Friday, September 28, 2007

The amazing Golden Ratio

The amazing Golden Ratio
Assem Deif
Al-Ahram Weekly
27 September - 3 October 2007

Issue No. 864


Artists reckon that the "Golden Ratio", also called the "Golden Section Phi" and nature's most astonishing number, is the ratio that controls the proportions of all beautiful objects, writes Assem Deif*

Historians trace the Golden Ratio back to Euclid, yet even before him it was governing the dimensions of monuments in ancient Egypt. The most pronounced of these is the Great Pyramid. The Great Pyramid is not the only structure from ancient Egypt that complies with constants like Pi or Phi; the Egyptians used the Golden Ratio in many ways both in the architecture of their temples and in their drawings. So whereas, the discovery of the "golden rule" was generally credited to the Greeks, the findings of such Egyptologists as De Lubicz and Fliders Petrie produced irrefutable proof that the Egyptians had a mathematical understanding of these constants, the ratios, not the symbol, 1000 years earlier.

There was much cross-culture between the Egyptian and Greek civilisations during the Hellenistic era, particularly in Alexandria where Egyptian and Greek scientists studied together at the Mouseion. Among them was Euclid. Historians call him Euclid of Alexandria without precluding the possibility he might have been Egyptian. It was in Alexandria that Euclid wrote The Elements.

Greek scholars were visiting Egypt even before the Mouseion was founded, including Thales, Socrates, Plato and Aristotle, and Pythagoras, who spent 22 years in Egypt and announced his theory only after leaving. Egyptians were aware of the triangle 3:4:5 which Pythagoras himself called the "Holy Triangle". Eight pyramids from the fourth and fifth dynasties have their inner triangle conforming to these ratios.


Giza Pyramids

leaf arrangements around a stem and plant branching in a fibonacci formation;

golden spiral

pascal triangle; panel of Khesi-Re

Friday, July 13, 2007

Introduction to Proporzione Divina

Proporzione Divina features commentary on art and music by architect and author Bevin Chu.

Proporzione Divina is Italian for "Divine Proportion." The Divine Proportion is also known as the golden ratio, the golden section (Latin: sectio aurea), golden mean, golden number, divine section (Latin: sectio divina), or golden proportion, and is represented by the Greek letter phi (φ).

The Golden Section

From Wikipedia, the free encyclopedia

The golden section is a line segment sectioned into two according to the golden ratio. The total length a+b is to the longer segment a as a is to the shorter segment b.

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887.

φ = (square root of 5 – 1)/2 = approximately 1.6

The Golden Rectangle

Many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

The Fibonacci Series

The irrational numbers of the golden section closely correspond to the whole numbers of the Fibonacci series, in which each number is the sum of the two preceding numbers.

The numbers of the Fibonacci series are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ... ad infinitum

This makes the Fibonacci series highly useful for architects and designers who wish to make their buildings and designs approximate the golden section, but must as a practical matter use whole number units.

Divine Immanence

The Golden Rectangle shows up in the most unexpected places.

For example, it shows up in the 16:10 aspect ratio of the newest, most advanced LCD monitors, which provide more comfortable viewing than older, now obsolescent LCD monitors with 4:3 aspect ratios.

The newer 16:10 aspect ratio is more ergonomic because it more closely matches the human being's natural field of view, formed by the two overlapping fields of view of two side by side human eyes.

Vesica Piscis: Latin for "bladder of the fish," two circles of the same radius whose centers lie on each others' circumference, aspect ratio of 3:2 corresponds to the fifth and fourth numbers in the Fibonacci series

The older 4:3 aspect ratio, by contrast, is noticeably less ergonomic because it chops off the left and right margins of that natural field of view.

ViewSonic Home Page (Traditional Chinese): "ViewSonic VG2230wm 22 inch multimedia liquid-crystal display, 16:10 golden ratio display ... high resolution golden ratio display 1680x1050"

The 16:10 aspect ratio of the newest LCD monitors reduces to 8:5, the seventh and sixth numbers in the Fibonacci series.