Geometry has two great treasures: one is the theorem of Pythagoras;
the other, the division of a line into extreme and mean ratio. The first
we may compare to a measure of gold; the second we may name a precious
jewel.
Kepler [1571-1630]
Part 1:
Some of the earliest references to the pleasures of mathematics are linked
with the name of the Greek philosopher, Pythagoras (569-500 B.C.), who
observed certain patterns and number relationships occurring in nature.
The explanation of the order and harmony of nature was, for Pythagoras,
to be found in the science of numbers. He conjectured that sympathetic
sounds were emitted by the heavenly bodies as they traveled their celestial
orbits. A taste for "the mysteries" led the ancient Greek to ascribe a
special significance to the dodecahedron: its twelve regular facets correspond
to the twelve signs of the zodiac. It was a symbol of the universe. Moreover,
each pentagonal face, being associated with the golden section, had a
special interest for the Pythagoreans. In this activity, we will explore
these "special" characteristics of a pentagon.
- In a new sketch window, construct a regular pentagon ABCDE,
labeled in the conventional manner. You may either create the construction
yourself or use the pre-recorded script (File...Open...Sample Scripts...Polygons...5byEdge.gss).
- Construct two of the pentagon’s two diagonals from point A, namely
AC and AD. Change the color of the diagonals to blue. Measure
the length of the diagonals (to the nearest thousandths). Manipulate
your pentagon while observing the diagonals lengths. Comment on your
findings.
- From point B, construct a diagonal that intersects both of the blue
segments, namely BE. Change the color of segment BE to red. Construct
the intersection points of the blue segments and the red segment and
label them as P and Q.
- Using the Distance command under the Measure menu, measure
the lengths of several "subsegments" along the diagonal BE,
using points B, P, Q, and E (e.g., the distance from B to Q). Do you
notice any relationship between these measurements? Compute the ratios
BE:BQ and BE:BP. What do you notice about these ratios as you manipulate
the pentagon? Explore several other ratios as the measure of
a segment to each of its subsegment parts. Comment on your findings.
The point of intersection P, of two of the diagonals, divides each diagonal
in the golden ratio. The point P divides BQ and BE internally and
QE externally in this ratio. The problem of finding the golden section of
a straight line is solved in the Elements Book II, Proposition 11.
It has therefore been a topic of interest to mathematicians for more than
twenty centuries.
Definition: Let a segment AB of length l be divided into
two segments by the point C (see figure below). Let the lengths of AC
and CB be a and b respectively. If C is a point such that
l:a as a:b, C is the "golden cut" or the golden section
of AB. The ratio of l/a or a/b is called the golden ratio.
In the terminology of the early mathematicians AB is divided by C in "extreme
and mean ratio." Kepler called it "the divine proportion."
Length (AB) = l
Part 2:
- There seems to be no doubt that Greek architects and sculptors incorporated
this golden ratio in their artifacts. The physical appearance of most
objects, whether they are furniture, plants, buildings or art work,
has some proportional aspect. The arms of a chair are most likely proportional
to the seat; or the windows on a building are proportional to the building's
size. Using the Geometer's Sketchpad, construct a rectangle in a new
sketch window. Manipulate your rectangle until its length and width
are the most "aesthetically pleasing" to you.
To construct
a perpendicular line:
- Select both a point and a straight object. The constructed perpendicular
line will go through the point and will be perpendicular to the
straight object.
- Choose Perpendicular Line under the Construct menu.
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- Compute the ratio of your rectangle's length to its width. Compare
your ratio with those of your neighbors. What did you notice about the
ratios? Discuss the range of the ratios computed by your classmates.
- The "divine proportion" appears in places from ancient Greek architecture
to ancient and modern paintings. More importantly however, the divine
proportion acquired fame from its common occurrence in nature. The ratio
was discovered in proportions of the human body, along with proportions
in the biology of plants. If the ratio of a rectangle's length to its
width is equal to the golden ratio, it is called a golden rectangle.
Manipulate your rectangle so that it is "golden." Compare your rectangle
to your neighbors. Are they congruent?
- Construct a golden rectangle, and investigate the ratio of
its length to its width. Manipulate your rectangle and observe its "golden
ratio". Comment on your findings.
To construct
a Golden Rectangle:
- Construct a square labeled ABCD.
- Construct the midpoint of the segment AB and name it point F.
- Construct a circle with center F passing through point C.
- Extend segment AB and label the points of intersection with the
circle G and H.
- Construct a line perpendicular to line AB passing through the
point H.
- Construct a line perpendicular to segment BC and passing through
point C.
- Construct the intersection of these last two lines and name it
point I. Construct segments BH, HI, and CI.
- Hide the circle; lines GH, HI, and DI; and points G and F.
- The resulting figure ADHI is a golden rectangle.
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- Using pencil and paper methods, formally compute the value of the
golden ratio using the construction above. (Hint: Let the width of the
rectangle equal one, compute its length. Then compute the ratio of the
rectangle’s length to its width.)
Part 3:
It was suggested in the early days of the present century that the
Greek letter f (Phi) - the initial letter
of Phidias’s name - should be adopted to designate the golden ratio.
Hence, f =
 .
- What numbers remain constant when they are squared? That is, for what
values of x is x2 = x ? When squaring other numbers,
they either get larger or smaller depending on their magnitude. One
definition of f is f
2 = f + 1. Mathematically
verify to square f , you simply add
1 to f . In fact, there are two numbers
with this property, one is f and the
other is closely related to f . Can
you figure out the other number with this property?
f 1 = 1
f 2 = 1 +
= 
f 3 = 1 +
= 1 +  =
f 4 = 1 +
 =
1 +  = 1 +
= 
- Did you recognize the numbers produced in the numerators and denominators?
Predict the next three terms of this sequence. How could you verify
your predictions? What do you notice about the ratio?
Part 4:
- The proportions of the well-known Parthenon support the influence
exerted by the golden rectangle on Greek architecture. Search and download
a photo of the Parthenon into the Geometer's Sketchpad. Resize and overlay
a golden rectangle showing the various "divine proportions" within its
architecture. An excellent source of architecture images is the University
of Wisconsin's Library of Art History images (http://www.wisc.edu/arth/index.html).
Use the search engine by selecting the Period Ancient Greece: Classical
and the Site Athens.
To copy
a picture and paste it into Geometer’s Sketchpad:
- Place your mouse on the picture and click the right hand mouse
button (on a Mac, click your mouse button and hold it down until
a window appears).
- Select Copy this image from the menu window.
- Return to the Geometer’s Sketchpad, and choose Paste from
the Edit menu.
- Click off of the picture to de-select the image.
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Since the ancient Greeks came to the realization that the golden rectangle
has an aesthetically pleasing quality, modern architects and artists
have adopted, maybe not the golden rectangle in its entirely, but the
golden section. Search and download a painting or photo which you think
demonstrates the golden rectangle or section. Try overlaying several
golden rectangles on the picture in your sketch window. Present your
findings in an oral or written report to your class. (Be sure to give
credit all photos/paintings in your report.)
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Verify this equation mathematically.