The Golden Ratio and the Fibonacci Numbers defined
The History of the Golden Ratio
The History of Fibonacci
Mathematical Properties of the Golden Ratio
Constructing a Golden Rectangle
The Secret of the Golden Ratio
The Golden Ratio and Beauty...
The Golden Ratio and Beauty in Humans
The Human Face
Phi is a mysterious number which has some related quantities and shapes, and it appears in the proportions of the human body, and other animals', in plants, in DNA, in solar system, in art and architecture, in music, etc. Nevertheless, it had been debated on whether or not it is a natural beauty-maker ever since the first renowned experience that a German physicist and psychologist Gustav Theodor Fechner conducted in the 1860s. His experience is simple: a person needs to choose the most pleasing rectangle among ten rectangles that are placed in front of him, and all have different ratios of length to width. The result shows that 76% of all choices are the three rectangles having ratio of 1.75, 1.62, and 1.50, which are really close to the value of pie (approximately 1.618). Is there enough evidence to show the result? Maybe not, many psychologist have repeated similar experiments ever since, the results are rather conflicting, and inaccurate. Later on, this topic has extended to the determination of the origin of facial attractiveness.
Dr. Stephen Marquardt is a former plastic surgeon, has used the golden section and some of its relatives to make a mask that he claims that is the most beautiful shape a human face can ever have, it used decagons and pentagons as its function that embodies phi in all their dimensions. Dr. Marquardt has been studying on human's facial beauty for many years. According to him, beauty this word can be defined as a mechanism to ensure humans recognize and are attracted by others. The most beautiful faces are just are the ones that are the most easily recognizable as human.
The Human Smile
A perfect smile: the front two teeth form a golden rectangle (which is said to be one of the most visually satisfying of forms, as it is formed with sides of 1 and 1.618). There is also a Golden Ratio in the height to width of the center two teeth. And the ratio of the width of the two center teeth to those next to them is phi. And, the ratio of the width of the smile to the third tooth from the center is also phi.
The Human Lungs
The windpipe divides into two main bronchi, one long (the left) and the other short (the right). This asymmetrical division continues into the subsequent subdivisions of the bronchi. It was determined that in all these divisions the proportion of the short bronchus to the long was always 1/1.618.
Facial Proportions and Human Health
Human health is affected by facial proportions. Biologically, people who have long faces have more chances of having breathing problem, suffering from sleep apnea. And People with shorter faces tend to have abnormal jaw development due to the excessive pressure on the jaw joint, suffering from headaches because their jaws are positioned in a manner that can restrict blood flow to the brain.