Information AboutGolden Ratio |
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Shapes proportioned according to the golden Ratio have long been considered Aesthetically pleasing in Western cultures, and the golden ratio is still used frequently in art and design, suggesting a natural balance between symmetry and asymmetry. The ancient Pythagoreans , who defined numbers as expressions of ratios (and not as units as is common today), believed that reality is numerical and that the golden ratio expressed an underlying truth about existence. DEFINITION Two quantities are said to be in the ''golden ratio'', if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if : where ''a'' is the larger part and ''b'' is the smaller part. Equivalently, they are in the golden ratio if the Ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e. if : Dividing the numerator and denominator of the second fraction by ''b'', we get : and replacing ''a''/''b'' by φ, we get : This becomes : or equivalently, : The solutions of this Quadratic Equation are : (Note: The above solutions can be obtained directly by the Quadratic Formula or by ''completing the square'': : : :which of course yields the same solutions as above.) Since φ is positive, we have : HISTORY The golden ratio was first studied by ancient mathematicians because of its frequent appearance in Geometry . There is evidence that it was understood and used as far back in history as Ancient Egypt . In the Great Pyramid Of Giza built around 2600 BC , the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the Secant of the angle θ. The above two lengths were about 186.4 and 115.2 metres respectively. The ratio of these lengths is the golden ratio 1.618. The largest Isosceles Triangle of the ''sriyantra'' design used in Ancient India , described in the '' Atharva-Veda '' (circa 1200 - 900 BC ) is one of the face triangles of the Great Pyramid in miniature, showing almost exactly the same relationship between π and the golden ratio as in its larger counterpart. The Ancient Greeks usually attributed its discovery to Pythagoras (or to the Pythagoreans, notably Theodorus ) or to Hippasus Of Metapontum . Hellenistic mathematician Euclid spoke of the "golden mean" this way, "a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". The golden ratio is represented by the Greek Letter ( Phi , after Phidias , a sculptor who commonly employed it) or less commonly by '''''' ( Tau , the first letter of the Ancient Greek root τ(ε/ο)μ– meaning ''cut''). A STARTLINGLY QUICK PROOF OF IRRATIONALITY Recall that we denoted the "larger part" by ''a'' and the "smaller part" by ''b'', and concluded that : This gives a startlingly quick proof that the golden ratio is an Irrational Number . An irrational number is one that cannot be written as ''a''/''b'' where ''a'' and ''b'' are Integer s. If ''a''/''b'' is such a fraction, in Lowest Terms , then ''b''/(''a'' − ''b'') is in even lower terms — a contradiction. Thus this number cannot be so written, and it is therefore irrational. ALTERNATE FORMS The formula can be expanded recursively to obtain a Continued Fraction for the golden ratio: : and its Reciprocal : : Note that the convergents of these continued fractions (1, 2, 3/2, 5/3, 8/5, 13/8, ..., or 1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci Number s. The equation likewise produces the continued Square Root form: : Also: : : : These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a Pentagram . If x agrees with to n decimal places, then agrees with it to 2n decimal places.
MATHEMATICAL USES s. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of Logarithmic Spiral . Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.]] that approximates the Golden Spiral.]] The number φ turns up frequently in Geometry , particularly in figures with pentagonal Symmetry . The length of a regular Pentagon 's diagonal is φ times its side. The vertices of a regular Icosahedron are those of three orthogonal Golden Rectangle s. The explicit expression for the Fibonacci Sequence involves the golden ratio: : The Limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the Fibonacci sequence is divided by its preceding number, it approximates φ. e.g., 987/610 ≈ 1.6180327868852. Alternatingly the approximation to φ is too small and too large, it gets better as the Fibonacci numbers get higher, and: |
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