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The Fibonnacci Series
Fibonnacci Series, in mathematics, series of numbers in which each member is the sum of the two preceding numbers. For example, a series beginning 0, 1 … continues as 1, 2, 3, 5, 8, 13, 21, and so forth. The series was discovered by the Italian mathematician Leonardo Fibonacci (circa 1170-c. 1240), also called Leonardo of Pisa. Fibonacci numbers have many interesting properties and are widely used in mathematics. Natural patterns, such as the spiral growth of leaves on some trees, often exhibit the Fibonacci series.A Brief History
The use of the Fibonacci sequence goes back to the year 1202, when Leonardo of Pisa wondered
how many pairs of rabbits would be produced in the n th generation if he used the
following
rule. Start with a single pair. Each pair produces a new pair in the next generation and
also in the
generation following before it dies.
Leonardo's father was nicknamed Bonacci - the good-natured one, and so Leonardo became
known as Fibonacci - son of the good-natured one. The sequence which is the solution to
this
rabbit problem thus became known as the Fibonacci sequence.
In the latter half of the nineteenth century, a French mathematician named Edouard Lucas
went on
to study the same recurrence with starting values 2 and 1. This version of the numbers
became
known as Lucas numbers. Lucas was the person who made the term Fibonacci numbers popular,
and was also a perpetrator of the Towers of Hanoi puzzle, which is related to subset
generation,
discussed in the subset section of AMOF.
Johannes Kepler noticed that if you take the ratio of consecutive Fibonacci numbers, the
ratio
approaches (1 + 51/2)/2. This value, called the golden number (or golden ratio), was known
to
the Greeks, who used it in their architecture.
Applications to Mathematics and other Areas
The Fibonacci and Lucas numbers appear in numerous mathematical problems. In fact, there
is
an entire journal dedicated to such topics. Fibonacci Quarterly is a mathematical
periodical
devoted entirely to the subject of mathematics related to Fibonacci sequences. It covers
topics
involving many branches of mathematics, including linear algebra and probability.
In computer science, there is a data structure called a Fibonacci heap which is at the
heart of
many fast algorithms which manipulate graphs.
The Fibonacci numbers appear in nature as well. The study of Phyllotaxis, or leaf
arrangements,
has made this evident. The arrangements of structures like florets, leaves and flowers,
seem to
have two systems of spirals. The number of clockwise spirals is usually different from the
number of counterclockwise spirals. In fact, these two numbers are usually two consecutive
values of the Fibonacci sequence.
For example, the florets in the head of a sunflower appear to have two systems of spirals,
both
beginning from the center. Counting these spirals on one particular sunflower shows that
there
are 55 in the clockwise direction and 34 in the counterclockwise. Other plants which
exhibit
similar structure include pineapples, daisies, cauliflowers, pinecones, ocotilloes, Joshua
trees
and some cacti, such as the species Opuntia.
The reason why this pattern occurs in nature has to do with what happens when buds
initially
appear on these structures. Each bud that appears repels others around it. As new buds
form,
they repel the most recent buds more strongly, while buds that have been formed longer are
repelled less. This naturally gives rise to a spiral pattern. Whether the buds eventually
become
seeds, twigs, petals, or leaves, the principle is the same.
The Fibonacci sequence also arises in the ancestry of male bees. A male bee (known as a
drone)
hatches from an unfertilized egg, while a female bee (known as a queen) hatches from a
fertilized
one. The following tree depicts the ancestry of a male bee in terms of how many ancestors
it had
of each sex. The number of bees in each row is a number in the Fibonacci sequence.
The Fibonacci numbers have been found in many other areas as well. In the
area of physics,
there are indications that the golden ratio and Fibonacci numbers are related to the
structure of
atoms and also to the spacing of planets in the solar system. Also, the golden ratio is
found in
Greek and Minoan architecture.
Explanation of AMOF Implementation
AMOF accepts only one input, n. This represents the nth Fibonacci number.
For output, there are three options:
0-1 Strings
Stair stepping
Pane Reflections
Choosing 0-1 Strings will produce all the length n bitstrings that do not contain
consecutive 1's.
Choosing Stair stepping will produce a series of 1's and 2's which represent the different
ways
to climb a set of n + 1 stairs. Each stair-stepping result corresponds to a 0-1 string.
Choosing Pane Reflections will produce a series of Reflection diagrams corresponding to
the
0-1 strings as illustrated in the Example section above.
In the Classroom
Many activities can be used in the classroom to generate and investigate Fibonacci
sequences.
One is to have students place 1 and 2 cent stamps across the top of a postcard (facing
with
correct side up) in different arrangements to make up a certain postage amount. The number
of
different arrangements will be a Fibonacci number. Also, the example of climbing stairs by
taking 1 or 2 steps at a time is a simple activity. Students can check their answers with
AMOF.
An activity suggested for investigating number sequences is to have students draw out the
difference between consecutive numbers in a sequence. They can then find patterns in these
differences which allow them to determine how the sequence is constructed. If this
activity is
tried with the Fibonacci numbers, students will notice that the sequence of differences is
the
same as the original sequence itself. This can give them a clue to the recursive nature of
the
sequence.
Explain the rabbit problem to students and have them come up with the sequence up to a
year.
Then ask them to complete the sequence up to the second year. This comes to 161 392.
Bringing in pine or fir cones for students to inspect is a good way to demonstrate that
Fibonacci
sequences occur in nature. This should give students a sense that mathematics is part of
their
world and not just something they are made to learn in school.
For students who are familiar with Pascal's triangle, the following exercise will
demonstrate the
connection between Pascal's triangle and Fibonacci numbers. Write out Pascal's triangle so
that
the leftmost 1's of each row are lined up with each other. Draw diagonal lines through
this
triangle and add up the values on the diagonals. Notice that they are Fibonacci numbers.
The golden ratio can be investigated by having students calculate the ratio of consecutive
Fibonacci numbers on a calculator. They will notice that this ratio tends to approximately
0.618.
For those in higher grades, discovery of functions and relations among Fibonacci and Lucas
sequences might be useful. Items to explore include:
FnLn = F2n
The sum of the squares of two consecutive Fibonacci numbers is a Fibonacci number.
Specifically, Fn2 + Fn + 12 = F2n + 1
Ln = Fn + 1 + Fn - 1
By pursuing the following sequence of formulas, students may arrive at a formula for Fn2 -
Fn +
kF n - k
Fn2 - Fn + 1Fn - 1 = (-1)n - 1
Fn2 - Fn + 2Fn - 2 = (-1)n
Fn2 - Fn + 3Fn - 3 = 4(-1)n - 1
There are several other ways to investigate Fibonacci sequences. One good reference for
ideas
is Readings for Enrichment in Secondary School Mathematics, a book produced by the NCTM.
Fibonacci sequences can also be a way to introduce recursion into the classroom. [NCTM]
contains several chapters about recursion on pages 149 - 170 which may be helpful. One
idea
given there is to have students use a spreadsheet to generate recursive sequences. This
gives
students experience with both recursion in mathematics and spreadsheets in computer
science.
For more Information on Fibonnacci Sequence, Please visit the following website:
http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
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