The Golden Ratio

In mathematics, the number that starts as 1.618 is better known by which name?

And the answer: the golden ratio.    

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Like pi, the golden ratio is an irrational number that can be found in geometry, nature, architecture, music and more. In short, if you divide a line into two parts, the golden ratio is found when Part B has the same ratio to Part A, as Part A does to the two combined.

Often abbreviated as "phi," the golden ratio is a non-repeating number whose decimal points span into infinity. Like pi, phi is an irrational number because it can not be written as the ratio of two integers.

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The golden ratio's irrationality was first discovered by the ancient Greek known as Euclid. As one of the founding fathers of modern mathematics, Euclid wrote a series of books called The Elements around 300 BCE. In it contained most of what was known about math at the time, and today is widely considered one of the most influential textbooks of all time. While the name "phi" or "golden ratio" wasn't present in its pages, Euclid presented the concept in all its novelty as the "extreme and mean ratio." Instead of thinking of it numerically, the concept was considered in relation to the whole: the ratio of the whole to the longer segment was the same as the ratio of the longer segment to the shorter line.

The golden ratio appears in geometry to represent a figure whose sides are of this specific proportional measure. Often times, these figures are dubbed for their aesthetically pleasing appearance and symmetry. For example, for triangles whose ratio of long side to short equals phi (i.e., the golden ratio), they earn the name of "golden triangle" or "sublime triangle." The angles of the triangle are 72, 72, and 36 degrees. This golden triangle can be divided and added in part to create similarly perfect pentagons, or a figure known as a "gnomon." Similarly, figures such as the golden rectangle can be divided again and again into smaller golden rectangles, which, when traced, result in a perfect spiral shape.

Learn more about the occurrences and use of this fascinating ratio below.


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