Phi - The Golden Ratio

The golden ratio is also called the golden mean, golden section,
divine proportion, golden proportion, golden number, and Phi.

Phi: Integrated Information Theory

Phi: The Golden Ratio -   Live Science - November 25, 2019

The golden ratio is one of the most famous irrational numbers; it goes on forever and canŐt be expressed accurately without infinite space. The number phi, often known as the golden ratio, is a mathematical concept that people have known about since the time of the ancient Greeks. It is an irrational number like pi and e, meaning that its terms go on forever after the decimal point without repeating.

Over the centuries, a great deal of lore has built up around phi, such as the idea that it represents perfect beauty or is uniquely found throughout nature. But much of that has no basis in reality.

Phi is closely associated with the Fibonacci sequence, in which every subsequent number in the sequence is found by adding together the two preceding numbers. This sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on. It is also associated with many misconceptions.

By taking the ratio of successive Fibonacci numbers, you can get closer and closer to phi. Interestingly, if you extend the Fibonacci sequence backward - that is, before the zero and into negative numbers - the ratio of those numbers will get you closer and closer to the negative solution, little phi -0.6180339887...

Though people have known about phi for a long time, it gained much of its notoriety only in recent centuries. Italian Renaissance mathematician Luca Pacioli wrote a book called "De Divina Proportione" ("The Divine Proportion") in 1509 that discussed and popularized phi, according to Knott.

Pacioli used drawings made by Leonardo da Vinci that incorporated phi, and it is possible that da Vinci was the first to call it the "sectio aurea" (Latin for the "golden section"). It wasn't until the 1800s that American mathematician Mark Barr used the Greek letter for Phi _ to represent this number.

As evidenced by the other names for the number, such as the divine proportion and golden section, many wondrous properties have been attributed to phi. Novelist Dan Brown included a long passage in his bestselling book "The Da Vinci Code" (Doubleday, 2000), in which the main character discusses how phi represents the ideal of beauty and can be found throughout history. More sober scholars routinely debunk such assertions.

For instance, phi enthusiasts often mention that certain measurements of the Great Pyramid of Giza, such as the length of its base and/or its height, are in the golden ratio. Others claim that the Greeks used phi in designing the Parthenon or in their beautiful statuary.

Markowsky pointed out in his 1992 paper in the College Mathematics Journal, titled "Misconceptions About the Golden Ratio": "measurements of real objects can only be approximations. Surfaces of real objects are never perfectly flat." He went on to write that inaccuracies in the precision of measurements lead to greater inaccuracies when those measurements are put into ratios, so claims about ancient buildings or art conforming to phi should be taken with a heavy grain of salt. The dimensions of architectural masterpieces are often said to be close to phi sometimes this means that people simply look for a ratio that yields 1.6 and call that phi. Finding two segments whose ratio is 1.6 is not particularly difficult. Where one chooses to measure from can be arbitrary and adjusted if necessary to get the values closer to phi.