 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 handson 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 6913 triangle
and is selfcontained 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
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 17, 2018.
The
official conference website
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 Mathjamstyle on popular and recreational mathematics, puzzles
or games. Dr Knott will be there giving a 5minute 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
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 26, 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 MiniMaths talks on:

Friday 30 June,
Saturday 1 July,
Saturday 16 September,
Saturday 14 October
These are free 1520 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 MiniMaths talks
Three free 20 minute minimaths 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
Online 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 fastloading
HTML version and a
another version using MathJax.
 20 June 2014

34 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 ErdösStraus 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 2term 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 MonnerotDumaine'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 recentlyfound stonesandpots game (mancalatype) 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 SternBrocot Tree.
It now a new section and calculator on the SternBrocot 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 Johnson11
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
www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci
to
www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci
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 Fibonacci^{2} Base system has been added to the
Fibonacci Bases page and has a new Calculator.
 31 July 2008
 Photos
from the 13^{th} International Conference on Fibonacci Numbers and Applications held at
Patras, Greece, 512 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 lookandsee 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 clearup 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 viceversa.
 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 cyclelengths 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
nonintersecting 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...) seedsperturn 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
FibonacciRelated 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 RightAngled 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.