- The 2018 International Fibonacci Numbers and Applications Conference
- will be in Halifax, Nova Scotia, Canada at Dalhousie University, July 1-7, 2018.
official conference website
has more details of accommodation, travel, submitting an abstract for a talk and how to register.
- 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 2018
- Proceedings of the 2016 International Fibonacci Conference at Caen, France.
The 2016 Conference Proceedings
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
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
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
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,
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.
Saturday 1 July,
Saturday 16 September,
Saturday 14 October
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.
- 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
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
- 20 July 2014
Many updates and expanded sections on the Mathematical Magic of the Fibonacci Numbers
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
3-4 July 2014: Festival of Maths and its Applications
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
- It now uses MathJax to improve the
appearance of expressions involving nested square roots
- a new section to explain the signs of
the trig functions in the four quadrants
- inclusion of (links to) a couple of Mathematica demonstrations and details of how to get the free
CDF player which makes the domonstrations interactive
- 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
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
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
- 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
- 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:
You will need to do this for my other pages at MCS at Surrey too until the site is back online.
- 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.
- 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
- 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
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:
- The Excel sheet that draws equiangular spirals
now has Phi changed to F, the expansion factor for the curve.
Explanatory text has been added and another box to show the constant angle between the radius
(the line from the centre to the point on the curve)
and the direction of the curve at that point, which is constant for all points, hence the name Equiangular Spiral.
A further box computes the F for a given angle.
- A new Exact Fractions Calculator has been added to the Mathematical Magic of the Fibonacci Numbers page.
It converts any proper fraction into an exact decimal fraction, identifying the
fixed and recurring parts (if any).
- A new year is here so this Weblog file (you're reading it!) has been shortened by archiving the Weblog notes
for 2005 and 2006.
- 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.
- 6 June 2007
- More on convergents to continued fractions and new sections:
simple geometrical interpretation
and a quick and easy method of calculating convergent
explains why the convergents are "best" fractional approximations and why nature
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?
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
- 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
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
- 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.