The golden ratio undermines the way that we see the world. When we look at shapes, be they a 2 dimensional rectangle, a playing card, a piece of art, or the complex features of a persons face, the beauty or "the pleasing factor" which we have, often depends on how close the dimensions of that shape approach the golden ratio.
As an example of this, decide which one of the pictures below you find more pleasing.
Most of us decide on (b). Picture (a) is too square, (c) too upright and (d) too long and thin.
Because of this "pleasing factor", the golden ratio is found throughout Art, Architecture and surprisingly Nature.
In Leonardo da Vinci's painting of an old man, which many experts believe was a self-portrait, we see superimposed on the picture, a square which has been subdivided into rectangles, some of which approximate golden rectangles. In his unfinished painting of "St Jerome", we can see a lion lying at his feet. The golden rectangle fits so perfectly around the figure that it is believed that the painting was made to conform to these proportions.
In Architecture, we need look no further than one of the most magnificent buildings ever built: the Parthenon on the Acropolis in Athens, to see evidence of the golden ratio. The temple was built in honour of the daughter of Zeus, Pallas Athene, the goddess of Wisdom, the patroness of the arts and crafts and the protectress of cities. Although the temple is built in the golden rectangle's proportions, it is unlikely that the 5th Century B.C. builders new of its existence.
The Greek mathematician, Pythagoras, suspected that somehow the way humans are built could be described using the golden ratio. He finally proved that each part of the body is in a definite golden proportion to all the other parts. For example, the ratio of the height of a man to the height of his navel is approximately the 1.618... .
Here's the golden spiral inside the golden rectangle:
Class 9 this year constructed their own golden rectangles and coloured them in Art. Many of them contained designs in each of the smaller rectangles. As an initial idea, we suggested that they might think about shading in a similar way to these:
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.
J. Kepler (1571 - 1630)