Download The individual (Fibonacci) and the culture (connections, applications

The individual (Fibonacci) and the culture (connections, applications of his sequence)
Part 1: The Mathematical Experience, Study Edition page 132, #5 (expanded).
A. You are asked to collect evidence for the following conjecture linking the Fibonacci
sequence to the golden ratio (φ): When φ is raised to a positive integer power, the result can
be written as A + Bφ where A and B are Fibonacci numbers. Find φ2, φ3, φ4 to gather evidence
for this conjecture. You are asked to guess (without calculating) A and B for φ5 and then
explain the pattern emerging from your calculations
B. You are asked to investigate various ways to construct golden rectangles and show a
geometric connection between them and Fibonacci numbers. See Instructions for students
(below).
Objectives of this exercise:
The Fibonacci numbers have interesting applications, not only to other mathematical objects
(like the golden ratio), but to object in our natural world. They provide a rich opportunity for
mathematicians to engage in “pattern-finding”, a very important activity in mathematics. They
also illustrate how the discovery or creation of some mathematical object can stimulate growth
in mathematics in different areas. This assignment is intended to:
1. To illustrate connections between the Fibonacci numbers and the golden ratio.
2. To give you experience in pattern finding.
3. To show how these two mathematical objects can be used to make connections between
geometry and arithmetic.
4. To help you understand how the contributions of an individual can affect the growth of
mathematics is a variety of ways.
Instructions for students:
A.
1. Before you begin, read about:
a. Leonardo of Pisa (Fibonacci) at http://www-groups.dcs.stand.ac.uk/~history/Biographies/Fibonacci.html
b. the Fibonacci sequence at http://mathworld.wolfram.com/FibonacciNumber.html
c. the golden ratio at http://mathworld.wolfram.com/GoldenRatio.html.
d. the golden rectangle at http://mathworld.wolfram.com/GoldenRectangle.html
2. When you compute φ2, φ3, and φ4 , use ½(1 + 5) as the value for φ. Show these
calculations when you submit your homework assignment
3. To show the evidence for the conjecture, you must write each of φ2, φ3, and φ4 in the
form A + Bφ where A and B are Fibonacci numbers. Recall that the Fibonacci sequence is 1, 1,
2, 3, 5, 8, 13, 21, 34, 55, ……etc. where each subsequent number is the sum of the previous two
numbers. To help with the next step, give each of these numbers in the sequence a name that
reflects its position in the sequence. For example: Let
F[1] = 1; F[2] = 1; F[3] = 2; F[4] = 3; F[5] = 5; F[6] = 8, etc. In other words F[n] = the number
“sitting” in the nth position of the sequence.
4. Guess (without calculating) A and B for φ5. Then explain how you figured it out. In other
words, explain the pattern that connects powers of the golden ratio to the Fibonacci number.
You should take into consideration the power you are raising φ to; and its relationship to the n
in the F[n] name of the number in the Fibonacci sequence.
B.
1. Use Euclid’s construction (see to create a golden rectangle).starting with a square with side
lengths 2 inches. Indicate the lengths of the sides of the golden rectangle and show why it is
“golden”.
2. Take your golden rectangle and “remove” a square with side lengths ½ inch from it.
Describe the rectangle that results. Is it “golden”? Why or why not?
3. Illustrate a geometric connection between golden rectangles and Fibonacci numbers. Start
with a square with side lengths 1 inch and add a square of the same size to form a new
rectangle. Continue adding squares whose sides are the length of the longer side of the
rectangle. Eventually the large rectangle formed will look like a Golden Rectangle - the longer
you continue, the closer it will be. What do you notice about the lengths of the longer sides of
successive rectangles?
Dr. Elena A. Marchisotto
Department of Mathematics
California State University, Northridge