Fibonacci Sequence Fibonacci Sequence The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1) The 3 is found by adding the two numbers before it (1+2), And the 5 is (2+3), and so on! Example: the next number in the sequence above is 21+34 = 55 It is that simple!

Here is a longer list: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ... Can you figure out the next few numbers? Makes A Spiral When we make squares with those widths, we get a nice spiral: Do you see how the squares fit neatly together? For example 5 and 8 make 13, 8 and 13 make 21, and so on. The Rule The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series ).

First, the terms are numbered from 0 onwards like this: n = 8 ... (Prove to yourself that each number is found by adding up the two numbers before it!) In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- ... pattern. It can be written like this: x−n = (−1)n+1 xn Which says that term "-n" is equal to (−1)n+1 times term "n", and the value (−1)n+1 neatly makes the correct 1,-1,1,-1,... pattern. History Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before! About Fibonacci The Man His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy. "Fibonacci" was his nickname, which roughly means "Son of Bonacci".

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo. Fibonacci Day Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know! What is the next number in the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34? - Quora Quora Written Jul 31, 2014 Well, how about first I tell you how to derive the next number first? The Fibonacci sequence is actually quite simple: add the last two terms of the sequence together to get the next one! So in your case here, 21+34 = 55. Thus, 55 is the next term.

If you know programming at all (good for everyone to know) do this (python language here) n = 101 while len(f)<n: f.append(f[-1]+f[-2]) What this says is, given the initial sequence [1,1], while the length (number of terms) of the sequence is below 101, append the list (add another term in the list) that is the sum of the last (f[-1]) and second to last (f[-2]) terms. Continue to do so until you have a hundred terms.

Here's the first 100 terms for you :) [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 632459

86, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074, 32951280099, 53316291173, 86267571272, 139583862445, 225851433717, 365435296162, 591286729879, 956722026041, 15

48008755920, 2504730781961, 4052739537881, 6557470319842, 10610209857723, 17167680177565, 27777890035288, 44945570212853, 72723460248141, 117669030460994, 190392490709135, 308061521170129, 498454011879264, 806515533049393, 1304969544928657, 211148507797805

0, 3416454622906707, 5527939700884757, 8944394323791464, 14472334024676221, 23416728348467685, 37889062373143906, 61305790721611591, 99194853094755497, 160500643816367088, 259695496911122585, 420196140727489673, 679891637638612258, 1100087778366101931, 177

9979416004714189, 2880067194370816120, 4660046610375530309, 7540113804746346429, 12200160415121876738, 19740274219868223167, 31940434634990099905, 51680708854858323072, 83621143489848422977, 135301852344706746049, 218922995834555169026, 3542248481792619150

75, 573147844013817084101] 1.4k Views · View Upvotes · Answer requested by What is Fibonacci sequence?

- Definition from WhatIs.com pneumatics The Fibonacci sequence is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. If the Fibonacci sequence is denoted F (n), where n is the first term in the sequence, the following equation obtains for n = 0, where the first two terms are defined as 0 and 1 by convention: F (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... In some texts, it is customary to use n = 1. In that case, the first two terms are defined as 1 and 1 by default, and therefore: F (1) = 1, 1, 2, 3, 5, 8, 13, 21, 34 ... The Fibonacci sequence is named for Leonardo Pisano (also known as Leonardo Pisano or Fibonacci ), an Italian mathematician who lived from 1170 - 1250.

Fibonacci used the arithmetic series to illustrate a problem based on a pair of breeding rabbits: "How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?" The result can be expressed numerically as: 1, 1, 2, 3, 5, 8, 13, 21, 34 ... Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. The branching patterns in trees and leaves, for example, and the distribution of seeds in a raspberry are based on Fibonacci numbers. A Sanskrit grammarian, Pingala, is credited with the first mention of the sequence of numbers, sometime between the fifth century B.C. and the second or third century A.D. Since Fibonacci introduced the series to Western civilization, it has had a high profile from time to time. In The Da Vinci Code, for example, the Fibonacci sequence is part of an important clue. Another application, the Fibonacci poem , is a verse in which the progression of syllable numbers per line follows Fibonacci's pattern.

The Fibonacci sequence is related to the golden ratio , a proportion (roughly 1:1.6) that occurs frequently throughout the natural world and is applied across many areas of human endeavor. Both the Fibonacci sequence and the golden ratio are used to guide design for architecture, websites and user interfaces, among other things. This was last updated in October 2016 Continue Reading About Fibonacci sequence Fibonacci Numbers Fibonacci Numbers by: Stephanie J. Morris Fibonacci numbers and the Fibonacci sequence are prime examples of "how mathematics is connected to seemingly unrelated things." Even though these numbers were introduced in 1202 in Fibonacci's book Liber abaci, they remain fascinating and mysterious to people today. Fibonacci, who was born Leonardo da Pisa "son of Bonaccio", gave a problem in his book whose solution was the Fibonacci sequence as we know it today. The now famous problem appeared in Liber abaci which was one of the first works on equations to be published by a European. Fibonacci's Problem: "Start with a pair of rabbits, (one male and one female) born on January 1. Assume that all months are of equal length and that : 1.

rabbits begin to produce young two months after their own birth; 2. after reaching the age of two months, each pair produces a mixed pair, (one male, one female), and then another mixed pair each month thereafter; and 3. no rabbit dies. How many pairs of rabbits will there be after one year?" Fibonacci's Solution: The Fibonacci Sequence! 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... So, at the end of the year, there will be 144 pairs of rabbits, all resulting from the one original pair born on January 1 of that year. Each term in the Fibonacci sequence is called a Fibonacci number. As can be seen from the Fibonacci sequence, each Fibonacci number is obtained by adding the two previous Fibonacci numbers together. For example, the next Fibonacci number can be obtained by adding 144 and 89. Thus, the next Fibonacci number is 233. The recursive definition for generating Fibonacci numbers and the Fibonacci sequence is: fn = fn-1 + fn-2 where n>3 or n=3 and 1.

n = the number of the term, for example, f3 = the third Fibonacci number; and 2. f 1 = f2 = 1 One of the most fascinating things about the Fibonacci numbers is their connection to nature. Some items in nature that are connected to the Fibonacci numbers are: - the growth of buds on trees - the pinecone's rows - the starfish - the petals on various flowers such as the cosmo, iris, buttercup, daisy, and the sunflower - the appendages and chambers on many fruits and vegetables such as the lemon, apple, chile, and the artichoke The Fibonacci sequence and the Fibonacci numbers also have many interesting mathematical properties. Mathematical Properties of the Fibonacci Sequence : fn = fn+2 - fn+1, By adding each of these terms, we get the desired result. Example: f1 + f2 + f3 = f5 - 1 1 + 1 + 2 = 4 = 5 - 1. III. The sum of the first n Fibonacci numbers with odd indices is f1 + f3 +...+ f2n-1 = f2n. The proof is similar to the proof in II.

Example: f1 + f3 + f5 = 1+ 2 + 5 = 8 IV. The sum of the first n Fibonacci numbers with even indices is f2 + f4 + ... + f2n = f2n+1 - 1. Proof: f1 + f2 + f3 + ... + f2n = f2n+2 - 1. When we subtract the result from III, we get the desired result. Example: f2 + f4 + f6 + f8 + f10 + f12 = 1 +3 + 8 + 21 +55 +144 = 232 V. The sum of all (fn+1)/ (fn ) converges to the Golden Ratio. 3/1 + 5/3 + 8/5 + 13/8 .... converges to ) / 2.

Proof that Rn converges to the Golden Ratio: Let R = lim Rn as n approaches infinity = lim f n+1 / f n as n approaches infinity = lim fn + fn-1 /fn as n approaches infinity = lim (1 + fn-1/ fn) as n approaches infinity = 1 + lim (fn-1 /fn ) as n approaches infinity = 1 + 1/ lim (fn-1 /fn ) as n approaches infinity = 1 + 1/R or R^2 = R + 1, R^2 - R - 1 = 0, R = ( Since Rn is positive, Rn = ( ) / 2 . Thus, Rn converges to the Golden Ratio. Number Theory Properties of the Fibonacci Sequence: Theorem 1 : If n is divisible by m, then fn is divisible by fm. Proof by Induction: Let n be divisible by m, i.e., n = m * k where k is some integer. Assume that fm * fk is divisible by fm . Consider fm * fk + 1. fm (k + 1) = fmk + m.

and fmk + m = fmk -1fm + fmkfm + 1 Since fmk -1fm is divisible by fm, fmk1fm + 1 is also divisible by fm. Theorem 2 : Consecutive Fibonacci numbers are relatively prime. Proof by Contradiction: Assume that there exist some two consecutive Fibonacci numbers say fn and fn+1 that have a common divisor d, where d is greater than 1. Thus, their difference fn+1 - fn = fn-1 will also be divisible by d. However, we know that f1=1 which is clearly not divisible by d. Thus, we have reached a contradiction. Therefore, consecutive Fibonacci numbers are relatively prime. The Euclidean Algorithm and the Fibonacci Sequence: Consider the following Fibonacci numbers: 1 = 1 * 1 fn = 1*fn-1 + fn-2 fn+1 = 1*fn + fn-1 Therefore, the Euclidean Algorithm can be used to express Fibonacci numbers. By reversing the process, (final step to first step), it can be seen that it is relatively prime. The Fibonacci sequence can also be used to test for primality, namely with the Lucas N+1 primality test.

The Fibonacci sequence can also be used on the spreadsheet. Using the spreadsheet, one can find any Fibonacci number. 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 So, if one wanted to know how pairs of rabbits there would be at the end of 25 months, the answer, 75,025, could be found using the spreadsheet. The Fibonacci numbers have many mathematical properties that are worthy of exploration in today's mathematics curriculum. They can be explored in many courses and mathematical contexts. Fibonacci numbers can easily provide many worthwhile mathematical tasks for students. Bibilography Dion, Gloria S. . Fibonacci Meets the TI-82: : Mathematics Teacher February 1995. Reston, VA. "Those Fascinating Fibonaccis!" NCTM: Student Math Notes.

January 1996, Reston, VA. The Fibonacci Numbers and Golden section in Nature - 1 Fibonacci Numbers and Nature This page has been split into TWO PARTS. This, the first, looks at the Fibonacci numbers and why they appear in various "family trees" and patterns of spirals of leaves and seeds. The second page then examines why the golden section is used by nature in some detail, including animations of growing plants. Contents of this page The icon means there is a You do the maths... section of questions to start your own investigations. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More .. Rabbits, Cows and Bees Family Trees Let's look first at the Rabbit Puzzle that Fibonacci wrote about and then at two adaptations of it to make it more realistic. This introduces you to the Fibonacci Number series and the simple definition of the whole never-ending series.

Fibonacci's Rabbits The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances. Suppose 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. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs will there be in one year? At the end of the first month, they mate, but there is still one 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 produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ... Can you see how the series is formed and how it continues? If not, look at the answer ! The first 300 Fibonacci numbers are here and some questions for you to answer. Now can you see why this is the answer to our Rabbits problem? If not, here's why. Another view of the Rabbit's Family Tree: Both diagrams above represent the same information. Rabbits have been numbered to enable comparisons and to count them, as follows: All the rabbits born in the same month are of the same generation and are on the same level in the tree. The rabbits have been uniquely numbered so that in the same generation the new rabbits are numbered in the order of their parent's number. Thus 5, 6 and 7 are the children of 0, 1 and 2 respectively. The rabbits labelled with a Fibonacci number are the children of the original rabbit (0) at the top of the tree. There are a Fibonacci number of new rabbits in each generation, marked with a dot.

There are a Fibonacci number of rabbits in total from the top down to any single generation. There are many other interesting mathematical properties of this tree that are explored in later pages at this site. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More .. The Rabbits problem is not very realistic, is it? It seems to imply that brother and sisters mate, which, genetically, leads to problems. We can get round this by saying that the female of each pair mates with any male and produces another pair. Another problem which again is not true to life, is that each birth is of exactly two rabbits, one male and one female. Dudeney's 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) in problem 175 in his book 536 puzzles and Curious Problems (1967, Souvenir press): 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? This is a better simplification of the problem and quite realistic now. But Fibonacci does what mathematicians often do at first, simplify the problem and see what happens - and the series bearing his name does have lots of other interesting and practical applications as we see later. So let's look at another real-life situation that is exactly modelled by Fibonacci's series - honeybees. Puzzle books by Henry E Dudeney Amusements in Mathematics , Dover Press, 1958, 250 pages. Still in print thanks to Dover in a very sturdy paperback format at an incredibly inexpensive price. This is a wonderful collection that I find I often dip into.

There are arithmetic puzzles, geometric puzzles, chessboard puzzles, an excellent chapter on all kinds of mazes and solving them, magic squares, river crossing puzzles, and more, all with full solutions and often extra notes! Highly recommended! 536 Puzzles and Curious Problems is now out of print, but you may be able to pick up a second hand version by clicking on this link. It is another collection like Amusements in Mathematics (above) but containing different puzzles arranged in sections: Arithmetical and Algebraic puzzles, Geometrical puzzles, Combinatorial and Topological puzzles, Game puzzles, Domino puzzles, match puzzles and "unclassified" puzzles. Full solutions and index. A real treasure. The Canterbury Puzzles , Dover 2002, 256 pages. More puzzles (not in the previous books) the first section with some characters from Chaucer's Canterbury Tales and other sections on the Monks of Riddlewell, the squire's Christmas party, the Professors puzzles and so on and all with full solutions of course! Honeybees and Family trees There are over 30,000 species of bees and in most of them the bees live solitary lives.

The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee's ancestors (in this section a "bee" will mean a "honeybee"). First, some unusual facts about honeybees such as: not all of them have two parents! In a colony of honeybees there is one special female called the queen. There are many worker bees who are female too but unlike the queen bee, they produce no eggs. There are some drone bees who are male and do no work. Males are produced by the queen's unfertilised eggs, so male bees only have a mother but no father! All the females are produced when the queen has mated with a male and so have two parents. Females usually end up as worker bees but some are fed with a special substance called royal jelly which makes them grow into queens ready to go off to start a new colony when the bees form a swarm and leave their home (a hive) in search of a place to build a new nest.

So female bees have 2 parents, a male and a female whereas male bees have just one parent, a female. Here we follow the convention of Family Trees that parents appear above their children, so the latest generations are at the bottom and the higher up we go, the older people are. Such trees show all the ancestors (predecessors, forebears, antecedents) of the person at the bottom of the diagram. We would get quite a different tree if we listed all the descendants (progeny, offspring) of a person as we did in the rabbit problem, where we showed all the descendants of the original pair. Let's look at the family tree of a male drone bee. He had 1 parent, a female. He has 2 grand-parents, since his mother had two parents, a male and a female. He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one. How many great-great-grand parents did he have? So the farther back you go in your Family Tree the more people there are. It is the same for the Family Tree of everyone alive in the world today.

It shows that the farther back in time we go, the more people there must have been. So it is a logical deduction that the population of the world must be getting smaller and smaller as time goes on! Is there an error in Dee's argument? If so, what is it? Ask your maths teacher or a parent if you are not sure of the answer! 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More ..

Fibonacci numbers and the Golden Ratio If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers: 1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666..., 8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538... It is easier to see what is happening if we plot the ratios on a graph: The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1·618034 , although we shall find an even more accurate value on a later page [this link opens a new window] . You do the maths... What happens if we take the ratios the other way round i.e.

we divide each number by the one following it: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ..? Use your calculator and perhaps plot a graph of these ratios and see if anything similar is happening compared with the graph above. You'll have spotted a fundamental property of this ratio when you find the limiting value of the new series! 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More .. The golden ratio 1·618034 is also called the golden section or the golden mean or just the golden number. It is often represented by a Greek letter Phi . The closely related value which we write as phi with a small "p" is just the decimal part of Phi, namely 0·618034.

Fibonacci Rectangles and Shell Spirals We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1). We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles. Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature.

Such spirals are seen in the shape of shells of snails and sea shells and, as we see later, in the arrangement of seeds on flowering plants too. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So points on the spiral are 1.618 times as far from the centre after a quarter-turn. In a whole turn the points on a radius out from the centre are 1.6184 = 6.854 times further out than when the curve last crossed the same radial line. Cundy and Rollett (Mathematical Models, second edition 1961, page 70) say that this spiral occurs in snail-shells and flower-heads referring to D'Arcy Thompson's On Growth and Form probably meaning chapter 6 "The Equiangular Spiral". Here Thompson is talking about a class of spiral with a constant expansion factor along a central line and not just shells with a Phi expansion factor. Below are images of cross-sections of a Nautilus sea shell. They show the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows.

The chambers provide buoyancy in the water. Click on the picture to enlarge it in a new window. Draw a line from the centre out in any direction and find two places where the shell crosses it so that the shell spiral has gone round just once between them. The outer crossing point will be about 1.6 times as far from the centre as the next inner point on the line where the shell crosses it. This shows that the shell has grown by a factor of the golden ratio in one turn. On the poster shown here, this factor varies from 1.6 to 1.9 and may be due to the shell not being cut exactly along a central plane to produce the cross-section. Several organisations and companies have a logo based on this design, using the spiral of Fibonacci squares and sometime with the Nautilus shell superimposed. It is incorrect to say this is a Phi-spiral. Firstly the "spiral" is only an approximation as it is made up of separate and distinct quarter-circles; secondly the (true) spiral increases by a factor Phi every quarter-turn so it is more correct to call it a Phi4 spiral. Click on the logos to find out more about the organisations.

This poppy seed head has 13 ridges on top. Fibonacci numbers can also be seen in the arrangement of seeds on flower heads. The picture here is Tim Stone's beautiful photograph of a Coneflower, used here by kind permission of Tim. The part of the flower in the picture is about 2 cm across. It is a member of the daisy family with the scientific name Echinacea purpura and native to the Illinois prairie where he lives. You can have a look at some more of Tim's wonderful photographs on the web . You can see that the orange "petals" seem to form spirals curving both to the left and to the right. At the edge of the picture, if you count those spiralling to the right as you go outwards, there are 55 spirals. A little further towards the centre and you can count 34 spirals. How many spirals go the other way at these places? You will see that the pair of numbers (counting spirals in curing left and curving right) are neighbours in the Fibonacci series.

Here is a picture of a 1000 seed seedhead with the mathematically closest seeds shown and the closest 3 seeds and a larger seedhead of 3000 seeds with the nearest seeds shown. Each clearly reveals the Fibonacci spirals: A larger image appears in the book 50 Visions of Mathematics Sam Parc (Editor) published by Oxford and also available for the Kindle. Click on the picture on the right to see it in more detail in a separate window. Here is a sunflower with the same arrangement: This is a larger sunflower with 89 and 55 spirals at the edge: Sunflower AllPosters.com The same happens in many seed and flower heads in nature. The reason seems to be that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage, all the seeds being the same size, no crowding in the centre and not too sparse at the edges. The spirals are patterns that the eye sees, "curvier" spirals appearing near the centre, flatter spirals (and more of them) appearing the farther out we go.

So the number of spirals we see, in either direction, is different for larger flower heads than for small. On a large flower head, we see more spirals further out than we do near the centre. The numbers of spirals in each direction are (almost always) neighbouring Fibonacci numbers! Click on these links for some more diagrams of 500 , 1000 and 5000 seeds. Click on the image on the right for a Quicktime animation of 120 seeds appearing from a single central growing point. Each new seed is just phi (0·618) of a turn from the last one (or, equivalently, there are Phi (1·618) seeds per turn). The animation shows that, no matter how big the seed head gets, the seeds are always equally spaced. At all stages the Fibonacci Spirals can be seen. The same pattern shown by these dots (seeds) is followed if the dots then develop into leaves or branches or petals. Each dot only moves out directly from the central stem in a straight line. This process models what happens in nature when the "growing tip" produces seeds in a spiral fashion.

The only active area is the growing tip - the seeds only get bigger once they have appeared. [This animation was produced by Maple. If there are N seeds in one frame, then the newest seed appears nearest the central dot, at 0·618 of a turn from the angle at which the last appeared. A seed which is i frames "old" still keeps its original angle from the exact centre but will have moved out to a distance which is the square-root of i.] Phyllotaxis : A Systemic Study in Plant Morphogenesis (Cambridge Studies in Mathematical Biology) by Roger V. Jean (400 pages, Cambridge University Press, 1994) has a good illustration on its cover - click on the book's title link or this little picture of the cover and on the page that opens, click on picture of the front cover to see it. It clearly shows that the spirals the eye sees are different near the centre on a real sunflower seed head, with all the seeds the same size. Smith College (Northampton, Massachusetts, USA) has an excellent website : An Interactive Site for the Mathematical Study of Plant Pattern Formation which is well worth visiting. It also has a page of links to more resources.

Note that you will not always find the Fibonacci numbers in the number of petals or spirals on seed heads etc., although they often come close to the Fibonacci numbers. You do the maths... Why not grow your own sunflower from seed? I was surprised how easy they are to grow when the one pictured above just appeared in a bowl of bulbs on my patio at home in the North of England. Perhaps it got there from a bird-seed mix I put out last year? Bird-seed mix often has sunflower seeds in it, so you can pick a few out and put them in a pot. Sow them between April and June and keep them warm. Alternatively, there are now a dazzling array of colours and shapes of sunflowers to try. A good source for your seed is: Nicky's Seeds who supplies the whole range of flower and vegetable seed including sunflower seed in the UK. Have a look at the online catalogue at Nicky's Seeds where there are lots of pictures of each of the flowers. Which plants show Fibonacci spirals on their flowers? Can you find an example of flowers with 5, 8, 13 or 21 petals? Are there flowers shown with other numbers of petals which are not Fibonacci numbers?

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More .. Pine cones Pine cones show the Fibonacci Spirals clearly. Here is a picture of an ordinary pine cone seen from its base where the stalk connects it to the tree. Can you see the two sets of spirals? How many are there in each set? Here is another pine cone. It is not only smaller, but has a different spiral arrangement. Use the buttons to help count the number of spirals in each direction] on this pine cone. You do the maths... Collect some pine cones for yourself and count the spirals in both directions. A tip: Soak the cones in water so that they close up to make counting the spirals easier. Are all the cones identical in that the steep spiral (the one with most spiral arms) goes in the same direction? What about a pineapple? Can you spot the same spiral pattern? How many spirals are there in each direction? From St.

Mary's College (Maryland USA), Professor Susan Goldstine has a page with really good pine cone pictures showing the actual order of the open "petals" of the cone numbered down the cone. Fibonacci Statistics in Conifers A Brousseau , The Fibonacci Quarterly vol 7 (1969) pages 525 - 532 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.

On the trail of the California pine, A Brousseau, The Fibonacci Quarterly vol 6 (1968) pages 69-76 pine cones from a large variety of different pine trees in California were examined and all exhibited 5,8 or 13 spirals. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More .. Leaf arrangements Also, many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. If we look down on a plant, the leaves are often arranged 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. Here's a computer-generated image , based on an African violet type of plant, whereas this has lots of leaves.

Leaves per turn The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one. If we count in the other direction, we get a different number of turns for the same number of leaves. The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers! For example, in the top plant in the picture above, we have 3 clockwise rotations before we meet a leaf directly above the first, passing 5 leaves on the way. If we go anti-clockwise, we need only 2 turns. Notice that 2, 3 and 5 are consecutive Fibonacci numbers. For the lower plant in the picture, we have 5 clockwise rotations passing 8 leaves, or just 3 rotations in the anti-clockwise direction. This time 3, 5 and 8 are consecutive numbers in the Fibonacci sequence. We can write this as, for the top plant, 3/5 clockwise rotations per leaf ( or 2/5 for the anticlockwise direction). For the second plant it is 5/8 of a turn per leaf (or 3/8).

The sunflower here when viewed from the top shows the same pattern. It is the same plant whose side view is above. Starting at the leaf marked "X", we find the next lower leaf turning clockwise. Numbering the leaves produces the patterns shown here on the right. The leaves here are numbered in turn, each exactly 0.618 of a clockwise turn (222.5°) from the previous one. You will see that the third leaf and fifth leaves are next nearest below our starting leaf but the next nearest below it is the 8th then the 13th. How many turns did it take to reach each leaf? Leaf The pattern continues with Fibonacci numbers in each column! Leaf arrangements of some common plants One estimate is that 90 percent of all plants exhibit this pattern of leaves involving the Fibonacci numbers.

Some common trees with their Fibonacci leaf arrangement numbers are: 1/2 elm, linden, lime, grasses 1/3 beech, hazel, grasses, blackberry 2/5 oak, cherry, apple, holly, plum, common groundsel 3/8 poplar, rose, pear, willow 5/13 pussy willow, almond where t/n means each leaf is t/n of a turn after the last leaf or that there is there are t turns for n leaves. Cactus's spines often show the same spirals as we have already seen on pine cones, petals and leaf arrangements, but they are much more clearly visible. Charles Dills has noted that the Fibonacci numbers occur in Bromeliads and his Home page has links to lots of pictures. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More .. Vegetables and Fruit Here is a picture of an ordinary cauliflower.

Note how it is almost a pentagon in outline. Looking carefully, you can see a centre point, where the florets are smallest. Look again, and you will see the florets are organised in spirals around this centre in both directions. How many spirals are there in each direction? These buttons will show the spirals more clearly for you to count (lines are drawn between the florets): Romanesque Broccoli/Cauliflower (or Romanesco) looks and tastes like a cross between broccoli 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. How many spirals are there in each direction? These buttons will show the spirals more clearly for you to count (lines are drawn between the florets): Here are some investigations to discover the Fibonacci numbers for yourself in vegetables and fruit. You do the maths... 2 knuckles Is this just a coincidence or not????? However, if you measure the lengths of the bones in your finger (best seen by slightly bending the finger) does it look as if the ratio of the longest bone in a finger to the middle bone is Phi?

What about the ratio of the middle bone to the shortest bone (at the end of the finger) - Phi again? Can you find any ratios in the lengths of the fingers that looks like Phi? ---or does it look as if it could be any other similar ratio also? Why not measure your friends' hands and gather some statistics? NOTE: When this page was first created (back in 1996) this was meant as a joke and as something to investigate to show that Phi, a precise ratio of 1.6180339... is not "the Answer to Life The Universe and Everything" -- since we all know the answer to that is 42 . The idea of the lengths of finger parts being in phi ratios was posed in 1973 but two later articles investigating this both show this is false. Although the Fibonacci numbers are mentioned in the title of an article in 2003, it is actually about the golden section ratios of bone lengths in the human hand, showing that in 100 hand x-rays only 1 in 12 could reasonably be supposed to have golden section bone-length ratios.

Research by two British doctors in 2002 looks at lengths of fingers from their rotation points in almost 200 hands and again fails to find to find phi (the actual ratios found were 1:1 or 1:1.3). On the adaptability of man's hand J W Littler, The Hand vol 5 (1973) pages 187-191. The Fibonacci Sequence: Relationship to the Human Hand Andrew E Park, John J Fernandez, Karl Schmedders and Mark S Cohen Journal of Hand Surgery vol 28 (2003) pages 157-160. Radiographic assessment of the relative lengths of the bones of the fingers of the human hand by R. Hamilton and R. A. Dunsmuir Journal of Hand Surgery vol 27B (British and European Volume, 2002) pages 546-548 [with thanks to Gregory O'Grady of New Zealand for these references and the information in this note.] Similarly, if you find the numbers 1, 2, 3 and 5 occurring somewhere it does not always means the Fibonacci numbers are there (although they could be).

Richard Guy's excellent and readable article on how and why people draw wrong conclusions from inadequate data is well worth looking at: The Strong Law of Small Numbers Richard K Guy in The American Mathematical Monthly, Vol 95, 1988, pages 697-712. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More .. Always Fibonacci? But is it always the Fibonacci numbers that appear in plants? I remember as a child looking in a field of clover for the elusive 4-leaved clover -- and finding one. A fuchsia has 4 sepals and 4 petals: and sometimes sweet peppers don't have 3 but 4 chambers inside: and here are some flowers with 6 petals: crocus A sunflower with 47 and 76 spirals is an illustration from: Quantitative Analysis of Sunflower Seed Packing by G W Ryan, J L Rouse and L A Bursill, J. Theor.

Biol. 147 (1991) pages 303-328 Variation In The Number Of Ray- And Disc-Florets In Four Species Of Compositae P P Majumder and A Chakravarti, Fibonacci Quarterly 14 (1976) pages 97-100. In this article two students at the Indian Statistical Institute in Calcutta find that "there is a good deal of variation in the numbers of ray-florets and disc-florets" but the modes (most commonly occurring values) are indeed Fibonacci numbers. A quote from Coxeter on Phyllotaxis H S M Coxeter, in his Introduction to Geometry (1961, Wiley, page 172) - see the references at the foot of this page - has the following important quote: it should be frankly admitted that in some plants the numbers do not belong to the sequence of f's [Fibonacci numbers] but to the sequence of g's [Lucas numbers] or even to the still more anomalous sequences 3,1,4,5,9,... or 5,2,7,9,16,...

Thus we must face the fact that phyllotaxis is really not a universal law but only a fascinatingly prevalent tendency. But the tendency has behind it a universal number, the golden section,which we will explore on the next page. He cites A H Church's The relation of phyllotaxis to mechanical laws, Williams and Norgate, London, 1904, plates XXV and IX as examples of the Lucas numbers and plates V, VII, XIII and VI as examples of the Fibonacci numbers on sunflowers. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More ..

References and Links an article, usually in an academic periodical a link to a web page Excellent books which cover similar material to that which you have found on this page are produced by Trudi Garland and Mark Wahl: Mathematical Mystery Tour by Mark Wahl, 1989, is full of many mathematical investigations, illustrations, diagrams, tricks, facts, notes as well as guides for teachers using the material. It is a great resource for your own investigations. Phyllotaxis An interactive site for the mathematical study of plant pattern formation for university biology students at Smith College. Has a useful gallery of pictures showing the Fibonacci spirals in various plants. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .. More ..

Navigating through this Fibonacci and Phi site The Lucas numbers are formed in the same way as the Fibonacci numbers - by adding the latest two to get the next, but instead of starting at 0 and 1 [Fibonacci numbers] the Lucas number series starts with 2 and 1. The other two sequences Coxeter mentions above have other pairs of starting values but then proceed with the exactly the same rule as the Fibonacci numbers. These series are the General Fibonacci series . An interesting fact is that for all series that are formed from adding the latest two numbers to get the next starting from any two values (bigger than zero), the ratio of successive terms will always tend to Phi! So Phi (1.618...) and her identical-decimal sister phi (0.618...) are constants common to all varieties of Fibonacci series and they have lots of interesting properties of their own too. The links above will take you to further pages on this site for you to explore. You can also just follow the links below in the Where To next? section at the bottom on each page and this will go through the pages in order. Or you can browse through the pages that take your interest from the complete collection and brief descriptions on the home page .

There are pages on Who was Fibonacci? , the golden section (phi) in the arts: architecture, music, pictures etc as well as two pages of puzzles . Many of the topics we touch on in these pages open up new areas of mathematics such as Continued Fractions , Egyptian fractions , Pythagorean triangles , and more, all written for school students and needing no more mathematics than is covered in school up to age 16. There are no earlier topics - this is the first.

Question: Which is the next number in the Fibronacci sequence: 1, 1, 2, 3, 5, 8, 13, .....?
Answer: 21