Although it was first defined and proved just a couple of millenia ago, the golden ratio has been around since the dawn of time. It is prevalent in nature, considered to be highly aesthetically pleasing, and is related to almost all other kinds of mathematics. Mathematically speaking, it is the ratio of 1 to the sum of half the square root of 5 and one half. To describe it as such, however, is to rob it of its intrinsic beauty. But it is not just visually pleasing; it is fascinating for it prevalence and it interesting properties. In this report we will look at its place in nature, art, and history.
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The first formal proof of the Golden ratio is found in Book II of Euclid's Elements, published approximately 300 B.C. Proposition 11 gives the geometric method to cut a straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment. While to the untrained ear this doesn't sound like the golden ratio, the unused piece that remains when you subtract the rectangle contained by the whole and one of the segments and the square on the remaining segment.
The fact that Euclid first wrote down the proof does not mean he created it; almost no one believes that his Elements was an original work. He merely cataloged it, although he did it with such methodology and precision that it is an accomplishment in and of itself. Many believe that the Pythagoreans, circa 500 B.C., knew the technique for finding golden sections. The Pythagoreans loved the pentagram, and it served as a symbol, or logo for them. To draw a perfect pentagram, however, you need to be able to construct a golden ratio.
If we start with a line AB, we can pick a point C that is the golden mean.
We draw an arc with center A and radius AB. We then pick the point D such that AC = CD = DB.
We construct a circle that pass through A, B, and D. We then continue BC until it intersects the circle at point E. Next we bisect the angle ABD and mark the point F where it intersects the circle. Connecting F and E gives a perfect pentagram.
The first known calculation of the golden ratio as a decimal was found in a letter written by Michael Maestlin to noted astronomer (and former student) Johannes Kepler in the year 1597. He gives "about 0.6180340" for the ratio, which is not far from the 'actual' decimal value of 0.61803398874989484821...
None of the people mentioned here referred to it as the "golden" ratio or section. The common term was "division in extreme and mean ratio". Pacioli introduced the term "divine proportion" and Clavius also used the term "proportionally divided". The first known use of the term "golden section" appears in a footnote in Die reine Elementar-Matematik by Martin Ohm: One is also in the habit of calling this division of an arbitrary line in two such parts the golden section; one sometimes also says in this case: the line r is divided in continuous proportion.