Golden Ratio Download Fixed
Kam Bergmann
The golden ratio, also known as the golden number, golden proportion, or the divine proportion, is a ratio between two numbers that equals approximately 1.618. Usually written as the Greek letter phi, it is strongly associated with the Fibonacci sequence, a series of numbers wherein each number is added to the last. The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on, with the ratio of each number and the previous number gradually approaching 1.618, or phi.
Phi does show up in other aspects of nature. Tree leaves and pine cone seeds tend to grow in patterns that approximate the golden ratio, and sunflower spirals and other seeds tend to hew close to phi. Phi allows for efficient distribution or packing, so leaves that grow in relation to the golden ratio will not shade each other and will rest in relation to one another at what is known as the golden angle.
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.
Phi can be defined by taking a stick and breaking it into two portions. If the ratio between these two portions is the same as the ratio between the overall stick and the larger segment, the portions are said to be in the golden ratio. This was first described by the Greek mathematician Euclid, though he called it "the division in extreme and mean ratio," according to mathematician George Markowsky of the University of Maine.
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 Φ (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.
But as 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, but as Markowsky discussed, 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.
Attempts to find phi in the human body also succumb to similar fallacies. A recent study claimed to find the golden ratio in different proportions of the human skull. But as Dale Ritter, the lead human anatomy instructor for Alpert Medical School (AMS) at Brown University in Rhode Island, told Live Science:
People have claimed that seashells, such as those of the nautilus, exhibit properties in which phi lurks. But as Devlin points out on his website, "the nautilus does grow its shell in a fashion that follows a logarithmic spiral, i.e., spiral that turns by a constant angle along its entire length, making it everywhere self-similar. But that constant angle is not the golden ratio. Pity, I know, but there it is."
While phi is certainly an interesting mathematical idea, it is we humans who assign importance to things we find in the universe. An advocate looking through phi-colored glasses might see the golden ratio everywhere. But it's always useful to step outside a particular perspective and ask whether the world truly conforms to our limited understanding of it.
In the book, Pacioli writes about mathematical and artistic proportion, particularly the mathematics of the golden ratio and its application in art and architecture. The book contains dozens of beautiful illustrations of three-dimensional geometric solids and templates for script letters in calligraphy.
The golden ratio, also known as the divine proportion, is a special number (equal to about 1.618) that appears many times in geometry, art, an architecture. The golden ratio is found when a line is divided into two parts such that the whole length of the line divided by the long part of the line is also equal to the long part of the line divided by the short part of the line.
Some shapes, such as dodecahedrons and icosahedrons, have inherent golden ratios in their dimensions and spatial positions of their intersecting lines. Some artists and architects believe that the golden ratio makes the most beautiful shapes. As a result the ratio can be found in many famous buildings and artworks, such as those by Leonardo da Vinci.
In a nutshell, the golden ratio, also known as the Golden Mean, Golden Section or the Greek letter phi, is a mathematical ratio that can be expressed algebraically. But it's also commonly found in nature when the ratio of two quantities is the same as the ratio of their sum to the larger of the two quantities.
Dylan Siemens, a United States Brewers Cup Champion and lead barista trainer for Onyx Coffee Lab, landed on a 1:16 ratio (22 grams of coffee to 350 ml of water) when brewing with our Stagg [X] Dripper.
The golden ratio has been used to analyze quantities found in nature, architecture, painting, and music. When used, it is often assumed to create an organic, balanced, and aesthetically pleasing composition, thought to be favored by the human eye.
However, it turns out that things are a little more complicated because, as line lengths get longer, line heights must also increase in order to maintain readability. For example, a 2004 study from the University of Reading, concludes that long line lengths need more interlinear spacing so that the eyes can easily locate the next line down. If you really want to design your typography with the golden ratio in mind, you can use a golden-ratio typography calculator, which will give you an ideal line height for a given font size and line width.
Some designers are fascinated with the golden ratio and use it to create and edit all sorts of interface-design elements. Others believe that the golden ratio is no more valid than any other method used to derive sizes and proportions. Regardless, the golden ratio can be a helpful reference to new visual designers or designers wanting to improve their skills with a concrete, mathematical approach.
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