Golden Rectangles and Spiral Growth

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Golden Rectangles and Spiral Growth
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Golden Rectangles

• A golden rectangle is a rectangle where the ratio
  of the sides is the golden ratio.
• We found that if we add a square onto the longer
  side of a golden rectangle we get another golden
  rectangle.
• This idea came from the realization that if we
  add a rectangle to the longer side of a Fibonacci
  rectangle we get another Fibonacci rectangle.

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The Golden Ratio

• This ratio between the sides of a golden
  rectangle is called the golden ratio.
• The ancient Greeks viewed this proportion as the
  ideal and employed it in much of their
  architedture and artwork.
• The golden ratio is often designated by the Greek
  letter Phi, the first letter in the name of the
  Greek sculptor and architect Pheidias.
• It was also popular among Renaissance artists,
  including Leonardo da Vinci.
• This value came up in Binets Formula that allows
  us to find any Fibonacci number, and seems to
  some strange connection to the Fibonacci
  sequence.

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The Golden Ratio

• We noticed that the ratios of consecutive
  Fibonacci numbers seemed to be approaching some
  value.
• We found the value of
• as a solution to the equation

• This means that

• We can use this to find higher powers of

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For example,
Continuing like this we get
and in general,
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Back to Golden rectangles and similar shapes

• Recall that two shapes are similar if the ratio
  of corresponding lengths are equal.
• If we add a square to the longer side of a golden
  rectangle we get a golden rectangle, i.e., a
  similar shape.
• When we can do this (start with one shape, add
  something to it and get a shape similar to the
  original), the added shape is called a gnomon to
  the original shape.

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Examples of gnomons
b

• A square is a gnomon to a golden rectangle

a
b
b
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A ring is a gnomon to a circle.
The new shape is still a circle.
Can a ring have a gnomon?
The result is a ring, but is it really similar to
the original ring?
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So not all shapes have gnomons and many shapes
can have several types of gnomons.
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Gnomic Growth

• Gnomonic growth is when growth occurs by the
  addition of a gnomon.
• That is, as the object grows it maintains the
  same relative shape, but increases in size.

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Gnomonic Growth
Start with a circle . . .
. . . and add a ring . . .
. . . and another ring . . .
. . . and another ring . . .
. . . and so on . . .
The result is always a circle, similar to the
original shape, only larger.
This is gnomonic growth.
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Spiral and Gnomonic Growth
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Suggested Problems
Chapter 9 5, 7, 8, 12, 15, 16, (see if you
can prove these using the recursive
definition) 19, 25, 27, 29, 31, 52 (for b,
rewrite FN in terms of FN-1 and FN1)
Next Linear, exponential, and logistic growth
models. (10.1-4)