Another common series of numbers in plants are the Lucas Numbers that start off with 2 and 1 and then, just like the Fibonacci numbers, have the rule that the next is the sum of the two previous ones to give:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..
But, no matter what two numbers we begin with, the ratio of two successive numbers in all of these Fibonacci-type sequences always approaches a special value, the golden mean, of 1.6180339... and this seems to be the secret behind the series. There is more on this and how mathematics has verified that packings based on this number are the most efficient on the next page at this site.
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
But the tendency has behind it a universal number, the golden section,which we will explore on the next page.
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 sequences3,1,4,5,9,... 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.
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.
References and Links
Excellent books which cover similar material to that which you have found on this page are produced by Trudi Garland and Mark Wahl:
means the reference is to a book (and any link will take you to more information about the book and an on-line site from which you can purchase it);
means the reference is to an article in a magazine or a paper in a scientific periodical.
indicates a link to another web site.
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.
Books by Trudi Garland:
Fascinating Fibonaccis by Trudi Hammel Garland.
This is a really excellent book - suitable for all, and especially good for teachers seeking more material to use in class.
Trudy is a teacher in California and has some more information on her book. (You can even Buy it online now!)
She also has published several posters, including one on the golden section suitable for a classroom or your study room wall.
You should also look at her other Fibonacci book too:
Fibonacci Fun: Fascinating Activities with Intriguing Numbers Trudi Hammel Garland - a book for teachers.
Mathematical Models H M Cundy and A P Rollett, (third edition, Tarquin, 1997) is still a good resource book though it talks mainly about physical models whereas today we might use computer-generated models. It was one of the first mathematics books I purchased and remains one I dip into still. It is an excellent resource on making 3-D models of polyhedra out of card, as well as on puzzles and how to construct a computer out of light bulbs and switches (no electronics!) which I gave me more of an insight into how a computer can "do maths" than anything else. There is a wonderful section on equations of pretty curves, some simple, some not so simple, that are a challenge to draw even if we do use spreadsheets to plot them now.
On Growth and Form by D'Arcy Wentworth Thompson, Dover, (Complete Revised edition 1992) 1116 pages. First published in 1917, this book inspired many people to look for mathematical forms in nature.
Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae) D Yanega, in Journal of Kansas Entomology Society, volume 69 Supplement, 1966, pages 98-115.
Because of the imbalance in the family tree of honeybees, the ratio of male honeybees to females is not 1-to-1. This was noticed by Doug Yanega of the Entomology Research Museum at the University of California. In the article above, he correctly deduced that the number of females to males in the honeybee community will be around the golden-ratio Phi = 1.618033.. .
On the Trail of the California Pine, Brother Alfred Brousseau, Fibonacci Quarterly, vol 6, 1968, pages 69 - 76;
on the authors summer expedition to collect examples of all the pines in California and count the number of spirals in both directions, all of which were neighbouring Fibonacci numbers.
Why Fibonacci Sequence for Palm Leaf Spirals? in The Fibonacci Quarterly vol 9 (1971), pages 227 - 244.
Fibonacci System in Aroids in The Fibonacci Quarterly vol 9 (1971), pages 253 - 263. The Aroids are a family of plants that include the Dieffenbachias, Monsteras and Philodendrons.
- Phyllotaxis - An interactive site for the mathematical study of plant pattern formation by Pau Atela and Chris Golé of the Mathematics Dept at Smith College, Massachusetts.
- is an excellent site, beautifully designed with lots of pictures and buttons to push for an interactive learning experience! A must-see site!
- Alan Turing
- one of the Fathers of modern computing (who lived here in Guildford during his early school years) was interested in many aspects of computers and Artificial Intelligence (AI) well before the electronic stored-program computer was developed enough to materialise some of his ideas. One of his interests (see his Collected Works) was Morphogenesis, the study of the growing shapes of animals and plants.
The book Alan Turing: The Enigma by Andrew Hodges is an enjoyable and readable account of his life and work on computing as well as his contributions to breaking the German war-time code that used a machine called "Enigma".
Unfortunately this book is now out of print, but click on the book-title link and Amazon.com will see if they can find a copy for you with no obligation.
- The most irrational number
- One of the American Maths Society (AMS) web site's What's New in Mathematics regular monthly columns. This one is on the Golden Section and Fibonacci Spirals in plants.
- 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.
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.