Sunday, July 15, 2007

The Golden Ratio: Pentagrams, Pythagoras and Venus

Some interesting facts about math, the pentagram, nature, and culture.

The first known uses of the pentagram (the simplest regular star polygon) are found in Mesopotamian writings dating to about 3000 BC. The 5 points represented the 5 astrological directions associated with Jupiter, Mercury, Mars and Saturn, and Venus (or Astarte, the local Goddess). The top or "above" point of the star symbolized Venus/Astarte - as she was revered as the "Queen of Heaven."

In fact, the pentagram (the simplest regular star polygon) has long been associated with the planet Venus, and the worship of the goddess Venus, or her equivalent... When viewed from Earth, successive inferior conjunctions of Venus plot a nearly perfect pentagram shape around the zodiac every eight years. (source)

At one time, the pentacle was used by Christians to represent the Five Sacred Wounds of Jesus. The ancient greeks and romans, among others, considered the pentacle a symbol of balance and health. The pentacle can be seen to resemble or symbolize the human body, the 5 points representing the head and 4 limbs. In the Tarot, the pentacle symbolizes the female suit of Earth - divinity manifesting in matter.

The golden ratio, (also known as the Divine Proportion, or φ (spelled PHI) can be expressed as:

φ = (1+√5)/2 ≈ 1.618

The Divine Proportion plays an important role in the the mathematical nature of the pentagram (among many, many other such phenomenon). In the pentagram each intersection of edges sections the edges in golden ratio: the ratio of the length of the edge to the longer segment is φ, as is the length of the longer segment to the shorter.

Also, the ratio of the length of the shorter segment to the segment bounded by the 2 intersecting edges (a side of the pentagon in the pentagram's center) is φ.

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

As the illustration shows:

\frac{\mathrm{red}}{\mathrm{green}} = \frac{\mathrm{green}}{\mathrm{blue}} = \frac{\mathrm{blue}}{\mathrm{magenta}} = \varphi .

The golden ratio has fascinated intellectuals of diverse interests for at least 2,400 years.

"Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics." ~ Mario Livio
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.

Architecture was also profoundly influenced by the golden ratio. The Egyptian Pyramids, Stonehenge, the Acropolis, the Great Mosque of Kairouan (built by Uqba ibn Nafi c. 670 A.D.), and the Tempietto chapel at the Monastery of Saint Peter in Montorio, Rome... these architectural wonders and many more are said to have been designed and built using the golden ratio.

Science has been influenced by the golden ratio as well. Jean-Claude Perez proposed a connection between DNA base sequences and gene sequences and the golden ratio [link]. Another such connection, between the Fibonacci numbers and golden ratio and Chargaff's second rule concerning the proportions of nucleobases in the human genome, was proposed in 2007.

one well known example of the golden ratio in popular culture can be found, of course, in a da vinci painting - in this case the vitruvian man. the painting illustrates the golden ratio as it is seen in the human body - for example, the ratio of the distance from your fingertips to your shoulder to the distance from your elbow will always be φ... same with the ratio between your hips, knees and feet. (look it up!)

The Greeks usually attributed discovery of the golden ratio to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.

Pythagoras and his Pythagoreans - along with being vegetarians and espousing the Pythagorean Theorum (who doesn't love a2 + b2=c2?) venerated the pentacle and the golden ratio. Pythagoras viewed the pentacle as a symbol of mathematical and natural perfection . One of the Pythagoreans' main symbols was the Pythagorean Pentacle (above), a pentacle inscribed within a pentagon.

It is said that the pentagram had a secret significance and power to the Pythagoreans, and was used not only as a symbol of good health, but as a password or symbol of recognition amongst themselves.

The Pythagoreans were also great nature lovers.

Pythagoreans celebrate sunrise
Fyodor Bronnikov, 1869

dedicated to the Divine Proportion