## What's New on Dr Ron Knott's Fibonacci and Phi Web site?

25 May 2006
When giving the talk below (May 12 2006) - which was about what Robert Langdon might have given in his lecture(s) on Symbolism In Art Class at Harvard, as given in the brief summary in chapter 20 on Dan Brown's "The Da Vinci Code" - I noticed several golden sections in the huge tapestry at the East end of Coventry Cathedral and checked them out on a print of Sutherand's huge work. What I found is detailed in a new section: Graham Sutherland's Tapestry in Coventry Cathedral on the Fibonacci Numbers and The Golden Section in Art, Architecture and Music page.
12 May 2006
An FRIDAY MAY 12th 2006 at Coventry Cathedral Do You Believe In The Da Vinci Code? where I talked about The Fibonacci Numbers and The Golden Section in Dan Brown's The Da Vinci Code' and other speakers addressed other issues raised by this popular book.
12 December 2005
Phi to 10,000 decimal places and to 25,000 binary places, computed using Maple, in separate text files with links to them from the Golden Section and Geometry page.
24 October 2005
After the section which shows that F(n)/F(n-1) tends to Phi as n gets larger, there is now a new section finding the ratios of F(n)/F(n-2), F(n)/F(n-3), ... and these ratios tend to Phi2, Phi3 etc.
16 August 2005
AAgh! The final section of the new the Fibonomials page is now included!
The large brackets are now rendered in a way that still allows Text Increase but should be visible now on all browsers.
6 August 2005
A new page! Finding recurrence relationships between the squares, cubes and all powers of Fibonacci numbers leads us to a table of numbers in many ways similar to Pascal's Triangle and the Binomal numbers (coefficients): they are called the Fibonomials.
28 May 2005
20 new formulae have been added to the Fibonacci and Phi Formulae page taking it to almost 240 formulae in total. The new ones are from A T Benjamin and J J Quinn's Proofs That Really Count (2003) and all the relevant Fib, Phi etc formula from that book are references on the new updated Formulae page. Benjamin's and Quinn's book proves each formula in a neat and fun way, using counting arguments based on domino patterns.
25 February 2005
Well in time for the next Pythagrean Triple Date (4/3/05), the Pythagorean Triples page has been updated with more sections, a calculator with very fast counting algorithms and a new Puzzles and Problems section.
8 February 2005
Two new pages:
The Fibonacci Sequence Visualiser is now online. It was designed for Casey Mongoven to use for his music composition and plots a series of numbers in a graph-like form.
Also, the Pythagorean Triangles page has been extended with two sophisticated calculators (no plug-ins or downloads needed).
7 January 2005
The sections on Pentagrams and Pentagons in the Two-dimensional Geometry and the Golden section or Fantastic Flat Phi Facts page have been simplified and updated.
25 November 2004
More on Index Numbers and Fibonacci Factors new section on the Mathematical magic of the Fibonacci Numbers page.
19 November 2004
More formulae added to Fib and Phi Formulae page. Note that the latest version of R Johnson's online paper (see references at the foot of the page) now includes an Appendix proving most of the formulae on the Fibonacci and Phi Formulae webpage.
17 November 2004
A new section has been added to the Mathematics of the Fiboancci Numbers page, where the Fibonacci number, F(i), has i as a factor: Fibonacci Numbers with Index number factor.
12 November 2004
The Fibonacci Number Trick now has an on-line "Bill" for you to practice the trick. The "trick" of appearing to have amazingly quick mathematical abilities depends on using the short-cut way to multiply any number by 11. The online calculator acting like "Bill" will provide as much practice as you need so you can do it flawlessly!
21 October 2004
Some details of California Polytechnic State University's plans for a new Engineering Plaza based on a Fibonacci spiral and other shapes has been added to the Modern Architcture section of the Fibonacci Numbers .. in Art and Architecture page
7 October 2004
A new Things To Do has been added to the Seed Heads section of the Fibonacci Numbers in Nature page.
New pictures of the fabulous Romanesque brocolli/cauliflower shown at the TSM 10 Conference in July at Peterborough and at the IMA NW Branch meeting in Manchester last night (6 Oct 04).
28 September 2004
Drawing on Dr Rachel Hall's Multicultural Mathematics pages (St Joseph's University, Philadelphia, USA) a new section on Fibonacci Numbers in Sanskrit poetry shows how the number series was known at least 70 years before Fibonacci wrote about them in 1202 AD.
25 September 2004
A new section Did Fibonacci invent his Series? on the Who was Fibonacci? page.
24 September 2004
A new section on three simple constructions of the golden section points (or gold points) from Kurt Hofstetter including a simple construction to find the "gold points" of a line using just your compasses.
24 August 2004
The complete text of the original version of Fibonacci's Rabbit Problem from Liber abaci from L E Sigler's English translation is now included on the Who was Fibonacci? page and more notes on this work and Sigler's other English translation of Fibonacci's Book Of Squares. Both books makes fascinating reading. I was impressed at the clarity of Fibonacci's mathematics and it opened my eyes to the sophistication of the mathematics of his day, using almost only words with no diagrams and no algebra - yet it is understandable and clear! The late Laurence Sigler has done an excellent job and must have been a real labour of love. Sigler's notes are invaluable too, putting Fibonacci's work into modern mathematical notation (with diagrams!). Fibonacci's Liber Abaci is worth buying for the huge variety of problems and solution methods the Fibonacci knew, in 636 pages, many of them with a practical application to solve.
5 August 2004
New formulae added to the Formulae page.
New flower pictures, leaves, seedheads on the Fibonacci Numbers In nature page.
15 July 2004
The Phi and a Trapezium (Trapezoid) section introduces Scott Beach's trisoceles trapezium that I've nicknamed the traphizium (or traphezoid if you speak American!).
2 July 2004
a new section on the index of the first Fibonacci number which has a given prime number p as a factor.
30 June 2004
Candido's and Aurifeuille's Indentities aded to the Formulae sheet
21 June 2004
The Eden Project in SW England (plants from many habitats housed in impresive biomes (geodesic domes) has a new Education Building in construction. It was designed using Fibonacci spirals. A new section on this is included in Fibonacci Numbers in Art, Architecture and Music: Modern Architecture
25 May 2004
I was surprised to learn that most numbers are palindromes (that is, they read the same forwards as backwards, e.g. 2002) in some base (e.g. 16 in base 3 is 121 and 24 is 44 in base 5). Of course every number is palindromic in base 1 (N is a list of N 1's) and in base (N-1) when N is just 11. Apart from those two bases, the numbers not palindromic in any base from 2 to N-2 are 2,3,4,6,11,19,47,53,79,103,137,139,149,163,167,179,223... (Sloane's A016038) where, after 6, they are all primes. So what about base Fibonacci (i.e the Zeckendorf representation) palindromes? The beginning of the list of Fibonacci palindromes is given in a new section Palindromic Fibonacci Representations on the Fibonacci Bases page. It looks like there should be a formula for this series - but what is it?
7 May 2004
New 3D stereo views of all the 5 Platonic solids have been added to the Golden Geometry of Solids page, with new solid views, and both the stereo and the solid images can be rotated using buttons. The stereo views use the same technique as those pictures of lots of dots that, when you go cross-eyed or alternatively, keep your eyes focussed in the distance, the eye sees a single image as if in 3 dimensions. They are also called "Magic eye" pictures or SIRDS (single image random dot stereograms).
29 April 2004
Three new Geometrical Gems have been added to the Flat Phi Facts page.
6 March 2004
An amazing diagram that shows how a circle of radius 2+2 Phi contains the angles 18, 30, 36, 45, 54, 60, and 72 degrees is now included on the Simple Trig values page.
5 March 2004
The Simple Trig values page and the Continued Fractions Calculator are now visible on the Fib and Phi Home page. Some sections expanded and two new Things To Do sections on the Introduction to Continued Fractions page.
3 March 2004
Two more patterns (from Ernesto La Orden, Madrid) added to the Table of Simple and Exact Trig Values
18 February 2004
Updated the section on the angles with exact trig expressions on the Exact Trig Values page;
added a new section on The √(2±Phi) pattern of expressons for 9, 18, 27, 36, 54, 64, 72 and 81 degrees.
11 February 2004
New formula for cos(12°), cos(24°) cos(48°) and cos(84°) added to Other Angles related to Phi section of 2D Geometry and the Golden Section page.
17 January 2004
The Phi and Trig Graphs section has been expanded to show how √phi and √Phi appear at the intersection points of the 6 trig functions.
22 December 2003
A new section on A surprising result about the REVERSE of a CF list has been added to the Introduction to Continued Fractions page, with some challenges for you to work on over this coming holiday season to find rules for the patterns and palindromes in the CFs of simple fractions.
29 November 2003
A new explanatory section on quadratic roots and continued fractions has been added in response to an interesting question from Anand Ramanathan: But what about the other solution to the quadratic.
An example of a Proof By Induction and an explanation of the methosd is now included on a new page relating to the first formula on the Phigits and Base Phi page.
A new section on the Formula for FIobnacci Numbers page: Find Fibonacci Number Fully discusses how to find Fib(n) efficiently as an exact integer even if it has many thousands of digits.
30 October 2003
New puzzle Steven and Todd on the harder Puzzles page.
New and updated sections and four new calculators on the A Formula for the Fibonacci Numbers page. Find the first few digits of and Fibonacci number and a quick test to see if a number is in the Fibonacci series or not.
22 October 2003
The Fibonacci bases and Other Ways of Representing Numbers has been updated with lots more information on using collections of the Fibonacci numbers that sum to n as a way of writing n (representing it). There are three new online calculators on that page too so you can painlessly investigate all sorts of properties of these collections, converting to and from different number bases, counting sets and bags (multi-sets) of Fibonacci numbers with a given sum, the Zeckendorf representations and other Fibonacci representation systems.
There are two new calculators on the The Golden section page.
Also, you can click on the calculator icons in the Contents sections of any page now to go straight to the calculators.
18 October 2003
A new section on Fib and Phi in Design and other miscellaneous and odd places has been added to the Art, Music and Architecture page (and the Turku power station section moved there too). If you can suggest more additions to this section, please do email me (address at foot of this page).
Which Fibonacci numbers are square? is answered in a new section on the Maths of Fibonacci Numbers page under Pythagorean Triangles.
Also, there is now an Interactive Calculator in the section Using the Fibonacci Numbers to make Pythagorean Triangles.
14 October 2003
The Formula page has a new section on Complex Number formula, 2 new references and some more formula from Knuth and Rabinowitz.
8 October 2003
All the Fibonacci and Golden Section pages are now HTML 4.01 compliant
3 October 2003
A new Harder Puzzle has been added (but it's not really too hard to prove) That's Odd!.
Detecting when N is a Fibonacci Number is a new section on the A Formula for the Fibonacci Numbers page.
25 September 2003
All Fibonacci and Phi pages now have the same style of Contents, headers etc. using Cascading Style Sheets supported by all the latest versions of the major browsers. If your browser has a problem displaying the pages please contact me at (shown this way to fool spambots, so replace AT by @ and DOT by . in the usual way).
Also, there is a major new Continued Fraction Calculator as a web page (using JavaScript, so ensure "Scripting" is on in your browser preferences). It is linked in to the Introduction to Continued Fractions page at the start of each Things To Do section.
15 September 2003
To mark this event, I've added a completely new page on General Fibonacci Series. It has several interactive calculators on the page to encourage you to do your own investigations without getting lost in the arithmetic. At present the page only covers one of the generalisations, choosing two different starting numbers, but even so the page is as full as any of the others at this site. Other generalisations (such as adding the previous 3 numbers rather than the latest two, etc) will follow some time in the future.
13 September 2003
A new section: Fibonacci Numbers and right-angled triangles is about Fibonacci numbers in triangles that are not Pythagorean. With thanks to Richard Van De Plasch who spotted the Lucas Fibonacci formula of 1876 that lies behind the idea in this section.
Are you going to be the 2,000,000-th visitor to the Home page? With close on 1000 Home-page visitors each day, the counter will pass the 2 million mark sometime today, tomorrow or Monday!
6 September 2003
A new section: Fibonacci and Phi in fashioning furniture has been added to the Art, Architecture and Music page.
3 September 2003
A new simpler method of writing down the sines and cosines of 15 and 75 degrees, using Ailles Rectangle, has been added to the Exact Trig Values page. I have some other simple forms to add soon when I can (find the time to) update my program to produce HTML fractions.
18 August 2003
Henri Lifchitz has recently found that Fib(397379) passes a large number of primality tests but has not definitely been proved prime yet. It has been added to the Largest Fibonacci Primes table as the largest probable prime. With thanks to Tony Noe.
14 August 2003
Two important new books are now available and highly recommended:
Fibonacci's Liber Abaci translated for the first time into English by L E Sigler, Springer Verlag (2002),672 pages.
This is the classic, historic and celebrated book that first popularized what we would now call Decimal Arithmetic and Algebra in Europe. Here we find the 10 digits and the decimal system of calculating (the title means The Book Of Calculating) and all the usual rules of arithmetic we meet in our first school days including, as an example in addition, the Fibonacci Rabbit Problem, that gives rise to the sequence known as the Fibonacci sequence in the late 1800's when E Lucas first invented the term.
New Visual Perspectives on Fibonacci Numbers by K Atanassova, V Atanassova, A Shannon and J Turner, World Scientific (Oct 2002)
introduces the idea of two intertwined Fibonacci-type series (2-Fibonacci series), recurrence trees and Gray codes, and a new Fibonacci vector as well as John Turner's goldpoint geometry (well known from his papers and presentations at the International Fibonacci Conferences and printed volumes) and fractals and tilings. Have a look at the publisher's description and the chapter titles.
These have been added to the Who was Fibonacci, Flat Phi Facts and Links and Bibliography pages.

9 August 2003
The Formulae page has been corrected so that fractions, binomial coefficients and symbols should appear in a form that all browsers should be able to display. They are shown on the next line:
alpha:α, beta:β, Phi:Φ, phi:φ; sigma:σ, tau:τ; infinity:∞, square-root:√
The Formulae page is now completely written in HTML so you can use the Text Zoom/View Larger feature of your browser to expand or shrink the whole page - which you cannot do with pages that use images for maths formulae. This should help visually impaired readers.
If you are sure that your browser is the latest version for your computer (and system) and you still don't see all the Greek letters and symbols above, then please email me fib AT ronknott DOT com, with your browser's name and version. I think all browsers now support these characters except
Mozilla 1.2.1 running under RedHat Linux 9
but I'd like to know of other exceptions too.
The old part of this FibLog (1999-2002) is now archived.
5 August 2003
Some redundant formulae removed from the Formulae page, and one formula corrected (sign missing).
Also, I realised the page you are now reading is a Blog (or 'Blog or WebLog) as it is a journal of what's happening at this part of my maths pages site, so ..... welcome to the Fib Blog or FiBlog ..... or should that be PhiBlog??!
10 July 2003
Some very nice, general Fibonacci and Lucas formulae from Bob Johnson added to the Fib, Phi and Lucas Numbers Formula page.
7 July 2003
A new simple proof from Tom E Ace that No neighbouring Fibonacci numbers have a common factor replaces the one using Cassini's Identity.
1 July 2003
The Introduction to Continued Fractions page has a new section Methods of finding continued fractions for square roots making clearer the algebraic process of finding an exact c.f. for any square root. The new Apple Safari browser for the Mac Os X revealed a problem with rendering some fractions on that page and this has been resolved too.
30 June 2003
Added a new section introducing the Fibonaccimals (Zeckendorf representations): Sumthing about Fibonacci Numbers
23 June 2003
The PDF version of the Formulae page is now in file FibFormulae.pdf and has been updated so the printing of background images (in order to show fractions properly) is no longer needed.
21 June 2003
Another Prime Curio and Neighbouring Fibonacci Numbers have no common factors have been added to the The Mathematical Magic of the Fibonacci Numbers page.
12 June 2003
Multimedia
On the Home page there is now a link to the BBC radio archive site so you can listen (on-line and at any time) to the third of Simon Singh's series of Five Numbers on the golden section on which I spoke. It was originally broadcast in 2002. [I shall never forget the day when we recorded my contribution -- 11 September 2001 -- because the news of the awful events at the World Trade towers in New York was just breaking as I travelled home.]
This is the latest form of interaction that this site uses to promote good communication, alongside interactive overlayed diagrams (e.g. pine cone spirals), animated diagrams (e.g. the Fibonacci Squares in a spiral), movies (e.g. golden section turns generating a flower head), and sound files (e.g. the golden string in sound) Geometer's Sketchpad animations (e.g. interactive demo of the golden angle), an interactive Fibonacci Number Calculator, and of course many pictures, tables, links to web sites and to books for follow up.
11 June 2003
Added section Phi and the Equilateral Triangle to the Flat Phi Facts page to show how a hexagon and equilateral triangle in a circle can be used to reveal a line with its golden section point.
The format of the Fibonacci and Phi Home page has been changed to make it a little more compact.
9 June 2003
Expanded the section on Polyominoes in the References and Links section of the The Golden Geometry of Solids or Phi in 3 dimensions page.
4 June 2003
• Added Catalan and d'Ocagne identities to the Formulae page (and pdf version). It seems printing the html formulae page from the Netscape 7 browser gives a PDF option which makes a much smaller file (161K) than Internet Explorer's (1100K) on my Mac. So I've found a use for my Netscape browser now !
• Four new photos have been added to the Fibonacci Numbers and Nature page: A lily and two of a passion flower in the Petals on Flowers section and a lettuce in the Vegetables and Fruit section.
• A new section of Fibonacci Primes has been added: Almost no primes next to Fibonacci's powers either!! which shows that not only are the neighbouring numbers of either side of Fibonacci numbers never prime, neither are the neighbours of the squares of Fibonacci numbers, nor cubes nor indeed any power except in very rare circumstances
1 June 2003
Robert Dawson has an interesting and simple argument that suggests there are only perhaps 20 Fibonacci numbers with an index number the same as their digit sums. The Digit Sums section of the Maths Of The Fibonacci Numbers page has been rewritten and extended to include his argument.
30 May 2003
Corrected the first special case of Vajda-20a (corrected the power of -1) on the Fibonacci Formulae page and updated the information on Vajda 20a's special cases, with thanks to David Webber of Glasgow University.
Also, abandoning Internet Explorer's "Save as PDF" option for printing the Formulae page, which produces a hugely bloated 1.8M PDF file, I used the old PrintToPDF Mac utility instead to get the new PDF version of the formulae page down to 84K!
28 May 2003
Tony Noe pointed out that some of the "primes" in the list (see 25 May 2003) turn out to be probably prime so I've updated the list to reflect this. I've included a new reference to some more Fibonacci Prime sources.
Also, my site has been awarded a MERLOT Classic award for Mathematics (one of 12 subject categories) for 2003. See the Awards section of the Home page.
25 May 2003
The table of known Fibonacci Primes on the Mathematics of the Fibonacci Series page has been extended by 3 new entries corresponding to the latest results up to March 2003. Now the largest known Fibonacci prime is Fib(201107) with 42029 digits.
11 May 2003
updated and extended a Things To Do Project on mathematical stamp collecting in the Flags Of The World section of the Fascinating Flat Phi Facts page.
3 May 2003
Jeremy Dunn told me that the volumes of the dodecahedron and icosahedron, two of the Five Platonic Solids, can be expressed in terms of Phi, so I've added this and other statistics to the Coordinates and other statistics of the 5 platonic Solids section. The Links and References section of that page has been expanded too.
28 April 2003
A search for dead links brought up about half a dozen which have now been corrected or removed. If you ever do find a link which does not work, please email me (see foot of page).
25 April 2003
Now the cube in a dodecahedron section also includes 5 cubes in a dodecahedron illustrated with buttons to reveal the separate and combined shapes together with an animation of the cubes-in-a-dodecahedron.
22 April 2003
Prof Susan Goldstine of Ohio State university has some excellent pinecone pictures and a link to her site has been added to the Pinecones section on the Fibonacci In Nature page.
She also suggested one more golden section relationship in the Platonic Solids section, that of a cube in a dodecahedron.
The Flags Of The World section of the Fascinating Flat Phi Facts page has had its broken links repaired and questions updated.
10 April 2003
I've given full references and links (for purchase) of Henry E Dudeney's 3 puzzle books: see Fibonacci Numbers in Nature page (after Fibonacci rabbits and Dudeney's Cows) and also on the Links and References page.
10 February 2003
Various small updates to references, a few more formulae on the Formulae page, (note that due to software changes on my computer, the PDF version is now 1.1M!), to layout, and a couple of small corrections have been implemented since the last What's New. A house move back to my native Lancashire (in November 2002) has taken up much of my time since then, but now I can catch up on some of the backlog of updates! Please note, my email is now FibandPhiATronknott.com (replace AT by @ in this address, of course). This form of description is an attempt to stop automatic sifting of my pages to harvest email addresses used by spam emailers. Get my updated contact details by clicking on my name as a link at the foot of all pages.
New sections: