 6
    18 x 12 = 216____________2+1+6 = 9
    19 x 12 = 228__2+2+8 = 12 = 1+2 = 3
    20 x 12 = 240____________2+4+0 = 6
    21 x 12 = 252____________2+5+2 = 9
    22 x 12 = 264__2+6+4 = 12 = 1+2 = 3
    23 x 12 = 276__2+7+6 = 15 = 1+5 = 6
    24 x 12 = 288__2+8+8 = 18 = 1+8 = 9

    http://science2art.tumblr.com/post/18398397422/72
    http://science2art.tumblr.com/post/18723310752/phi-369

  3. Venkatesh says

    I don’t understand why is it called ‘hindu-arabic’ system of numbers. Hindus invented it,
    arabians just copied it.

  4. says

    You cannot see Light, only the result of its presence is discernible. Every elemental of our existence here is the message of that media whereon and wherein we exist. Every ounce of our being and becoming is the answer to our question of who and what we are partaking of in this environ called Life. The Ancients gave us a veritable school filled with the lessons etched at every turn within their elements and principles of design — complexity. One great and grand lesson is the realm of genetics. Everywhere we find the numbers of above in this article and the expressions of the biological building blocks to our sheaths (means of motivation in a physical state) whereby movement in this physicality is Leveraged. Only the idiot takes all this for granted and it seems our world is currently filled with such practitioners of sloth.

    I have a cat that I look at daily in wonder. It is highly sophisticated and is indeed a message from the being that assembled all the ingredients to its fruition (a great geneticist). That cat is perfect in every detail and sits giving me love and knowledge that I would not even know, had I not invested in loving that cat. The stripes of the Zebra are fractals in perfection and therein one may find a message to humanity as to the knowledge and wisdom of the ancient of days (Archons), who gave to us — not unlike the Gods of ancient Egypt (Zep Tepi). They gave us Genetic Wisdom. Those who passed on a massive amount of Wisdom some 12,900 years ago held within their artifacts the secrets of a knowledge far beyond our silly little offerings of this present expression. R. A. Schwaller de Lubicz, has only scratched the surface of their messages to us and we cannot even understand him. What hope do we have? Those of us who write here have a charge…we must work together to amplify the secrets of our existence.

  5. Charles Marcello says

    Fibonacci’s Mathematical Conceptual Understanding can be found outside our planet as well. It’s a very simple mathematical process once humanity finally understands our planet has absolutely nothing whatsoever to do with our reality as we perceive it outside our own limited conceptual understanding here on Earth. When standing within a box, if you don’t know you’re living within a box, or that living inside a box is all you accept, you will never know you are standing within a box, or boxes can always exist within other boxes… stated another way… Turtles on top of Turtles, or Jacob’s Ladder… The same is true when viewing our universe with extremely limited concepts. We must look beyond our own window to see what is really happening outside our extremely limited environment. Fibonacci mathematical conceptual understanding of existence is happening everywhere in our universe, whether humanity is ready to look outside and apply those lessens within our own box or not. Sometimes the best way to learn how things actually work, is to watch how something is actually working. Truth can be found outside our own existence/experiences. Not every child has to touch the flame. Learning and Truth from a distance is everywhere.

    When looking outside our own limited environment/window we see other Solar Systems and Galaxies are moving is several directions at exactly the same time. An entire Galaxy is ONE unit, that entire unit is always moving. An entire Solar System is One Unit, once an entire unit is moving, its all moving at exactly the same speed in at least one direction from one end to the other… These truths are repeatable and repetitive… Just like in a flower, or more correctly… just like our physical bodies in motion. Size is relative, truth is not. Mathematics are and truth is… everything else can and in most cases are an illusion.

    Perhaps that is why every ancient culture passed a message forward warning humanity, beware of perception. Because I question everything, not accept or deny… yet question… I believe the warning, beware of perception… has nothing to do with things that go bump in the night… rather, it has everything to do with warning mankind if you only trust in your perceived/perceptual understanding of existence, you will waste decades or even hundreds of years not seeing the truth of our reality. The Pyramids of Giza tells so many stories… some completely unbelievable. Like… I’d be willing to bet when the Great Pyramid was built they used exactly 2 million, 621 thousand, 440 blocks… and I mean exactly… in order to pass yet another message forward in time. A message so beautifully awesome, when humanity finally grows up and stops all our childish concepts, us itty-bitty earthlings will finally begin to see our reality for the first time. Fibonacci teaches us how to finally see and measure our reality. While Three Dimensional Mathematics teaches us the truth of our place within each those measurements. Turtles on top of Turtles, boxes within boxes within boxes… and how it all “flows” together.

    –Charles Marcello

  6. Alex says

    Ron, I agree about 137. I discovered that 137 is one of the key numbers embedded in the design of the Pyramid of Khafre (The Second Pyramid). The article about it (and more) will be posted soon.

    Alex

  7. says

    Well, I have seen a million of these posts from those who are inclined with a great love of the numbers. I love them too, but what is beyond the numbers? 137 remains my number and I suppose it always will. Not that one is, but that whereby Is, is.

    Ron O.

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