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]

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 built-in custom tool (File...Open...Program Files...Sketchpad...Samples...Custom Tools...Polygons.gsp).

• Construct two of the pentagon’s 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.

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

• 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.

• 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.

• 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:

• 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?

• Another definition of Phi is "a number which is one more than its reciprocal"; that is, f = Verify this equation mathematically.
  
  
• Look again at this last equation: f = . We could define f with a recursive definition. How can we define f as a continued fraction?

  If we stop the continued fraction for f at various stages, we get values which approximate f :

• 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?

• 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.