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The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ... yet appear in the fruit and veg you eat. How? Why?

Fibonacci numbers & nature Where do the Fibonacci numbers appear in pineapples, cauliflower, apples and so and many other seeds and fruits?
Puzzles: easier Some puzzles where the answers are, unexpectedly, the Fibonacci Numbers ...but why?
Puzzles: harder Some more puzzles involving the Fibonacci numbers
Maths magic of Fib nbs There are some amazing pattterns in the Fibonacci numbers...here are a few
Formulas for Fib nbs How to compute any Fibonacci numbers without finding all those before it first
The Fibonacci Number Base Instead of using powers of 10 to write our numbers, why not use the Fibonacci numbers?
Fibonomials The Binomial numbers but using Fibonacci numbers are the Fibonomials!
Who was Fibonacci? Leonardo Pisano or "Fibonacci" - the man and his book that made Decimal numbers popular across Europe
Fibonacci numbers and Pi Yes Pi can be computed using the Fibonacci Numbers:
Fake Fibonaccis Series that look like they might be the FIbonacci numbers - but are fakes.
Fibonacci Number formulas If you've found an equation involving Fibonacci numbers, see if it is already known in this large list of formulas
The Fibonacci String: the Rabbit sequence The golden string is
1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ...
a sequence of 0s and 1s that is closely related to the Fibonacci numbers and the golden section.
The first 500 Fibonaccis, factorised Here they are in full
The Lucas Numbers
Another series that always appears where the Fibonacci numbers do:
- The first 200 Lucas numbers and factors
- the Lucas numbers
General Fibonacci series What if we began with some other nunmbers with the Fibonacci rule? They all relate to the Golden Section! Also two tables containing all the natural numbers but with some amazing extra properties related to the Fibonacci numbers and Phi: the Wythoff and the Stolarsky arrays
Fib, Phi and the Arts Some artists and builders have use Phi to make beautiful things
Links & bibliography Links to other sites, books, etc.

The golden section numbers appear in nature too and in the most symmetric 3D solids. They are

0·61803 39887... = phi = ϕ and
1·61803 39887... = Phi = Φ = 1 + phi = 1/phi

Phi in nature Why is Phi is the reason the Fibonacci numbers appear in fruits and flowers?
Phi, Fibonacci, calculating it
2D Geometry and Phi From the pentagon to Penrose's Darts and Kites, all using Phi
3D solid shapes and Phi The 5 most symmetric solids and Phi
Phi, powers The powers of Phi without any multiplication
The Phi number base Suppose we used powers of Phi instaed of powers of 10 to write numbers...
Phi to many dps and here it is!
Phi Formulas A list of many formulas connection Fibonacci numbers and Phi

Find Phi using + and / only
Finding Phi using + and sqrt
Finding Phi using + number and 1/x
Convert to Base phi
Powers and multiples of Phi
Rounding Phi
The Mancala Game
The first 200 dps of Phi

Primes & factors The almost-primes and a message sent to space

Primes & Factors
Sums of Divisors σ(n)
Eulcid's Algorithm

All kinds of number bases Not just binary and base N but Factorial, Fibonacci, Binomial and Negative Bases too
Convert to base 2,3,...
Find n in base B
Convert to Base phi
Factorial Base and Permutations
Binomial Representations
Bases which are Fractions
Chinese Remainders
The Fibonacci number bases More on the Fibonacci number base
Zeckendorf Representation
Think Of A Number using Fibonacci Numbers
Fibonacci Partitions
Fibonacci Base using negative indices
Fibonacci-squares nunmber base
The Phi number bases How about an irrational number as a base?
All bases Converter
Palindromes in any base
Palindromes Numbers such as 12921 and 444 are palindromes and have some amazing secrets. Did you know any number is a sum of just 3 palindromes?

Palindromes in any base
Palindromic sequences
Palindromes in multiple bases
Palindromic Fibonacci Reps (Zeckendorf)
The maths magic of the number 2025

Polygonal (Figurate) numbers Triangular, Square, Pentagonal ... Numbers
More on Figurate Numbers Central polygonal numbers, matchstick figures, 3D solid shapes and higher dimensions
Polytopes (higher dimensional shapes)

Triangular Numbers
Square Number Calculator
Rectangular, Oblong and Primes
Polygonal Numbers
Find polygonal shapes for a number
Find sum of triangular numbers for n
Polygonal Number sums
Many-shape Polygonal numbers
Polygonal runsums
Centered Figurate Numbers
Plain and Centred multi-Polygonal Numbers
3d Pyramid Numbers

Runsums
Numbers which are the sum of a run of consecutive whole numbers
- More on Runsums
Runsums and Triangular Numbers Counting runsums Runsum Tables Runsums of given length Number with many runsums
Oddrunsums If we sum odd numbers in sequence we get some very pretty number patterns
Odd runs for a sum Odd runs for any power Odd runs for fractions
Integer Sums: partitions How many ways can you write a given number as a sum? How is it related to paying with coins?

Partitions and Compositions Sets and Bags Partitions with a given product
Early bird numbers The sequence formed from joining all the numbers together in order: 123456710111213...
Natural Number Sequence (Early Birds)

Rounding, floor and ceiling functions
Fractions & decimals - their periods and patterns and in non-decimal bases.
Fraction - Accurate Decimal converter
Recurring Decimal to Fraction
Same Denominator Reduced Fractions
Maximal period
Similar sized Recurring Decimals
Series to Decimal Fractions
Fractions in any number base
The Farey fractions and a Tree: Two ways of arranging all fractions
Farey Fractions
The Stern-Brocot Fractions Tree
Egyptian fractions The Egyptians only had unit fractions of the form 1/n. How did they use them?
To and From EFs
Shortest EF
Fixed Length EFs
Productive EFs
Harmonic Numbers
Continued fractions: an introduction An unusual method of writing fractions that has many advantages.
Fraction - Continued Fractions
CF for expressions
Square-roots to CFs
CFs for roots of a quadratic
CFs- best approximations Fraction for a calue
The Silver means
Evaluate CFs with negative terms
Exact Fractions and CFs
A Converter between fractions Fractions to and from Decimal, Continued Fractions from and to Fractions with an expression evaluator and many in-built functions and all to as many decimal places as you like!

Pythagorean tri Right-angled triangles with integer sides, e.g. 3, 4, 5
Test a triple: is it a PT?
Generate PTs by two fractions
PTs from the m,n formula
PTs from a Fibonacci sequence
PTs from Fibonacci series
The UAD Tree of all PTs
PTS with a given area to perimeter ratio
PTs in any order
PTs with a given angle
PTs and EFs for 2/n
Counting PTs
Sums of Squares
Pythagorean triangles from Fibonacci series
Trilinear & barycentric coordinates An introduction to trilinear and barycentric coordinates, links to Clark Kimberling's Encyclopedia of Triangle Centers (ETC) with automatic lookup.
Triangles:Coordinates converter
Triangle solver Find all the angles and sides of a triagle given just some of them
Also other integer-sided triangles such as Heronian triangle, triangles on a grid, etc.
from coordinates:
from sides and angles
Triangles with integer sides
Triangles with integer coordinate vertices
Exact trig values Which angles have a simple and exact value for their sine, cosine or tangent?

Linear Recurrences and Generating Functions A catalogue for all series on the pages at this site
Lambda Calculus Combinators and Reductions Define combinators and reduce them
Models of Arithmetic with Combinators
A Random numbers/cards/coins/letters generator
Coin change
Galton Quincunx Simulator dropping balls bouncing off pegs to show the Normal (Bell) curve
Herschel's Perpetual Calendar Can be printed to make a calendar for any year.
Maths jokes

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An interactive GCSE self-test An interactive test with new questions each time, worked solutions and marking on the test online
Originally developed for University of Exeter
A-level tests on 80 topics 80 topics each generating a new online test, with marking and worked solutions
Originally developed for MEI

Lock and roll (like Yahtzee) roll 5 dice and find the best scores
Got it! (Like Countdown) pick 5 numbers to make a given value using + − × /

All pages are at https://r-knott.surrey.ac.uk
About Dr Ron Knott
Ron Knott's BBC programmes
- Simon #Singh’s Five Numbers: the Golden Ratio
- In Our Time, BBC Fibonacci Numbers
Thanks to...
These pages are all my own work but referencing the work of many other mathematicians whose work I have linked to or given references to on each page.
On several pages I have used Mike McLaughlin's BigNumber software to extend Javascript numbers and arithmetic to very large numbers in the Calculators.
I have written a library of maths functions in Javascript to extend Mike's BigNumbers that you are free to link to and use.
* BigNumber expressions
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