Transcript Excursions in Modern Mathematics Sixth Edition

Excursions in Modern
Mathematics
Sixth Edition
Peter Tannenbaum
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Chapter 9
Spiral Growth in Nature
Fibonacci Numbers
and the Golden
Ratio
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Spiral Growth in Nature
Outline/learning Objectives
To generate the Fibonacci sequence and
identify some of its properties.
 To identify relationships between the
Fibonacci sequence and the golden ratio.
 To define a gnomon and understand the
concept of similarity.
 To recognize gnomonic growth in nature.

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9.1 Fibonacci Numbers
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The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…
The Fibonacci numbers form what mathematician call
an infinite sequence– an ordered list of numbers that
goes on forever. As with any other sequence, the
terms are ordered from left to right.
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The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…
In mathematical notation we express this by using the
letter F (for Fibonacci) followed by a subscript that
indicates the position of the term in the sequence. In
other words, F1 = 1, F2 = 1, F3 = 2, F4 = 3, ...F10 =55,
and so on. A generic Fibonacci number as FN.
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The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…
The rule that generates Fibonacci numbers– a
Fibonacci number equals the sum of the two
preceding Fibonacci numbers– is called a recursive
rule because it defines a number in the sequence using
other (earlier) numbers in the sequence.
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Fibonacci Numbers (Recursive Definition)
•FN = F N-1 + F N-2 (the recursive rule)
FN is a generic Fibonacci number. F N-1 is the Fibonacci
number right before it. F N-2 is the Fibonacci number
two positions before it.
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Fibonacci Numbers (Recursive Definition)
• F1 = 1, F2 = 1 (the seeds)
The preceding rule cannot be applied to the first two
Fibonacci numbers, F1 (there are no Fibonacci numbers
before it) and F2 (there is only one Fibonacci number
before it– the rue requires two), so for a complete
description, we must “anchor” the rule by givng the
values of the first Fibonacci numbers as named above.
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Is there an explicit (direct) formula for computing Fibonacci
numbers?
Binet’s Formula
F
N
N
N





1 5 
 1  1 5

  2    2  
 5  
 
 
Binet’s formua is an example of an explicit formula– it allows us
to calculate a Fibonacci number without needing to calculate all
the preceding Fibonacci numbers.
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9.2 The Golden Ratio
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The Golden Ratio
We will now focus our attention on the number
  1  5 / 2 one of the most remarkable
and famous numbers in all of mathematics.
The modern tradition is to denote this number
by the Greek letter  (phi).

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
Spiral Growth in Nature
Powers of the Golden Ratio
  FN   FN 1
N
In some ways you may think of the preceding formula as
the opposite of Binet’s formula. Whereas Binet’s
formula uses powers of the golden ratio to calculate
Fibonacci numbers, this formula uses Fibonacci
numbers to calculate powers of the golden ratio.
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9.3 Gnomons
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The most common usage of the word gnomon
is to describe the pin of a sundial– the part that
casts the shadow that shows the time of day.
In this section, we will discuss a different
meaning for the word gnomon. Before we do
so, we will take a brief detour to review a
fundamental concept of high school geometry–
similarity.
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We know from geometry that two objects are
said to be similar if one is a scaled version of
the other.
The following important facts about similarity of
basic two-dimensional figures will come in
handy later in the chapter.
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Triangles: Two triangles are similar if and only
if the measures of their respective angles are
the same. Alternatively, two triangles are
similar if and only if their sides are proportional.
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Squares: Two rectangles are always similar.
Squares: Two rectangles are similar if their
corresponding sides are proportional.
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Circles and disks: Two circles are always
similar. Any circular disk (a circle plus all of its
interior) is similar to any other circular disk.
Circular rings: Two circular rings are similar if
and only if their inner and outer radii are
proportional.
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We will now return to the main topic of this section– gnomon.
In geometry, a gnomon G to a figure A is a connected figure
which, when suitably attached to A, produces a new figure
similar to A. Informally, we will describe it this way: G is a
gnomon to A if G&A is similar to A.
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Gnomons of Squares
Consider the square S in (a). The L-shaped figure G in (b) is a
gnomon to the square– when G is attached to S as shown in
(c), we get the square S´.
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Gnomons of Circular Disks
Consider the circular disk C with radius r in (a). The O-ring G
in (b) with inner radius r is a gnomon to C. Clearly, G&C
form the circular disk C´ shown in (c). Since all circular disks
are similar, C´ is similar to C.
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Gnomons of Rectangles
Consider a rectangle R of height h and base b as shown in (a).
The L-shaped G shown in (b) can clearly be attached to R to
form the larger rectangle R´ shown in (c). This does not, in
and of itself, guarantee that G is a gnomon to R.
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Gnomons of Rectangles
The rectangle R´ is similar to R if and only if their
corresponding sides are proportional. With a little algebraic
manipulation, this can be simplified to b y
h
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
x
Spiral Growth in Nature
9.4 Spiral Growth in
Nature
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In nature, where form usually follows function,
the perfect balance of a golden rectangle
shows up in spiral-growing organisms, often in
the form of consecutive Fibonacci numbers. To
see how this connection works, consider the
following example.
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Stacking Squares in Fibanacci Rectangles
Start with a 1-by-1 square [square 1 in (a).
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Stacking Squares in Fibanacci Rectangles
Attach to it a 1-by-1 square [square 2 in (b)].
Squares 1 and 2 together form a 2-by-1
Fibonacci rectangle. We will call this the
“second-generation” shape.
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Stacking Squares in Fibanacci Rectangles
For the third generation, tack on a 2-by-2
square [square 3 in (c)]. The “thirdgeneration” shape (1, 2, and 3 together) is
the 2-by3 Fibonacci rectangle in (c).
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Stacking Squares in Fibanacci Rectangles
Next, tack onto it a 3-by-3 square [square 4 in
(d)], giving a 5-by-3 Fibonacci rectangle.
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Stacking Squares in Fibanacci Rectangles
Then tack o a 5-by-5 square [square 5 in (e)},
resulting in an 8-by-5 Fibonacci rectangle.
We can keep doing this as long as we want.
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Conclusion
 Form
follows function
 Fibonacci numbers
 The Golden Ratios and Golden
Rectangles
 Gnomons
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