Golden Ratio, Golden Mean, Golden Section, Divine Proportion

The golden ratio is also called the golden section or golden mean. Other names include extreme and mean ratio, medial section, divine proportion, divine section golden proportion, golden cut, and golden number. 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.

Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which can be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.

Golden Ration  Wikipedia

The Golden Mean and Aesthetics  
PhysOrg - December 3, 2013

What do the facades of the National Gallery in London's Trafalgar Square and the Sydney Opera House in Australia have in common? Most people would suggest the two buildings have few similarities - the former is symmetrical and classically proportioned, whereas the latter is modern, organic and curvilinear. An architect might offer the provocative answer that both buildings feature compositions of geometric shapes, combing circles, squares and rectangles. But this answer is not the reason the two are linked in popular culture. Instead they can be found on the lists of buildings that allegedly rely on the Golden Mean, described below, to achieve a perfect aesthetic composition. The Golden Mean - also known as the Golden Section or the Divine Proportion - is a mathematical concept that is typically traced to the 15th century, a period in which geometry served both practical and symbolic purposes. It is a ratio that defines a recurring relationship between a larger element and a smaller subset of that element.

The Golden Ratio in Architecture

The Vitruvian Man is a drawing created by Leonardo da Vinci circa 1490. It is accompanied by notes based on the work of the architect Vitruvius. The drawing, which is in pen and ink on paper, depicts a male figure in two superimposed positions with his arms and legs apart and simultaneously inscribed in a circle and square. The drawing and text are sometimes called the Canon of Proportions or, less often, Proportions of Man.

The Golden Ratio (1.618), and Fibonacci too ...
If you take any two successive (one after the other) Fibonacci Numbers,
their ratio is very close to the Golden Ratio.

The successive numbers in the Fibonacci sequence quickly converge on Phi

In the News ...

Golden Ratio Discovered in the Quantum World   Epoch Times - January 20, 2010
The golden ratio, which is equal to approximately 1.618, can be found in various aspects of our life, including biology, architecture, and the arts. But only recently was it discovered that this special ratio is also reflected in nanoscale, thanks to researchers from the U.K.'s Oxford University. Their research, published in the journal Science on Jan. 8, examined chains of linked magnetic cobalt niobate (CoNb2O6) particles only one particle wide to investigate the Heisenberg Uncertainty Principle. They applied a magnetic field at right angles to an aligned spin of the magnetic chains to introduce more quantum uncertainty. Following the changes in field direction, these small magnets started to magnetically resonate.

Researcher explains mystery of golden ratio   PhysOrg - December 21, 2009

The Egyptians supposedly used it to guide the construction the Pyramids. The architecture of ancient Athens is thought to have been based on it. Fictional Harvard symbologist Robert Langdon tried to unravel its mysteries in the novel The Da Vinci Code. "It" is the golden ratio, a geometric proportion that has been theorized to be the most aesthetically pleasing to the eye and has been the root of countless mysteries over the centuries. Now, a Duke University engineer has found it to be a compelling springboard to unify vision, thought and movement under a single law of nature's design.