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

15 December 2021:
Number Bases : updated and extended:
all with Calculators for your own investigations.
3 November 2021:
The A Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates page has been updated.
9 March 2021
There is a new section on Fibonacci Factors of F(2n)
The Reference page with many Fibonacci and Phi Formulae has had some more results added


13 January 20201
There is a new section in the More on Polygonal Numbers page on the Aztec Diamonds

1 January 2021
A page on the fun and unusual facts about the number 2021, with puzzles too! 2021
April 2020
Ramanujan would have been 100 in April 2020 :
to commemorate this, there is a new page introducing from scratch

Integer Sums or Partitions of an integer

with Calculators; leading to a simple equation
involving the Pentagonal Numbers
that even Euler found remarkable and could not explain

24 July 2019
University of Surrey Open Days

Saturday 14 September 2019
Sunday 13 October 2019

Dr Ron Knott will be hosting the "You do the Maths..." informal sessions throughout the day
in the Mathematics Department (top floor, AA Building)
Come and say Hi, try some hands-on maths and meet current students and staff.

See this map of the campus
and other info you need is here.

A Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates

This page is designed to link into Clark Kimberling's excellent site:
Encyclopedia of Triangle Centers (ETC)

It provides an automatic lookup of a centre using Peter Moses lists for the 6-9-13 triangle
and is self-contained with an introduction for those new to trilinear and barycentric coordinates.

5 January 2019
The Primes and Factors page has been updated. It computes primes, factors and provides examples of Goldbach's conjecture in its "strong" form (from 4 on, every even number is a sum of 2 odd primes) and "weak" form (every odd number from 9 onwards is a sum of 3 primes) ... and to start your new year, a joke from Facebook's Easy Beautiful Maths group
i 1 right angled triangle has hypotenuse 0!
Happy New Year!
2 November 2018
The Polygonal and Figurate Numbers page has been updated and contains only polygonal number topics.
For 3D solid shapes and other 2D shapes including central polygonal number and shapes using matchsticks see the new page : More on Polygonal Numbers
22,23 June, 15 September 2018
University of Surrey OPEN DAYS with interactive sessions on "Is everything in Maths Always TRUE or FALSE? or When maths goes wrong!" about puzzles and paradoxes in Logic suitable for a general audience but of particular relevance to students applying to Maths Degree courses
March 2018
The 2018 International Fibonacci Numbers and Applications Conference will be in Halifax, Nova Scotia, Canada at Dalhousie University, July 1-7, 2018.
The official conference website in new window has more details of accommodation, travel, submitting an abstract for a talk and how to register.
26 January 2018
MathsJam and IMA joint meeting open to all
Manchester MathJam and the NW Branch of the Institute of Mathematics and its Applications are having a joint meeting on Wednesday January 31 starting at 5:15pm after refreshments from 5pm in the Frank Adams Room 2 of the Alan Turing Building at Manchester University on Oxford Road, postcode M13 9PL.
It is FREE and open to all and will include several 5 minute presentations Mathjam-style on popular and recreational mathematics, puzzles or games. Dr Knott will be there giving a 5-minute presentation on new results in Number Palindromes.
Update to Fractions page
The Farey Fractions page has been updated to include the Ford Circles, a beautiul way to visually represent all fractions. The References now has details of a new and very useful book "Neverending Fractions".
6 November 2017
Proceedings of the 2016 International Fibonacci Conference at Caen, France.
The 2016 Conference Proceedings in new window and all papers, conference report, list of attendees and Problems Section are now available online.
The 2018 Conference:
will be in Halifax, Nova Scotia, Canada, July 2-6, 2018. More details to follow soon.
15 August 2017
To celebrate one of the Pythagorean dates this year (15/8/17 in UK or 8/15/17 in USA) the Pythagorean Triangles page has been updated with a Sums of Squares Calculator to find Pythagorean Quadruples and longer lists of square numbers whose sum is a square plus lots of new Puzzles and Problem in a new section
17 May 2017
The Egyptian Fractions page has been updated and some of the Calculators now allow for very long integers.
19 April 2017
A workaround to the Mac iOS 10.3 Javascript bug which caused the new Calculator to hang (see 13 March entry below) has been installed :-)
30 March 2017, 3 April 2017
An update to Apple mobile iOS 10.3 on 30 March and 10.3.1 on 3 April has a JavaScript bug which causes the new Calculator (see 13 March) to hang. This affects iPad and iPhones only and appears on ALL browsers on those devices using the new Calculator WITH THE UPDATE ONLY. Previous versions of iOS (10.2) do not cause this problem and it does NOT appear on any browser on Mac laptops and desktop computers.
Until this is resolved, please use the older version on iPad and iPhone.
25 March 2017
The new Calculator (see next item) now includes in inverse trig functions (arcsin, arccos and arctan) and inverse hyperbolic functions (arcsinh, arccosh, arctanh).

The Exact Trig Values for Simple Angles now includes a new section on how to use the new Calculator (mentioned in the 13 March 2017 item) for evaluating trig functions.

13 March 2017
Updates continue on various pages and there is a revised section on reverse and other transforms of continued fractions here

There is a new Decimal↔Fraction↔Continued Fraction Calculator, completely revamped to use very long decimal numbers, exact decimal fractions including the recurring part, CF terms and a periodic part together with an Expression Evaluator with as many decimal places as you might wish.
The old Fraction Calculator and Continued Fraction Calculator will remain for now but will eventually be replaced by this new and extended version.

The University of Surrey Open Days in 2017 will again host Dr Ron Knott's popular Mini-Maths talks on:
Friday 30 June,
Saturday 1 July,
Saturday 16 September,
Saturday 14 October
These are free 15-20 minute "fun" talks on maths suitable for a general audience and held in the Mathematics Department on top floor of the AA building: see this PDF map of the campus.
On each day Ron will be giving three different presentations at 11:45, 12:30 and 14:15.
For more details of how to get to the University of Surrey in Guildford, here is some travel information for train, coach and car. Everyone welcome!
15 September 2016
of special interest to all maths teachers and for all new PGCE students but open to all, is the IMA NW Student Teacher Lecture "A Date with Mathematics" about the rich maths properties of the numer 2016 and some hints on keeping maths interesting and fun in the classroom.
This talk is at 4:30pm in room B4.4 in Manchester Metropolitan University on Thursday 15 September (in building 77 on this PDF campus map, next door to the Contact Theatre on Oxford Road)
13 March 2017
Updates continue on various pages and there is a revised section on reverse and other transforms of continued fractions here.
There is a new version of the CF Calculator, completely revamped to use decimal numbers, Cf terms and fractions with as many decimal places as you might wish. The old version will remain for now but will eventually be replaced by the new and extended version
12 September 2016
Updates and a new section on Every number starts some Fibonacci number on the Maths of the Fibonacci Numbers page;
Fibonacci Numbers Calculator version 3, extended and updates
9 September 2016
University of Surrey Open Days Mini-Maths talks Three free 20 minute mini-maths talks in the Mathematics department, 10:30, 11:30 and 1pm as part of the University of Surrey Open Days.
After encouraging numbers of attendees this year, the Maths department plans to continue these free 20 minute general talks on Open Days next year...watch this space!
18 July 2016
The Formula for Fibonacci Numbers page has been updated with an extended section on finding the index number i of the nearest Fibonacci number Fib(i) less than or equal to a given number.
16 January 2016
Updates to the Runsums pages: Introductory page, More on Runsums, updates to the Calculators and Tabulator
1 January 2016
To celebrate the new year, here is a new page on the number 2016 with lots of fun facts and picture and a puzzle or two aimed at the general reader - no advanced maths needed!
It was inspired by a search for 2016 in Neil Sloane's On-line Encyclopedia of Integer Sequences.
23 July 2015
Updates and new sections added to the Egyptian Fractions page, including two puzzles.
Various updates to several other pages.
1 March 2015:
A new page on Polygonal Numbers is launched today.
Promised at the Fibonacci Conference in June 2014, it is now live, ready for the IMA NW Sixth Form Lecture at Manchester Grammar School at 2pm Monday 2 March 2015.
11 November 2014:
Updates and new hints and answers on the Pythagorean Triangles (PTs) page; a new section on a fascinating connection between the Fibonacci method of generating PTs and a series of arctangent formulas for π
28 October 2014
Updates, bug fixes and a new Galton Quincunx simulator (balls dropping through an array of pins to produce the Bell curve) on The Mathematical Magic of the Fibonacci Numbers page.
3 August 2014
Updates to the Fibonacci and Phi Formulae page include a new section on the Fibonomials
20 July 2014
Many updates and expanded sections on the Mathematical Magic of the Fibonacci Numbers page.
Launch of two experimental pages on Linear Recurrence Relations related to the topics on these pages: Fibonacci series:General, Lucas, others; Phi; Fractions:Continued, Egyptian, Farey; Figurate Numbers (with separate pages coming soon on this new topic!).
These are still experimental and not fully checked but comments on the style and organisation etc would be appreciated.
There is a fast-loading HTML version and a another version using MathJax.
20 June 2014
IMA at 50 IMA50 image 3-4 July 2014: Festival of Maths and its Applications
The IMA are celebrating their 50th anniversary with two days of free talks (but you need to register) for schools and the general public - and lots more - from 10am to 4pm, 3 & 4 July 2014 in the Alan Turing Building at the University of Manchester, UK.
Further details here
I shall be talking on Food, Figures and Phi at 10:30am on Thursday July 3rd
and explaining my diagram that appears in the book celebrating this event: 50 Visions of Mathematics, Sam Parc (Editor) published by Oxford, 2014. Read more on Fibonacci Numbers in Nature: seed heads
15 May 2014
Various updates and new sections, one new Calculator, on the Fractions and Decimals page.
9 May 2014
New section on How to turn a Series into a decimal fraction on the Fractions page;
new format for maths display;
automated generation of Contents Of This Page introduced on some pages.
25 April 2014
The Fibonacci Calculator has been updated to version 1.3 A fuller account of the "Amaze me" button and the two number tricks is now on the Lucas Numbers page.
12 December 2013
Bug fixes and updates on the Egyptian Fractions page and an expanded section on the Erdös-Straus Conjecture that it is always possible to write 4/n as 1/x + 1/y + 1/z
29 October 2013
A new section Ratio of Area to Perimeter has been added to the Pythagorean Triangles page, thanks to Frank Rubin
1 October 2013
Inspired by an email from David Terr, his discovery of some pattens in certain powers of fractions are included in a new section on the patterns in CFs of some forms of Proper Fractions on the Introduction to Continued Fractions page
2 September 2013
Upgrading of Exact Trig Values page
16 August 2013
Updates and corrections to lists of perimeters shared by exactly 2 (3,4,...) Pythagorean triangles and Primitive triangles. Update to the Prefix Triples with new results from John MacMahon.
10 July 2013
Expanded the Continued Fractions page to include more on Noble numbers and added a new section on Near Noble numbers. The notation has now been made consistent with the Calculator Continued Fraction Calculator so that a CF is a list of numbers and a final periodic parts is enclosed in [ and ] brackets, as was used by the Calculator.
1 July 2013
Updated links to OEIS on Pythagorean Triangles page
31 May 2013 Updates to Ailles Rectangle to show an alternative image with the sizes.
Change to one table for consistency of arrangement of cosines and sines.
24 May 2013
Updates to the Pi and the Fibonacci Numbers page to include a table of nice 2-term arctan relationships in the More formulae for two angles section
3 March 2013
The Fractions Calculator has been updated and upgraded so that now if works correctly on Google Chrome and also now computes whether fractions terminate or recurr and all the details in any base not just decimal.
8 January 2013
Fib and Phi Formulae updated and corrected;
All links to Neil Sloane's Online Encyclopedia of Integer Sequences now point to the correct sequence on its new site.
24 June 2011
Updates to the Pythagorean Triangles page now includes a section on the three excircles and how the Fibonacci method of generating pythagorean triangles makes several formulae simpler, together with more on primes in pythagorean triples
13 May 2011
More images of Alexis Monnerot-Dumaine's rabBIT Fractal and mine, with detailed instructions of how to make them.
20 April 2011
Major update and extention with new sections added to The RabBIT Sequence (the Golden string) and a new comprehensive calculator.
A new section covers patterns that can and cannot occur in the Golden String (10110101101....), where they occur and some surprising results on these matchings. Another new section discusses the recently-found stones-and-pots game (mancala-type) that gives the spectrum of Phi. Finally, a new fractal curve based on the golden string is also included.
18 April 2011
Updates to the Pythagorean Triangles page with new sections on another simple fraction method of generating Pythagorean triples and the Calculators on this page now have buttons to lengthen or shorten the results area which makes the calculators more usable on mobile phones and iPad which do not provide scroll buttons on internal sections sometimes
4 February 2011
Some new formula for higher powers added to the Fib and Phi Formulae page
28 December 2010
Several small updates on several pages and some additional results about areas of Pythagorean triangles are included, as well as some interesting facts about Fibonacci numbers and Pythagorean triangle areas.
4 October 2010
The Continued Fraction Calculator has had three new functions added to use in input: Fibonacci: F(n), Lucas L(n) and General FIbonacci G(a,b,n);
Two new buttons manipulate the CF : to reverse it and to take its reciprocal
The second results area can be collapsed to make more room for the main Results.
1 October 2010
The old Farey Fractions page has been extended and renamed to fareySB.html with a new title: Fractions in the Farey Sequences and the Stern-Brocot Tree. It now a new section and calculator on the Stern-Brocot Tree of Fractions.
22 September 2010
A new section debunking the myths about the positions of hands on a clock forming golden rectangles has been added to the Fibonacci Numbers and The Golden Section in Art, Architecture and Music page.
23 July 2010 The Fibonacci Quarterly has now been scanned electronically and made available online except that the latest editions are only accessible to those who subscribe to the journal.
5 May 2010
Several people have asked me for an explanation of the 63=64 puzzle so I have added a "spoiler" button which will reveal a diagram showing the answer.
18 January 2010
More formulae added to the Fibonacci and Phi Formula page, in particular, almost all of the Hoggatt (1969) formulae are now included, along with a change of style (formatting) of the formulae and references. This is to facilitate distinguishing I2 from 12 in formula references because Hoggatt's formulae names all begin with I not 1.
11 January 2010
More Vajda formulae added and corrected the second form of Johnson-11 on the Fibonacci and Phi Formula page
9 December 2009
Some updates to the section on reversed CFs on An Introduction to Continued Fractions
28 September 2009
The URL of this site has now changed from

The original URL will redirect to the new site automatically for the foreseeable future.
My contact page and details of all my Maths web sites is still the same at www.ronknott.com.

23 September 2009
We're back - with apologies for the 6 days when this site was inaccessible due to major updates to Surrey University website computers.
If it ever happens again try this:
  • use Google and search for "Fibonacci" to find the URL to this FIbonacci website at www.mcs.surrey.ac.uk/...
  • then use Google's Cached link to retrieve a copy.
  • Animations, links, movies should all work but links to other pages at the Surrey Fibonacci site will not; follow the above procedure to access them individually.
You will need to do this for my other pages at MCS at Surrey too until the site is back online.
9 August 2009
Large tables have been made scrollable on the Pythagorean Triangles page;
New section on Smallest Denominators of Egyptian Fractions.
9 July 2009
More on the Pythagorean triples formed by prefixing an extra digit on the front such as 5 12 13 with a "1" on the front gives the (valid) triple 15 112 113
29 June 2009
Two corrections to the Fibonacci and Phi Formulae page plus some more Generating Functions.
23 June 2009
A New section on Generating Functions for various Fibonacci, Lucas and Generalised Fibonacci series has been added to the Fibonacci and Phi Formulae page; the PDF version has been updated too.
6 June 2009
A new puzzle Another one! of the rods variety (compositions of integers).
8 March 2009
Penrose Tilings has a new animation Dissecting the Sharp and Flat Triangles.
27 February 2009
New graphs of the numbers of primitive and all Pythagorean triangles
1 February 2009
Updated and revised Pythagorean Triangles page:
27 January 2009
A new section on Continued Fractions connecting π, Φ and e also refers to the latest revised and updated edition (the 6th) of Hardy and Wright's classic book Introduction to the Theory of numbers.
9 January 2009
A new section on an irregular tiling similar to the Penrose tilings, but even simpler: Another Irregular tiling using Phi: the Amman Chair
30 December 2008
Happy New Year!
The mark a New Year, there is now a new panel on the Home page with links to my other Maths pages.
The Runsums Calculator has been updated and extended.
7 November 2008
The Fibonacci2 Base system has been added to the Fibonacci Bases page and has a new Calculator.
31 July 2008
Photos from the 13th International Conference on Fibonacci Numbers and Applications held at Patras, Greece, 5-12 July 2008
3 July 2008
A new section on the Introduction to Continued Fractions page on a nice method of converting a continued fraction to a simple fraction
30 June 2008
The Fibonacci Representations page has some a new section Using only the Negatively Indexed Fibonacci numbers. The Some names have been changed and
17 June 2008
An email from John Bibby gave me the idea of a new Fibonacci magic trick using the Fibonacci (or Zeckendorf) representations of number. It is a Think Of A Number trick and an online calculator will produce the cards you need for the trick.
3 June 2008
The Two fractions method of generating all Pythagorean triangles is now included, with a calculator
19 May 2008
Many updates and more new sections on the Pythagorean Triangles page
28 April 2008
More on Pythagorean dissections added
18 April 2008
A new section on visual proofs (proof without words or look-and-see proofs) of Pythagoras' Theorem has been added to the Pythagorean Triangles page. Also, a primary student from Australia, Penny Drastik, has found some new and smaller solutions to one of the Pythagorean Triangles Jigsaw (dissection) puzzles!
14 April 2008
The Pythagorean Triangles page has been expanded with many new features.
11 February 2008
The Egyptian Fractions page has been updated with extended calculators and a complete list of the 2/n table of 4000 years ago as in the Rhind Mathematical Papyrus, with comments on that list.
4 February 2008
A new Calculator to find all the best fractions that approximate a given value is now on the Introduction to Continued Fraction page.
17 January 2008
Pythagorean triangles updates:
8 January 2008
Two new goodies and a clear-up for the New Year:
26 December 2007
S Vajda's wonderful book Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications, is now back in print in a Dover reprint.
21 December 2007
A new Phigits (Base Phi) Calculator converts from decimal (whole numbers) to base Phi and vice-versa.
6 December 2007
A new section on Remainders after division or Fibonacci MOD n also called the Pisano Periods is on the The Mathematical Magic of the Fibonacci Numbers page with a new Calculator so you can investigate these cycle-lengths for yourself.
29 November 2007
I was on Melvyn Bragg's In Our Time on BBC Radio 4 today (November 29, 2007) when we discussed "The Fibonacci Numbers". Click on the link if you want to Listen Again or to download the 45 minute programme as a podcast.
5 November 2007
There is now an even simpler proof that Phi is irrational.
A new section on Fixed Length Egyptian Fractions has been added to the Egyptian Fractions page and the online calculator in that section now finds fractions of a given length.
1 October 2007
A new section on a formula for the number of primitive Pythagorean triangles with a given inradius is on the Pythagorean Triples page.
29 September 2007
Another minor update to the Pythagorean Triple Generator and Fast Count Calculator.
The FibLog entries for 2003 and 2004 archived (in a separate file).
27 September 2007
The Pythagorean triangles page has been updated with more information on Pythagorean triangles with a given inradius, and a bug in the Triple Generator and Fast Count Calculator (computing total counts now works) has been corrected.
26September 2007
The Easier Puzzles page has been rearranged to regroup the puzzles. Most puzzles now have a mathematical note following them saying how the problem in desribed in more formal mathematical terms. It is surprising the wide variety of areas of mathematics (graph theory, number theory, permutations, compositions, strings, etc) where the Fibonacci Numbers appear!
25 September 2007
And yet another new Easier Fibonacci puzzle! This is based on finding matchings in a path graph of n vertices, or Chairs in a Row: The Chatty version. Note that the links and titles of the other puzzles in the Seating Arrangements section have been renamed.
24 September 2007
Another new puzzle has been added to the Easier Fibonacci Puzzles page: Chess Match Seating. The snail of 20 Sept 07 is now Fibonacci's Ravenous Rabbit! This is the problem of finding a perfect matching in P(2)xP(n) - a lattice on 2n points.
20 September2007
A new puzzle has been added to the Easier Fibonacci Puzzles page: Fibonacci's Rabbits and the Two Lettuce Rows inspired by the latest copy of the Fibonacci Quarterly. This is the problem of finding non-intersecting walks that "never look back" in a P(2)xP(n) lattice.
4 August 2007
After a very enjoyable trip to the Eden Project in Cornwall at the end of July, there are new photos of the Education Centre (the Core) and the sculpture at its heart (the Core) as well as some innovative applications of the Fibonacci Numbers to the design of quilts shown to me by two Falmouth ladies present at my talk there. photos
6 June 2007
More on convergents to continued fractions and new sections: a simple geometrical interpretation and a quick and easy method of calculating convergent fractions further explains why the convergents are "best" fractional approximations and why nature uses the Fibonacci ratios to best approximate the irrational value of Phi.
The Continued Fraction Calculator now has a button to compute all the best approximations (up to an accuracy of 0.00001).
30 May 2007
Several updates to the Introduction to Continued Fractions page:
a new CF for e has been added; a new section on other "best" approximations; more on cogs and orreries.
More pictures of the 4 and 7 spirals in the recently added Always Fibonacci? section;
Some emails I have received now cast doubt on whether the golden section does occur in Beethoven's Fifth symphony and this is mentioned now on the Art and Music page.
10 May 2007
a new section on the first page (Fibonacci Numbers in Nature) shows pictures of flowers and plants where the Fibonacci numbers do not appear.
2 May 2007
Several new interactive demonstrations of seedheads showing why phi (0.6180339...) seeds-per-turn is the ideal packing for seeds on a seedhead including an Excel spreadsheet and several links to Mathematica's Demonstration Project animations (you don't need to have Matheamtica to see them).
6 March 2007
A new section on Vajda's formulae for the factors of F(kt) and L(kt) has been added to the Fibonacci and Phi Formulae page
5 March 2007
A new sunflower picture with 34 and 55 spirals as been added to the Fibonacci Numbers in Nature page.
27 February 2007
Two new sections have been added to the Introduction to Continued Fractions page: More Silver mean Properties and Fibonacci-Related Continued Fractions on CFs of the form [1,1,1,...,1,a], which was reported in the latest Fibonacci Quarterly of August 2006.
22 February 2007
A new diagram in the Leaves Per Turn section of the Fibonacci Numbers in Nature page shows that the sunflower leaves of an actual sunflower are almost exactly where they should be if each leaf is 0.618 of a turn further round than the previous leaf.
16 February 2007
The Two Dimensional Geometry and the Golden Section page has been reorganised to make it more accessible and ordered by shape (the links to the sections have not been changed) and a new section has been added: Phi and Right-Angled Triangles.
13 February 2007
Two new Fibonacci Puzzles have been added to the Simple Fibonacci Puzzles page and a new formula for F(n) in terms of any generalised Fibonacci series G(n) on the Fibonacci and Phi Formulae page. Dead links have been updated across the site.

The Fiblog for 2005 and 2006
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