There are many misconceptions and misrepresentations about the golden ratio. Some look too fervently for patterns and say it exists where it really doesn't. Some whose goal is to debunk golden ratio myth say it doesn't exist where it really does, missing the obvious and … More on Myth and Misconception

# 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 Article: Math or Myth? Fact or Fiction?

#### From the Ukraine to the classroom to Las Vegas

## Renaissance Art Composition and the Ukranian Parliament Fight

The Divine Proportion of Renaissance Art is found in less than Divine circumstances. A recent photo of a fight in the Ukranian Parliament is generating … More on this article

## Common Core Curriculum Math Standards for the Golden Ratio

Welcome educators, teachers, parents and students to the Golden Ratio Academy page. The Golden Ratio appears not only in nature and the arts, but also in the … More on this article

## Da Vinci and the Golden Ratio – Las Vegas style

Elvis Presley, Celine Dion, Penn & Teller, the Rat Pack and many others have found their way to Las Vegas. Now Leonardo Da Vinci and the Golden Ratio are … More on this article

#### Applications for Better Art, Design and Composition

## Leonardo Da Vinci, Salvator Mundi and the Divine Proportion

In 2011, the discovery of a lost painting by Leonardo da Vinci was announced to the world. This painting, Salvator Mundi, had been in the art collection of King Charles I of England in 1649, was auctioned in 1763 and then lost for many years. Its recovery was led by Robert Simon, an art historian and private art … 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

## The UN Secretariat Building, Le Corbusier and the Golden Ratio

Some claim that the design of the United Nations headquarters building in New York City exemplifies the application of the golden ratio in architecture. Debunkers of the golden ratio say no, that this is just another groundless myth to be dispelled. Let's look at the history of its design, the sources of the claims and … 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

## Product Design and Marketing applications of the Golden Mean

Phi is used in the design of many consumer products. Phi has been used to bring beauty, balance and harmony to some of the world's greatest art and architecture. It is also used to add style and appeal in the marketing and design of everyday consumer products. The applications are endless, but are illustrated by … 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

## Beauty, the perfect face and the Golden Ratio, featuring Florence Colgate

What determines the beauty of a perfect face? We may never answer the question "how many angels can dance on the point of a needle," but how about this one: "How many Divine proportions ratios are there in the face of an angel?" The YouTube video and information in this article may give … More on Beauty

## Facial Beauty and the "New" Golden Ratio (or is it just 1.618 in disguise?)

University study declares a "new" golden ratio for facial beauty but validates Phi, the Golden Ratio, as the basis for perceptions of beauty. A university study (PDF) by Pamela M. Pallett, Stephen Link and Kang Lee at the University of Toronto and University of California, San Diego … 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 Number, Golden Proportion, Golden Mean, Golden Section, Divine Proportion and Divine Section. 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 … More on 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

#### 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

## 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