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!"

Featured Articles: Math or Myth? An overview and the evidence

phi golden ratio 1.618 overview

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 … More on Math, Myth and Truth

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Applications for Better Art, Design and Composition


Google Logo and the Golden Ratio in Design

New Google logo design change still finds visual harmony using the Golden Ratio. Here's a version without the arrows for a clearer view: There's a good chance you've used Google to search the Internet lately, maybe even in getting to this site. With over a billion people visiting per month, Google obviously … More on Art and Design


Raphael and the Golden Ratio in Renaissance Art

Raphael was one of three Master artists of the Renaissance Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance and lived from 1483 to 1520. He is recognized as one of the three great masters of that period, accompanied by Michelangelo and Leonardo da Vinci. His … More on Art and Design


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


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


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

Stock Market and FOREX Trading Analysis


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


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?


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?

What is Phi? (The Basics of the Golden Ratio)

Phi for "Neo-Phi-tes:" Phi ( Φ = 1.618033988749895... ), most often pronounced fi like "fly," is simply an irrational number like pi ( p = 3.14159265358979... ), but one with many unusual mathematical properties.  Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation. Phi is the basis for the Golden Ratio, Section or Mean The ratio, or proportion, determined by Phi (1.618 ...) was known to the Greeks as the "dividing a line in the extreme and mean ratio" and to Renaissance artists as the "Divine Proportion"  It is also called the Golden Section, Golden Ratio and the Golden Mean. Phi, like Pi, is a ratio defined by a geometric construction Just as … More on Phi

What is the Fibonacci Sequence (aka Fibonacci Series)?

Leonardo Fibonacci discovered the sequence which converges on phi. In the 12th century, Leonardo Fibonacci wrote in Liber Abaci of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi.  This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . The ratio of each successive pair of numbers in the sequence approximates phi (1.618. . .) , as 5 divided … More on Phi

The Golden Section / Golden Ratio

GoldenNumber.Net explores the appearance of Phi, 1.618 (also known as the Golden Ratio, Golden Mean, Golden Section or Divine Proportion, in mathematics, geometry, life and the universe and shows you how to apply it, and its applications are limitless: Art Architecture Design of any kind – Graphics, logos, products, fashion, web sites and more Photo composition, photo cropping matting and framing Personal beauty and facial or dental cosmetic procedures to enhance beauty Stock market and FOREX analysis The Golden Section is a ratio based on a the number Phi, 1.618... The Golden Section or Ratio is is a ratio or proportion defined by the number Phi (= 1.618033988749895... … More on Phi

Appearances in Life and Nature

Is the Nautilus shell spiral a golden spiral?

Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. 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. … More on Nature

Appearances in the Cosmos


Stars that Pulsate to the Golden Ratio

A research article called "Strange Nonchaotic Stars" published at the American Physical Society reported the discovery of a class of white-blue variable stars that pulsate in a fractal pattern at frequencies close to the golden ratio. Download the paper in PDF format … More on the Cosmos


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


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