The Golden Proportion

"What is beauty?". Is beauty only in the eye of the beholder, or are there some absolute values? Beauty is a mystery! "I know not what beauty is, but I know that it touches many things", Dürer .If we study the beauty of nature, teeth, or art we will discover a common principle running throughout. This common principle is the universal recognition of pleasant proportion. We all have a natural understanding of good proportion much in the same way as we know how to divide a line in half or erect a perpendicular. We easily agree that an object of art has good or bad proportion, or that this face looks too long, or too short and out of proportion. This magical connecting thread of proportion, known since antiquity, is non other than the Golden Proportion, a phenomenon related to beauty. The following notes illustrate just a few facets of the mystery and magic of this proportion. The understanding of this concept will enable us to take a first step on a journey of discovery into an unexpected dimension of beauty which affects our lives at every turn. It is one of the building blocks of beauty that we can easily apply to our dental work, confident of success.


The Golden Ratio
Throughout history, the ratio for length to width of rectangles of 1.61803 39887 49894 84820 has been considered the most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The space between the collumns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece. Phidias widely used the golden ratio in his works of sculpture. The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle.

Many artists who lived after Phidias have used this proportion. Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings. The famous "Mona Lisa". Try drawing a rectangle around her face. Are the measurements in a golden proportion? You can further explore this by subdividing the rectangle formed by using her eyes as a horizontal divider. He did an entire exploration of the human body and the ratios of the lengths of various body parts.

You can do the following explorations by using a protractor to draw on paper, or by using Geometer's Sketchpad.

Draw a regular pentagon (to get you started recall that the interior angles have measure 108 degrees) and also draw in one diagonal of the pentagon. Measure the length of one side of the pentagon and measure the length of the diagonal. What is the ratio of the side to the diagonal?

A German psychologist by the name of Gustav Fechner studied the crosses in graveyards and discovered an interesting fact about their dimensions. Measure the upper and lower portions of the main stem of the cross which is cut by the crossbar. What is the ratio of the lengths of these two portions?

Draw an isosceles triangle with base angles equal to 72 degrees. Measure the length of the shorter side and the two legs, which of course have the same length since this is an isosceles triangle. What is the ratio of the lengths?


Input the number "five" into your calculator. Raise it to the .5 power, then mulitiply by .5
*What do you get?*
Square your result and subtract 1.
*What do you get?*
Take the reciprical of your result and add 1.
What is your final result?
Does this number look familiar? Why do you think this occurs?
Sometimes we refer to this magic number as Phi, using the greek symbol.