What's New on the Fibonacci and Phi Web site: Archive for 1999 to 2002

20 July 2002
The Numerical relationships between Phi and its powers table has been corrected (the continued fractions with negative components were wrong and have been removed) and extended to powers in the range -9 to 9.
5 July 2002
Two new sections have been added to the Lucas Numbers page dealing with a relationship on Fib(2n)
26 June 2002
A new puzzle has been added to the Easier Puzzles page in the Puzzle about puzzles section. I rediscovered Smith College Mathematics Department's interactive site for the mathematical study of plant pattern formation by Pau Atela or Chris Golé of the Mathematics Dept at Smith College, USA. You must see it! Follow the new link added to the Fibonacci Numbers in Nature - Links section. Updated the seedPlotMAPLE file which plots the seed heads and animations so that it is compatible with the latest Maple version.
4 March 2002
A new investigation on the Digit Sums of Fibonacci numbers has been added to the Mathematics of the Fibonacci Series page, following a suggestion by Michael Semprevivo.
6 February 2002
Updates to Fibonacci Numbers and the Golden Section in Art, Music and Archtiecture page, include new references and a new photo of Turku Power Station showing the Fibonacci Numbers, with grateful thanks to Dr. Ching-Kuang Shene for letting me use his photograph here.
22 January 2002
Please note that my email address has changed to
because my Surrey university email is not available at present and the Altavista POP email service is no longer free.

A new continued fraction investigation on fractions formed from neighbouring square numbers has been added to the Introduction to Continued Fractions page - with thanks to Anthony Shaw for first spotting these patterns.

Fernando Hirujo tells me that Donald Knuth's third edition of "Fundamental Algorithms" (volume 1 of The Art Of Computer Programming) has more correctly stated the full name of Al-Khwarizmi from whom the word algorithm is derived. So I've corrected my reference in the Algorithm section of the Fibonacci biography page to reflect this.

6 November 2001
The 2 pages which were tables of factorized Fibonacci numbers from 0 to 100 and from 101 to 300 have now been combined into one page and rechecked using Maple.
The table from 301 to 500, unfactorized but with prime Fibonacci numbers indicated, remains the same.
Carmichael's Theorem (see note for 2 Nov 2001) applies to the Lucas Numbers (except for Luc(6)=18).
The Factorization of the first 200 Lucas Numbers page has been updated and extended to include 101-200 and also now shows characteristic prime factors also.
5 November 2001
Shane Findley of Dover, USA, brings a new observation about Fib(p) where p is a prime number and Carmicael's Theorem (see the additions for 2 November 2001): all the factors of Fib(p) where p is a prime number appear to be "new" (that is, they have not appeared as factors of any earlier Fibonacci numbers). See Fib(prime) and Carmichael's Theorem on the Mathematics of the Fibonacci Series page.
2 November 2001
  1. The Fibonacci Numbers up to Fib 500 are now colour coded: prime Fibonacci numbers are colour-coded and every Fibonacci number's unique prime factors that have not occured as a factor of any smaller Fibonacci number are shown in a separate colour too.
  2. These factors are shown in green up to Fib 200 and they are introduced in a new section on Fibonacci numbers and special prime factors on the Mathematics of the Fibonacci Numbers page.
  3. There is another new section on the same page pointing out that no Fibonacci number is next to a rpime (except for a couple of initial exceptions): No primes next to Fibonacci's!
  4. On the Easier Fibonacci Puzles page, yet another version of the Chairs In A Row puzzle has been included (and the 2 others have new names) called Chairs in a Row: The Antisocial Version.
9 October 2001
The Fibonacci Primes section has been updated to include all known prime Fibonacci numbers including the latest result of April 2001.
Simon Singh presented the third of five weekly programmes today on Numbers on BBC Radio 4 (9.30am to 9.45am BST). The first two were on Zero and Pi and today's was on the Golden Ratio where you would have heard me making a contribution. The final two are on the complex number "i" (October 16) and finally, Infinity (October 23). You can listen to them live and online (anywhere in the world) by clicking on the "Listen to Radio 4 Live" button on the BBC Radio 4 web page.
4 October 2001
I have found another pinecone that has a different arrangement of spirals than that already shown in the Pinecones section of the Fibonacci Numbers in Nature page. It has been added in with buttons to show the spirals.
27 September 2001
Formula Vajda-25a has been corrected on the Fibonacci, Phi and Lucas Formulae page and on the pdf version and split into two separate formulae. Two new formula have been added (between Vajda-11 and Vajda-12) communicated to me by Nick Vorobtsov from Russia.
15 September 2001
An interactive section on the cauliflower has been added to the Fibonacci Numbers in Nature page.
13 September 2001
The Exact Trig values page has been updated with a complete solution to the question of which angles (that are fractions of a whole turn) are expressible using only square-roots, due to Gauss.
10 September 2001
Shane Findley has as given me a fascinating and simple formula for pi/4 which complements the one on the Fibonacci Formula for Pi page. Whereas the Fibonacci formula uses only the odd-indexed Fibonacci numbers, Shane's uses only the even-indexed Lucas numbers! I've provided a full justification for his formula too using Wetherfield's useful notation of {t} for arctan(1/t).
27 August 2001
Fangming Shi of Ontario, Canada, pointed out that there is an important restriction on the formula for calculating the next Fibonacci number from the previous one on the Fibonacci Formula page. I've also changed the formula to a simpler one.
11 August 2001
A new Fibonacci prime is reported on the Mathematics of the Fibonacci series page.
A new Fibonacci Forgery: A simpler formula using Phi? that makes a nice application of logs.
29 July 2001
Following on from the update shown for 7 June 2001, I've put together a list of all the trig function values that have a simple exact expression that I know. It is an example of using a continued fraction generator to spot those trig values with a repeation (periodic part). I am not convinced that this is the complete list. What's missing? Email me if you can help.
23 July 2001
At last! I've replaced the poor quality pinecone picture with a new one. The two sets of spirals are now shown by pressing buttons.
4 July 2001
To celebrate the American 4th of July, I have included an incredibly beautiful picture of an Illinois prairie Coneflower from Tim Stone in the Seed head section of the Fibonacci Numbers in Nature page - thanks Tim!
7 June 2001
I've always been interested in exact expressions for angles and have expanded the section on the angles whose sine or cosine involve Phi or phi. I'd like to expand this some more, so if you know of exact expressions for angles that are not listed, let me know and let's see if we can get a list of them! But see the entry above for 29 July 2001.
Another relevant reference to the debate about Pi or Phi being used deliberately in the dimensions of the Great Pyramid has been added to the Phi and the Egyptian Pyramids? section.
8 May 2001
THe section about the History of Phi has been updated to debunk and provide some real data for discussion about whether the Egyptians used the golden section in the design of some pyramids. See Phi and the Egyptian Pyramids? which corrects some misleading statements I had on that subject. Thanks to Gyula Priskin for helpful information on this.
3 May 2001
Using the series expansions of cos(x) and the radian-measure formula cos(Pi/5)=Phi, we can find a formulae for Phi purely in terms of pi! We can do the same for phi too. A new section on this has been added to the Fascinating Flat Phi Facts page, which also corrects an error in the angle formulae.
Apologies that I still have lots of emails to reply to (I'm getting to yours soon!) not helped by the fact that I lost all emails sent during March and April! Sorry, but please can you resend your email if you sent one during that time and it'll be added to the queue awaiting a reply. Thanks!
29 March 2001
A couple of people have emailed me about the ratio of consecutive Fibonacci numbers getting close and closer to Phi and why that is. So I've added a new section in The Golden Section - The Number and Its Geometry which proves this, updating the graphs too.
There have been lots of small changes to add more references and update others, with two new awards added to the growing collection on the Home page.
A reference to Blair Kelly's excellent pages on the complete factorisation of all Fibonacci and Lucas numbers up to Fib(1000) and Lucas(1000) and partial results beyond has been added to the Fibonacci factors tables at this site.
24 January 2001
The list of Fibonacci numbers from Fib(101) to Fib(300) now has Louis Mello's' full factorisations of Fib(201) to Fib(300) [none of which are prime]. Thanks Louis!
Several minor changes have been made to other pages.
For those of you who have emailed me - thanks! Please accept my apologies if I have not yet replied to you. I have about 150 emails waiting for a reply at present going back to January 2000 :-( but don't let that stop you writing if you want to. Short emails get a fast response, often the same day. It's the longer ones or those with lots of detail that I have to find the time for - and I don't get paid for this now I'm no longer a lecturer! Thanks for your patience and I *do* appreciate all your emails.
8 November 2000
Dimtry Portnoy has suggested another Fibonacci Puzzle about Boat Building which I have added to the Easier Fibonacci Puzzles page.
16 October 2000
Dave Wood has found a Fibonacci Number Trick.
27 September 2000
Minor updates to Fibonacci Biography (new link added), Benford's Law (new link added).
New section on the Golden String page: The number of additions when computing f(n) by the recursive definition.
22 June 2000
The Fibonacci Primes has been extended a little (again) and now has two sections: Fibonacci Primes and A Prime Curio.
21 June 2000
The Fibonacci, Golden Mean and Golden String pages have been checked for broken links by Linkguard's free service. It seems an excellent resource, one I shall be using regularly.
It picked up some dead links on the References page, the Art links and the Fibonacci in Nature page, all now corrected or removed.
12 June 2000
A nice application of the Fibonacci Representation of a number has been included on the Fibonacci Bases page.
The Fibonacci primes have been re-organised, and expanded on the Maths of the Fibonacci Series page.
31 May 2000
Do you want to be informed automatically when there are changes to this site? Try out SpyOnIt's free service (see the logo at the foot of the Fibonacci Home Page).
The changes have been averaging no more than one or two per month recently.
14 May 2000
Two more puzzles have been added to the Easier Fibonacci Puzzles page.
10 April 2000
The references on Fibonacci's life and times have been extended on the Who was Fibonacci? page.
31 March 2000
Floor van Lamoen's student, Roy Nauw of Kloetinge, the Netherlands has found a new variation on the Fibonacci Jigsaw puzzle. The principle is the same as the 64=63 and 63=65 jigsaw "proofs" but it is based on a different underlying Fibonacci formula. With thanks to them both for permission to use the puzzle on this web site, it is now included on the Harder Fibonacci Puzzles page.

The Google Search Engine is, in my view, the best on the Web! It always produces *just* the page I was after as the first in its list, due to advanced searching algorithms based on the popularity of a web page. You can now try it out both to search this site and to search the web, on the Home Page.
Google now replaces the BeSeen search box.

3 March 2000
Look out for The Fibonacci Numbers poster on the London Underground!
WWW references to Eric's Treasure Trove of Mathematics have now been updated to go to the Wolfram site.
Also, a terrific demonstration of Geometer's Sketchpad facilities is now online, illustrating what happens when you dynamically alter the angle between seed on a seedhead and why the golden section angle gives the best seed packing. I developed this demonstration (with the aid of Sketchpad lecturer and expert: Annie Fetter) while the Math Forum for a Summer School in June 1998.
24 February 2000
Fib(101) to Fib (200) are now completely factorized on Fibonacci numbers 101-300.
The page of Formulae for Fibonacci and Lucas numbers and Phi has been updated and is now available in PDF format.
2 February 2000
Prof Ramon Glex-Regueral of Madrid detected two (typographical) errors in the Factor Table of the first 100 Fibonacci Numbers, affecting factors of Fib(95) and Fib(96). These have now been corrected. The format of the table was also altered to use x instead of * for multiplication and to show powers as proper superscripts e.g. 23 instead of 2^3.
30 November 1999
Small changes to The Mathematical Magic of the Fibonacci Numbers page.
Updated the seedPlotMAPLE file which plots the seed heads and animations so that it is compatible with the latest Maple version. Reduced the size of the Calculator gif file (185 bytes only now!) and altered the map of Pisa.
The Amazing Mathematical Object factor URL has been corrected on the first Puzzles Page.
13 November 1999
The Art links on the The Golden Section in Art, Architecture and Music page have been updated. The Uffizi Gallery site seems to have been down for a few days. If this persists, I'll remove its URL.
29 October 1999
Rob Baxter noticed a simple and intriguing way of generalizing Phi to find other numbers with the property that when they are squared, their decimal part remains the same, just like Phi = 1.618033.. and Phi2 is 2.618033.. . This is added as a new section on Similar Numbers on the (re-titled) The Golden Section - the Number and Its Geometry page.
1 October 1999
In order to make the Fibonacci Calculator easier to find, a link icon Fib calculator has been added to the (revised) Fibonacci, Phi and Lucas "rulers" used on many of the Web pages here. The rulers now appear uniformly as:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..Calculator

1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..Calculator

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..Calculator

29 August 1999
The order of pages has been changed to put more complex pages later and to make the structure of the site more apparent. Navigation areas at foot of each page changed to reflect the new Contents structure.
26 August 1999
Study Web has chosen another page at this site for an "Academic Excellence" award - the Who was Fibonacci page. Two new sections have been added: Fibonacci's Mathematical Books and References to Fibonacci's Life and Times.
14 August 1999
FreeFind have provided a search this site facility added to my Fibonacci and Phi Home page. It also includes an excellent Map of the Fibonacci and Phi pages at this site in three formats - Outline and List, which are alphabetical, and Table. It is free and seems to be better at finding subjects via keywords than the BeSeen searcher. On the down side, it includes banner adverts (not under my control) - if you want one for your site, see the FreeFind home page.
Which do you find is the better searcher? Please do email me with your opinions.
1 August 1999
This site now has most book references linked to Amazon.com's site where you can find more information about the book and even order it online, often with a discount! You can order any book, not just those listed. By so doing you will be helping me to continue to develop this site (I am not paid to do so) as I will earn a small commission from every order you placed from a link on this site.
23 July 1999
Amended section on bees on the Fibonacci Numbers and Nature page. Doug Yanega, an Entomologist, had already discovered that there are more female bees than males in honeybee colonies, and corrected my impression that this was true of all bee species.
This site was included in University of Wisconsin-Madison's Internet Scout Report for July 21st.
7 July 1999
Added: A more complete Flags site on to the Flat Phi Facts page.
Added: A reference to the Rabbit constant evaluated to 330 dps on the Continued Fractions page.
Added: A reference to Phi to 10,000,000 decimal places on the Golden Section page.
4 June 1999
Do you remember that when you had to line up at school, it was almost always in alphabetical order, so you always stood next to the same one or two people every time? Here is a new puzzle about choosing a group from the class that has no alphabetical neighbours in it, so the group members get to meet different people in the class for a change. How many such groups are there in a class of n people? A Fibonacci number of course!
25 May 1999
A new section on Pythagorean triangles and the Fibonacci Numbers has been added to the The Mathematical Magic of the Fibonacci numbers. Take any 4 consecutive Fibonacci numbers, such as 1,2,3 and 5. From these there is a simple method to generate the Pythagorean triangle 5,12,13. In fact, we can start from any two numbers a and b and find the next two as if they were in a Fibonacci-like series ("add the latest two to get the next") and we have a,b,a+b,a+2b. All Pythagorean triangles (or triples) can be generated in this way!
21 May 1999
A Search engine for this Fibonacci and Golden Section site!
As an experiment, I am trying out the BeSeen search engine facility which can search just this Fibonacci site for any topic you like.
It will give you any Web pages here that contain the word(s) of your search. Open a page shown in the list and then use your browser's "Find..." facility (under the "Edit" menu on Internet Explorer and Netscape Navigator) to find where the words occur on that page.

I'd be grateful for any feedback with this new service so please let me know if you like it or if you get problems with it (tell me what browser you are using, version number and what kind of computer you used)! See for email and other ways to contact Ron Knott.

1 May 1999
The Fibonacci and Phi Formulae
has been updated and expanded. Formulae are arranged to show similarities.
26 April 1999:
The promised large collection of Fibonacci and Phi Formulae is here!
It uses HTML Tables to format the formulae and only a couple of tiny images for special symbols so it loads very fast and is viewable on all browsers. In making up this page, errors were discovered in the large collection of such formulae in Dunlap's book. The formulae have been checked by computer so these are bound to be error-free :-) !
The JavaScript Fibonacci Calculator 1.2
can find all the digits of several thousand Fibonacci and Lucas numbers and the Rabbit sequence as well as the first few digits of Fib(1,000,000) and beyond! It is all on one Web page - no need to download software, just click on the link above.
How random are the initial digits of Fibonacci Numbers?
The Mathematical Magic of the Fibonacci Numbers page now has a new section on how the initial digits of the Fibonacci numbers are distributed. You might think that a randomly chosen Fibonacci number would be as likely to begin with a "1" as with a "9" - but you'd be wrong. They much prefer a 1, then 2 is the next most popular and so on in order. Rarely do they start with 9.
And on the topic of randomness....
A new mathematical constant has been discovered
by Divakar Viswanath of Mathematical Sciences Research Institute, Berkeley, California. The ratio of successive numbers in the Fibonacci (and Lucas and generalized Fibonacci sequences) gets closer to Phi the further we go along the series. But suppose that instead of always adding the last two terms to get the next, we randomly added or subtracted the previous from the one before that. The numbers in such a series would go up and down but do they also tend to a limit on the average? Viswanath has found that they do and discovered that ratio constant to be 1.13198824... . Read about it in Keith Devlin's Devlin's Angle, a column on the Mathematical Association of America (MAA) site. [Sorry for the bad link - this should work now!] You can read Viswanath's paper with more details (but these are research papers and at postgraduate mathematics level) at his web site (the link is broken - 19 June 2000).
The Lucas Numbers page
continues to be updated and expanded (with an improved way of presenting mathematics on a web page without using images).
This Fibonacci site was started in 1995 and submitted to major search engines in March 1996 (which is when the counter on the Home page was started too). The date at the foot of each page is its date of the latest change.
The latest changes
Archive of FibLog for 2003 and 2004

Back to FIB Home Fibonacci Home PageCalculator

Valid HTML 4.01! © 1996-2007 Dr Ron Knott