The Golden Ratio: Phi, 1.618

This site is dedicated to sharing the best information on Phi, the number 1.618, with insights from dozens of Contributors on a broad range of topics.  We're here to help you understand its unique properties in mathematics and geometry, appreciate its many appearances in life and the universe, and benefit by applying it in art, design, beauty, financial markets and more. Share your thoughts on the Discussion Page or on any of the article posts. Enjoy the "phinomenon!"

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Applications for Better Art, Design and Composition

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Da Vinci and the Divine Proportion in Art Composition

Leonardo Da Vinci has long been associated with the golden ratio. This association was reinforced in popular culture in 2003 by Dan Brown's best selling book "The Da Vinci Code." The plot has pivotal clues involving the golden ratio and Fibonacci series. In 2006, the public awareness of the association grew when the … More on Art and Design

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Logo Design using the Golden Ratio

Phi is often applied in product logos. From Renaissance artists of the 1500's to graphic artists of today, phi is recognized for its ability to give a sense of aesthetic appeal in balance and harmony of design.  Product logos represent an image that must make a positive and memorable impact on the conscious and … More on Art and Design

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Aston Martin, James Bond and the Golden Ratio

What do James Bond, Aston Martin and the Golden Ratio have in common? James Bond, also known as 007, drove an Aston Martin DB5 in the movies GoldFinger and GoldenEye, and Aston Martin is now boasting its application of the Golden Ratio in the design of its latest DB9 and Rapide S automobiles. The Aston Martin Rapide … More on Art and Design

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Golden Ratio in Art Composition and Design

"Without mathematics there is no art,"  said Luca Pacioli, a contemporary of Da Vinci. Just as the Golden Section is found in the design and beauty of nature, it can also be used to achieve beauty and balance in the design of art.  This is only a tool though, and not a rule, for composition, but still a good Art 101 … More on Art and Design

Stock Market and FOREX Trading Analysis

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Stock Market Analysis, Phi and the Fibonacci Sequence

Human expectations occur in a ratio that approaches Phi. Changes in stock prices largely reflect human opinions, valuations and expectations. A study by mathematical psychologist Vladimir Lefebvre demonstrated that humans exhibit positive and negative evaluations of the opinions they hold in a ratio that … More on Trading

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Foreign Exchange Markets (FOREX) and Trading

Phi relationships appear in foreign currency price movements. It has long been known that phi and Fibonacci relationships appear in the stock markets.  The foreign exchange market, or Forex, is the largest market in the world.  The simplest definition of foreign exchange is the changing of one currency to … More on Trading

What determines our perceptions of beauty?

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Facial Analysis and the Beauty Mask

"Beauty is in the phi of the beholder." It has long been said that beauty is in the eye of the beholder and thought that beauty varies by race, culture or era.  The evidence, however, shows that our perception of physical beauty is hard wired into our being and based on how closely the … More on Beauty

What is Phi?

Golden Ratio Overview

What makes a single number so interesting that ancient Greeks, Renaissance artists, a 17th century astronomer and a 21st century novelist all would write about it? It's a number that goes by many names. This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Proportion, Golden Mean, Golden Section and Divine Proportion.  It was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in "De Divina Proportione" in 1509, by Johannes Kepler around 1600 and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.” With the movie release of the “The Da Vinci Code”, the quest to know … More on Phi

Appearances in Life and Nature

The Nautilus shell spiral as a golden spiral

Nautilus shell spirals have phi proportions, but not as you may have heard. There is a fair amount of confusion, misinformation and controversy over whether the graceful spiral curve of the nautilus shell is based on the golden mean.  Some say yes, but offer no proof at all.  Some show examples of spirals, but incorrectly assume that every equi-angular spiral in nature is a golden spiral.  One university professor says no, but only compared the nautilus spiral to … More on Nature

Appearances in the Cosmos

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Phi and the Solar System

The dimensions of the Earth and Moon are in Phi relationship, forming a Triangle based on 1.618. The illustration shows the relative sizes of the Earth and the Moon to scale. Draw a radius of the Earth (1). Draw a line from the center point of the Earth to the center point of the Moon (square root of … More on the Cosmos

Unique Properties in Geometry

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Phi and Geometry

Phi (Φ) was described by Johannes Kepler as one of the "two great treasures of geometry." (The other is the Theorem of Pythagoras.) Phi appears in many basic geometric constructions. 3 lines: Take 3 equal lines.  Lay the 2nd line against the midpoint of the 1st.  Lay the 3rd line against the midpoint … More on Geometry

Unique Properties in Mathematics

Mathematics of Phi, 1.618, the Golden Number

Phi, Φ, 1.618…, has two properties that make it unique among all numbers. If you square Phi, you get a number exactly 1 greater than itself: 2.618…, or Φ²  = Φ + 1. If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than itself: 0.618…, or 1 / Φ = Φ - 1. These relationships are derived from the dividing a line at its golden section point, the point at which the ratio of the line (A) to the larger section (B) is the same as the ratio of the larger section (B) to the smaller section (C). This relationship is expressed mathematically as: A = B + C, and A / B = B / C. Solving for A, which on both sides give us this: B + C = B²/C Let's say … More on Mathematics