The Golden Ratio

1
The Golden Ratio
In this chapter, weve been discussing ratios
and proportions. Remember, that a ratio is simply
a comparison of two numbers. For the next two
days, we are going to be studying a special
ratio, called The Golden Ratio. It appears many
places in nature, music, architecture and art.
First, we are going to define the golden ratio
and then we are going to look at two of its
special applications.
2
Defining the Golden Ratio
1. You and a partner should pick up the
worksheet titled Where Are the Golden
Rectangles? and a ruler from your teacher.
Before beginning step 2, you and your
partner should pick one rectangle from each
set that you like the best, or find the most
pleasing. 2. Measure the length and width of
each rectangle to the nearest millimeter.
You can write the results directly on the
worksheet. 3. One person should open Spreadsheet
1, while the other reads the directions on
the next page.
Spreadsheet 1
3
Completing Spreadsheet 1 Finding the Golden
Rectangle

• Step 1) Fill in the length of each rectangle in
  Column B.
• Step 2) Fill in the width of each rectangle in
  Column C.
• Step 3) In cell D2, type in the formula
  B2/C2
• Step 4) Click on cell D2 again. Under the Edit
  menu at the top of the screen, choose Copy.
  Your cell is now outlined.
• Step 5) Click on cell D3. Holding your left
  mouse key, drag down the remainder of column D
  (through row 13). The column should now be
  shaded black.
• Step 6) Under the Edit menu, choose Paste. What
  has happened in each of the cells? Was this
  expected?
• Step 7) Repeat steps 4-6 for the E column. Your
  formula for cell E2 should be C2/B2

4
Defining the Golden Ratio
Looking at Spreadsheet 1, which rectangles in
column D have the same value? What about column
E?
Rectangles B, H, and J are called Golden
Rectangles because their ratio of length to width
(or width to length) approximates the Golden
Ratio. (Were these the rectangles that you and
your partner preferred from each set?) The Golden
Ratio is an irrational number (like Pi) and is
denoted by the Greek letter, Phi. When dividing
the length by the width, the approximate value of
the golden ratio is 1.618. When the width is
divided by the length, the value is approximately
0.618. Now that weve defined the golden ratio,
we are going to look at two of its
applications. (Your partner should close
Spreadsheet 1 by clicking on the x at the very
top right corner of the screen and proceed to
this slide.)
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Application 1 Architecture
The ancient Greeks believed that the Golden
Rectangle was one of the most pleasing shapes to
the eye. Thus, many of their buildings, such as
the Parthenon (pictured above center) are
composed of golden rectangles. Likewise, many
modern works of architecture, such as the United
Nations Headquarters, are also made up of golden
rectangles. For our next project, we are going to
explore the golden rectangles that make up the
Parthenon in a little more detail.
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Application 1 Architecture
Pick up the worksheet titled Find the
Parthenons Golden Rectangles from your teacher.
Your job is to outline as many golden rectangles
as you can find in the architecture of the
Parthenon. To do this, you will want to measure
the length and width of each rectangle and then
find the ratio of length divided by width. To
make this process go more quickly, you may want
to type the lengths and widths into a spreadsheet
and then calculate the ratio using a formula.
Ive started a spreadsheet for you. You just need
to type in the appropriate formula in column
C. See Spreadsheet 2. How many golden rectangles
did you find?
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Application 2 Art
In addition to architecture, the golden ratio can
be found in many places in the human body. When
one measures various body parts in relation to
the whole (such as the length measured from the
top of the head to the floor divided by the
length measured from the navel to the floor), a
golden ratio is found. Mathematically, the ratio
looks like this Many artists are aware of
this fact and use mathematics and the golden
ratio when planning the composition of their
paintings (or sculptures) to make the
work seem more realistic.
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Application 2 Art
For our last project, we are going to explore the
golden ratios that occur in human beings.

• Pick up the worksheet titled My Body
  Measurements and a tape measure from the
  teacher. You and your partner should measure each
  other and fill in sections A and B for each part
  on the Data Recording Sheet. You will then use a
  spreadsheet to calculate the remaining pieces.
• Use the metric system when choosing your units.
• Please note that the units used for the entire
  height of the person (such as meters or
  centimeters) do not have to be the same as those
  used to measure the parts of the finger
  (millimeters). Just make sure that you are using
  a common measure for each separate part.
• Once you have filled out sections A and B, type
  in the appropriate formulas to complete the
  spreadsheet below. (There are two areas to
  complete, one for your measurements and one for
  your partners.)
• Spreadsheet 3

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Application 2 Art
Did you and your partner discover any golden
ratios? In which parts were your measurements
pretty close to the golden ratio? Are there any
areas that were way off the mark? In
conclusion, we have looked very briefly at the
Golden Ratio and some of its applications. If you
would like more information about the Golden
Ratio, please feel free to visit the following
website http//mathforum.org/dr.math/faq/faq.gold
en.ratio.html The site provides an introduction
to the topic, including its relationship to the
Fibonacci sequence, as well as links to many
other websites.
Homework Complete the Beautiful Faces
worksheet that was handed out in class.
(Teachers you may access this worksheet at the
Illinois State Board of Education website
http//www.isbe.net/ils/math/stage_I/7A_7B_7C_8BI.
pdf