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).
New flower pictures, leaves, seedheads on the Fibonacci Numbers In nature page.
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).
The Simple Trig values page is now a link on the Fibonacci Home page.
added a new section on The √(2±Phi) pattern of expressons for 9, 18, 27, 36, 54, 64, 72 and 81 degrees.
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.
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.
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.
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.
Detecting when N is a Fibonacci Number is a new section on the A Formula for the Fibonacci Numbers page.
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.
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.
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!
- 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.
alpha:α, beta:β, Phi:Φ, phi:φ; sigma:σ, tau:τ; infinity:∞, square-root:√
If they are not all showing, your browser may not be the latest version so please update your browser or else view the PDF version which is a compact 208K so should download quickly.
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.
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??!
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.
The format of the Fibonacci and Phi Home page has been changed to make it a little more compact.
- 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
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!
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.
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.
New sections:
- Fibonacci in Films on the Fibonacci Numbers in Art, etc page
- Every number is a factor of a Fibonacci number on the Mathematical Magic of the Fibonacci Numbers page
