Honeybees Fibonacci numbers and Family trees





There are over 30,000 species of bees and in most of them the bees live solitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee's ancestors in this section a bee will mean a honeybee . First, some unusual facts about honeybees such as not all of them have two parents In a...
 Controversial Issue
 Fibonacci Calculator
 A formula for Phi using a continued fraction
 Purely Trigonometric Formula for Fibn
 A quote from Coxeter on Phyllotaxis
 Rectangle Triangle dissection Problem
 A series Paper
 A simple definition of
 Algorithm
 An Application of the Fibonacci Number Representation
 An Application to the Solar System
 An easy way to Multiply
 An Icosahedron in an Octahedron
 And another
 And The Fibonacci Series
 And x2 5483220372 2 3416277185 31415926535
 Another formula for the Fibonacci numbers
 Another way to generate The Rabbit sequence
 Another way to make the Fibonacci Rabbit sequence
 Approximating Root 2 using Fractions
 Are any Platonic solids spacefilling
 Base Phi Representations
 Basic G Identities
 Basic Golden Ratio Identities
 Best Fractions for Pi
 Binets Formula for negative n
 Binets formula for noninteger values of n
 Bipyramids as dice
 Calculating the next Fibonacci number directly
 Calculator Method 1 Invert and
 Calculator Method 2 Add 1 and take the squareroot
 Can we write Phi as a fraction
 Computers use The Rabbit sequence
 Constructing the golden section phi
 Contents  2
 Contents of this Page  2 3 4 5
 Contents of this page 6 7
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 Cycles in the Lucas numbers
 Decagons
 Definitions and Notation
 Do quasicrystals occur in nature too
 Does the Golden String ever repeat
 Each side of the rectangle is divided in the same ratio This ratio is Phi 16180339 ie 11618 or 06181
 Euclid
 Europe
 Extend this table by a few more rows Do the values look like they are integers always What integers do they Lucas if they are hint Yes They are the Lucas numbers again
 F1 f1
 Factors
 Fibonacci and Lucas Summations
 Fibonacci and Music
 Fibonacci and Poetry
 Fibonacci and the Mandelbrot
 Fibonacci Fingers
 Fibonacci memorials to see in Pisa
 Fibonacci Numbers 333333333333333333
 Fibonacci Numbers and Branching Plants
 Fibonacci numbers and the Golden Number
 Fibonacci Numbers and the Golden Section
 Fibonacci paper
 Fibonaccis Mathematical Books
 Fibonaccis Rabbits
 Flags of the World and Pentagram stars
 Fnk Fnk x
 Footnote 2 Shapes for Fair Dice
 General Summations
 Generalised Fibonacci Series in the Fibonacci System
 Golden Ratio with Fibonacci and Lucas
 Gougli
 Hear the golden sequence too
 Here is another method to generate the Rabbit sequence but this time using the bits we threw away above the fractional parts of the multiples of
 Here n means the factorial of n which means the product ofall the whole numbers from 1 to n
 Heres how to construct point G using setsquare and compasses only
 His names
 Historical Note Binets Formula or de Moivres
 How many digits does a number have
 0
 If we consider further calls of fn for n5 and above
 Info  2
 Integers as sums of powers of
 Ir and ir
 Is there any significance in the value of tanx where tanxcosx
 Isohedral shapes
 L
 Lets call this the Expanding 1s Process
 Linear Relationships
 Lining up the Rabbits
 Links and References
 Links on Fractals
 Making a Paper Knot to show the Golden Section in a Pentagon
 More Links and References
 Music and
 Numbers  2
 On the Phis Fascinating Figures page the Things To Do in the Numerical Relationships between Phi and its Powers section asked you to investigate what happens when instead of subtracting the powers of Phi and phi as in the formula for Fibn above we added t
 Order 2 Fibonacci and Lucas Relationships
 Order 2 G Formulae
 Other angles related to
 Other bases
 Other numbers with patterns in their CFs
 Other WWW links on Phyllotaxis the Fibonacci Numbers and Nature
 Other WWW pages on Fibonacci and his series
 Our decimal system
 Patterns in the Fibonacci representations
 Pentagrams contain this triangle
 Petals on flowers
 Phi and 3dimensional geometry
 Phi and another Isosceles triangle
 Phi and Powers of Pi
 Phi and the Fibonacci numbers
 Phi and the Pentagon Triangle
 Phi and the Root5 Rectangle
 Phi and Trig graphs
 Phi is not a fraction
 Phi to 2000 decimal places
 Pi
 Pine cones
 Plato
 Points 30 edges 20 faces
 Powers of
 Quasicrystals
 Quasicrystals and
 R Mtheta where M Phi2Pi
 References  2
 References and Links  2
 References on Fibonacci and Golden Section
 References to articles and books
 Replacing every M in one month by MN in the next and similarly replace every N by M
 Seed heads
 Similar numbers
 Solving Quadratics with Continued Fractions
 Squares
 Summations with Binomial Coefficients
 Tends to
 The difference between x2 and x is N a whole number
 The dodecahedron of side 2Phi has coordinates
 The Dual of a Solid
 The Fibonacci base system
 The Fibonacci Rectangles and Shell Spirals
 The Fibonacci series
 The Fibonacci Series
 The first 100 Fibonacci numbers completely factorised
 The Golden Section and
 The Golden section in architecture
 The Greeks Kepler and the Five Elements
 The importance of the gcd of a and b is that it tells us how to put the fraction ab into its simplest form which is to divide the top and the bottom by the gcd The resulting fraction will be the simplest form possible So
 The Lucas Numbers
 The most irrational number
 The Phi line Graph
 The rabbit sequence defined using the whole part of Phi multiples
 Thefirst 2000 bits of the Rabbit Sequence
 To compute f3
 Trigonometry and
 Tt H2nl
 Useful book references
 Using these golden rectangles it is easy to see that the coordinates of the icoshedron are as given above since they are 0 1 phi phi 0 1 1 phi
 We can prove that ABBC is the golden ratio
 What is the equation of the Golden spiral
 What is the golden section or
 What we really meant to ask was how to do this using only powers of Phi and not repeating any power in the sum which is what we did in the examples above
 Where to
 Who was Fibonacci
 Why does this work
 Why is the Golden section the best arrangement
 WWW Links
 You will find that your right hand is copying the original sequence but at something like 06 of the speed actually at 0618034 of the speed