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Fibonacci Numbers and Nature

by

John R. Simmons

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On a dark night, we can often see a band of light stretching across the sky. This band is the Milky Way galaxy -- a gigantic collection of stars, gas and dust. Far beyond the Milky Way, there are billions of other galaxies -- some similar to our own and some very different -- scattered throughout space to the very limits of the observable universe.

Types of Galaxies

Astronomers classify galaxies into three major categories. Spiral galaxies look like flat disks with bulges in their centers and beautiful spiral arms. Elliptical galaxies are redder, more rounded, and often longer in one direction than in the other, like a football. Galaxies that appear neither disk-like nor rounded are classified as irregular galaxies.

Spiral Galaxies

Spiral galaxies usually consist of three components: a flat disk, an ellipsoidal bulge and a halo. The disk contains a lot of interstellar gas and dust, and most of the stars in the galaxy. The gas, dust and stars in the disk rotate in the same direction around the galactic center at hundreds of kilometers per second and are often arranged in striking spiral patterns. The bulge is located at the Centrex of the disk and consists of an older stellar population with little interstellar matter. The near-spherical halo surrounds the disk, and is thought to contain copious amounts of dark matter : matter that acts gravitationally like "normal" matter but that can't be seen! Astronomers infer the presence of this dark matter by the motions of stars and gas in the disk of the galaxy, as well as older stellar populations in the halo like globular clusters. The young stars in the disk are classified as stellar population I, and the old bulge and halo stars as population II.

Astronomers classify spiral galaxies according to their appearance by using the Hubble scheme. Those with pronounced bar structures in their centers are called "barred spirals" and are classified "SB" (examples are given in brackets). Galaxies with conspicuous bulges and tightly wound spiral arms are called "Sa" ( Sombrero galaxy ) or "SBa" ( NGC 3185 ). Galaxies with prominent bulges and pronounced spiral arms are classified as "Sb" ( M31 , M81 ) or "SBb" ( M95 , NGC 4725 ). Other spirals with loose spiral arms and a small bulge are classified as "Sc" ( M33 , M74 , M100 ) or "SBc" ( M83 , M109 ), (Curious About Astronomy? Ask an Astronomer) .
curious@astro.cornell.edu

(via Astronomy Picture of the Day ) M51: The Whirlpool Galaxy. The Whirlpool Galaxy is a classic Sc type spiral galaxy visible with binoculars in the constellation of Canes Venatici. It is 23 million light years distant and is one of the brightest and most picturesque galaxies on the sky. The smaller galaxy appearing below and to the left is behind M51 and is thought to be responsible for the spiral structure of M51.

The Location in the Sky of M51, the Whirlpool Galaxy

A Closer View

Still Closer

An Estimate of one and two Spirals of M51

To see a GSP sketch showing the measurements approximating the spirals click below.

Click

M51 has two (Fibonacci number) spirals in one direction. To see a GSP sketch showing the measurements approximating the spirals, click below.

Click

Logarithmic Spirals Isabel and M51
Credit: Comparison and M51 image copyright Brian Lula ; Hurricane Isabel, courtesy GHCC, NASA

Explanation: Uncomfortably close hurricane Isabel (left) and 30 million light-year distant galaxy M51 actually don't have much in common. For starters, Isabel was hundreds of miles across, while M51 (the Whirlpool Galaxy ) spans about 50,000 light-years making them vastly dissimilar in scale, not to mention the extremely different physical interactions which control their formation and evolution . But they do look amazingly alike, both exhibiting the shape of a simple and beautiful mathematical curve known as a logarithmic spiral , a spiral whose separation grows in a geometric way with increasing distance from the center. Also known as the equiangular spiral , growth spiral, and Bernoulli's spiral or spira mirabilis , this curve's rich properties have fascinated mathematicians since its discovery by 17th century philosopher Descartes . Intriguingly, this abstract shape is much more abundant in nature than suggested by the striking visual comparison above. Logarithmic spirals also describe, for example, the arrangement of sunflower seeds , the shapes of nautilus shells, and ... cauliflower .

Pine cones

Here is a picture of an ordinary pinecone seen from its base where the stalk connects it to the tree.

This pine cone shows the Fibonacci Spirals clearly. The number of spirals in either direction are two consecutive Fibonacci numbers.

You will occasionally find pine cones that do not have a Fibonacci number of spirals in one or both directions. Sometimes this is due to deformities produced by disease or pests but sometimes the cones look normal too. This article reports on a study of this question and others in a large collection of Californian pine cones of different kinds. The author also found that there were as many with the steep spiral (the one with more arms) going to the left as to the right. Pineapples and Fibonacci Numbers P B Onderdonk The Fibonacci Quarterly vol 8 (1970), pages 507, 508.

The Human Body

DNA fibonacci spiral structure

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 a coincidence.

Human Hand

Typical Human Body

The golden ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside M51, the Whirlpool Galaxy, 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.

For a review of the General Fibonacci series go to:

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Fibonacci Numbers and Nature by John R. Simmons _____________________________________________________________________________________________ On a dark night, we can often see a band of light stretching across the sky. This band is the Milky Way galaxy -- a gigantic collection of stars, gas and dust.

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Scoperta la serie di Fibonacci sulla facciata di una chiesa di Pisa

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Scoperta la serie di Fibonacci sulla facciata di una chiesa di Pisa

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università

Secondo un docente dell'Università di Pisa "Le geometrie dell'intarsio contenuto nella lunetta sopra il portale sono un richiamo esplicito al pensiero del matematico pisano"

PISA - È un richiamo esplicito alle scoperte del primo grande matematico dell'Occidente cristiano, Leonardo Fibonacci, ed è riemerso grazie a un recente restauro che ha riportato i marmi della Chiesa di San Nicola in via Santa Maria all'antico splendore. Un originale studio di Pietro Armienti, docente di Petrologia e Petrografia dell'Università di Pisa, recentemente pubblicato sul Journal of Cultural Heritage, ha permesso di interpretare le eleganti geometrie dell'intarsio della lunetta sopra l'originario portale principale come un riferimento alla celebre successione numerica individuata dal matematico pisano.

I segni del tempo

"Per secoli i segni del tempo avevano reso illeggibili gli intarsi della facciata della chiesa, la cui costruzione, che risale al XIII secolo, viene da molti attribuita a Nicola Pisano - commenta il professore - Dopo il restauro, il messaggio scolpito nella lunetta del portale è emerso in tutti i suoi dettagli e ci ha permesso di dimostrare che il pregevole manufatto, che ha comportato il lavoro congiunto di matematici, teologi, artigiani, celebra le intuizioni che segnarono a Pisa la nascita di una scuola di pensiero capace di trasformare la visione medievale del mondo e di fare della città la culla della pensiero scientifico moderno".

La serie dentro i segni

Secondo l'interpretazione del professor Armienti, le eleganti simmetrie dell'opera sono un richiamo diretto alle scoperte del matematico pisano: "Se si assume come unitario il diametro dei cerchi più piccoli dell'intarsio, i più grandi hanno diametro doppio, i successivi triplo, mentre quelli di diametro 5 sono divisi in spicchi nei quadratini ai vertici del quadrato in cui è inscritto il cerchio principale, quello centrale ha diametro 13 mentre il cerchio che circoscrive i quadratini negli angoli ha diametro 8. Gli altri elementi dell'intarsio disposti secondo tracce circolari individuano circonferenze di raggio 21 e 34, infine il cerchio che circoscrive l'intarsio ha diametro 55 volte più grande del circolo minore. 1,2,3,5,8,13,21,34,55 sono i primi nove elementi della successione di Fibonacci".

"L'intarsio è un abaco"

Per Armienti, il riferimento non potrebbe essere più esplicito e collega direttamente l'intarsio all'opera del grande matematico o a una cerchia di suoi diretti collaboratori o allievi: "L'intarsio di fatto è un abaco per rappresentare numeri irrazionali come p o il rapporto Aureo f, oltre che per calcolare con un'ottima approssimazione i lati dei poligoni regolari inscritti nel cerchio diametro maggiore. Si tratta dunque di un importante monumento la cui presenza era stata concepita per l'educazione delle élites, secondo il programma della filosofia scolastica: un dono prezioso della sapienza degli antichi giunto dopo ottocento anni di oblio e la cui presenza va valorizzata".

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"Scoperta la serie di Fibonacci sulla facciata di una chiesa a Pisa"

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università PISA - È un richiamo esplicito alle scoperte del primo grande matematico dell'Occidente cristiano, Leonardo Fibonacci, ed è riemerso grazie a un recente restauro che ha riportato i marmi della Chiesa di San Nicola in via Santa Maria all'antico splendore.

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by Keiren 904 Views

The fibonacci spiral appears not only in the perfect nautilus shell...

...but in events and objects viewed from afar.

An energy system in the shape of a fibonacci moves with limited losses. Hurricane Irene.

What is the Fibonacci Sequence?

The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is a recursive sequence, generated by adding the two previous numbers in the sequence.: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...

Here is a good video explanation from SciShow. He points out that plant sections, petals, and rows of seeds almost always count up to a Fibonacci number.

If you were to draw a line starting in the right bottom corner of a golden rectangle within the first square, and then touch each succeeding multiple square's outside corners, you would create a fibonacci spiral.

How Common Is The Fibonacci Sequence in Nature?

The fibonacci appears in the smallest, to the largest objects in nature. It is a way for information to flow in a very efficient manner. Here, a microscopic view of the ovary of an Anglerfish. Nikon's It's a Small World Competition.

Spirals are the most common galaxy shape. Galaxies group together in superclusters and superclusters group together in walls. These walls or filaments of numerous superclusters, gravitationally-bound and separated by large areas of void, are the largest known structures in the universe.

The Milky Way's dust obstructs us from seeing the depth of these filaments or sheets, so we do not yet know the exact shape of these walls. More information can be found at www.spacetelescope.org.

Cancer cell division. This composite confocal micrograph uses time-lapse microscopy to show a cancer cell (HeLa) undergoing cell division (mitosis). The DNA is shown in red, and the cell membrane is shown in cyan. The round cell in the centre has a diameter of 20 microns.

This is part 1 of three-part video series from "recreational mathematician" Vi Hart, explaining the mathematics behind the Fibonacci Sequence. Part 1 shows how you can draw the sequence and shows how it actually on pinecones and pineapples.

18 Amazing Examples of the Fibonacci Sequence in Nature

1) Chicken Egg

Fibonacci as starting point of life.

2) Romanesque Broccoli

Romanesque broccoli is a striking example of the fibonacci. Each nub is a Fibonacci spiral of its own.

3) Aloe Plant

Spiral aloe. Numerous cactus display the Fibonacci spiral. You can see how each set of leaves spiral outward.

4) Buena Mulata Pepper

This pepper has grown into a Fibonacci Spiral.

5) Sunflower

Sunflower. www.thestrong.org. The Fibonacci spiral is a little more subtle in this photo, but you can still see the spiral in the unopened disk florets.

6) Rock Daisy

Marlborough Rock Daisy by Sid Mosdell. Again, the spiral is visible in the disk florets of the flower.

7) Pinecone

All pinecones display a fibonacci sequence. The umbo on pinecones increase in size as you move outward, displaying a Fibonacci spiral.

8) Panther Chameleon

The tail of these creatures naturally curls into a Fibonacci spiral.

9) American Giant Millipede

American giant millipede. The fibonacci is thought to be the design of least resistance.

10) Monarch Caterpillar

A monarch caterpillar about to form a chrysalis.

11) Pangolin

Fibonacci and armor = very safe. The Pangolin is able to protect its soft underbelly by forming a Fibonacci spiral.

12) Double Fibonacci

This flower exhibits two Fibonacci spirals. You can faintly see how the spirals form from the center of the opened disk florets.

13) Koru

Fibonacci in spores. A fiddlehead or koru.

14) Snails And Fingerprints

Snails and fingerprints. Images are from www.123rf.com, and originally "artcatalyst.blogspot.com/2011/04/fibonacci-sequence-mathematics-nature.html" (respectively). Both have a distinct Fibonacci spiral.

15) Famous Art

Fibonacci in "The Great Wave Off Kanagawa." It seems even famous art can't escape the Fibonacci sequence.

16) Falling Water

Water falls into the shapes of a Fibonacci during numerous events. Another example would be a vortex.

17) Population

One blogger has applied the Fibonacci sequence to population density and land mass. In Africa the majority of highly populated cities fall on or close to where the spiral predicts.

18) Shell Fossil

A Shell Fossil with the Fibonacci sequence. You can see as the shell grew, a Fibonacci spiral was formed.

Who was Fibonacci?

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Though Fibonacci first introduced the sequence to the western world in 1202, it had been noted by Indian mathematicians as early as the sixth century.

The Fibonacci defines how the density of branches increases up a tree trunk, the arrangement of leaves on a stem, and how a pine cone's scales are arranged. Yet you will not see the Fibonacci everywhere, as nature has many different methods and shades of survival.

Hang Fibonacci In Your Home

These prints from Art.com can be printed at any size you like-they'll frame them for you or you can print directly to canvas. We've had really good luck with their prints; shipping is fast and the prints are good quality. These start at around $25 each.

This is "The Great Wave," by Katsushika Hokusai. A stunning example of the Fibonacci spiral in art.

Fibonacci Spiral by Seymour. If you like a more simplistic look, this drawing of the Fibonacci spiral may be more your style.

Nautilus Shell by Babar760. A natural depiction of the Fibonacci spiral, great for someone who enjoys math and nature.

Some stock traders are using the Fibonacci sequence as an attempt to "crack" the stock market, by selling or buying when certain sequences appear on stock charts. Not recommended!

There's also a Fibonacci betting system. The idea here is to start with an even money bet, like red/black in roulette. You keep moving up the Fibonacci sequence every time you lose a bet on the idea that, eventually, you'll win and recoup your money. Also not recommended!

The Best Books about Fibonacci and the Fibonacci Sequence

The Golden Ratio: The Story of PHI, the World's Most Astonishing Number by Mario Livio

Growing Patterns: Fibonacci Numbers in Nature by Sarah and Richard Campbell

The Golden Section: Nature's Greatest Secret by Scott Olsen

The Fabulous Fibonacci Numbers by Alfred Posamentier and Ingmar Lehmann

Blockhead: The Life of Fibonacci by Joseph D'Agnese and John O'Brien (children's book, named a Mathical Honor Book April 2015)

Fascinating! A must watch!
Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series.

Check out this Custom Fibonacci Spiral Generator - chromatism.net

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by Keiren 904 Views The fibonacci spiral appears not only in the perfect nautilus shell... ...but in events and objects viewed from afar. An energy system in the shape of a fibonacci moves with limited losses. Hurricane Irene. The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected.