![]() Spirals and the Golden Section, Animation Golden Rectangle and Fibonacci numbers One of many interesting ways of how phi can be expressed.Ī stamp with a series of 7 golden rectangles and a logarithmic spiral (Scott 805, Michel 1337). The Pentagon, and the Pythagoras Pentagram Symbol, The symbol of the Golden Section Phi (derived from Pheidias), Plato's Dodecahedron, Archimedes Pi, the Vitruvius Man. often the Greek symbol phi is used (due to Mark Barr since 1909 used this symbol as the initial of Pheidias, the Greek sculptor, who is supposed to have used this in the Parthenon ( Livio 2002) ![]() The smaller rectangle has sides with ratio 1-x : 1 since this is the same as the ratio for the big rectangle, one finds that x 2 = x+1 and thus x = (1+sqrt(5))/2 = 1.618033989. The Greeks were thus able to see geometrically that the sides of R have an irrational ratio, 1 : x. ![]() Note that the golden Rectangle can be approximated by adding squares like a spiral with a edge length: 1, 1, 2, 3, 5, 8, 13, 21, 34.the Fibonacci numbers that approach the golden ratio always better if we consider the value of the ratios 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21. The sides are in the "golden proportion" (1 : 1.618034 which is the same as 0.618034 : 1) has been known since it occurs naturally in some of the proportions of the Five Platonic Solids Thus a smaller square can be removed, and so on, with a spiral pattern resulting. The golden rectangle R, constructed by the Greeks, has the property that when a square is removed a smaller rectangle of the same shape remains. There are some who say that Leonardo da Vinci (1452 – 1519) used the name sectio aurea for the golden section even much earlier but this name was not used by others. The name Golden section probably is due to Martin Ohm from his Mathematics book (1835) (in German “Der Goldene Schnitt” ) maybe chosen due to the comment of Kepler. 500 BC, the wife of Pythagoras is assumed to have written a book Theorem of the golden mean. Pythagoras may have obtained this knowledge when he visited Egypt. It was probably known much earlier in Egypt as some ratio of lengths and heights of Pyramids suggest. The golden section was found by the Pythagoreans who used the Pentagram formed, by the diagonals of a regular Pentagon, as a symbol of their school. If AB = 1 and AG = x then GB = 1-x and 1/x = x/(1-x) and it follows x 2 = 1-x, i.e. Euclid used this phrase to mean the ratio of the smaller part of this line, GB to the larger part AG (ie the ratio GB/AG) is the SAME as the ratio of the larger part, AG, to the whole line AB (ie is the same as the ratio AG/AB).ĪB/AG = AG/GB. The first we may compare to a measure of gold the second we may name a precious jewel.Įuclid in Book 6, Proposition 30 shows how to divide a line in mean and extreme ratio which we would call "finding the golden section G point on the line". Geometry has two great treasures: one is the Theorem of Pythagoras the other, the division of a line into extreme and mean ratio.
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