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The Golden Mean 1 5 1.618033989... , called the Golden Mean, or the Golden 2 2 Section, has many interesting and useful properties, some of which have been known since the time of the ancient Greeks. The number 1. A regular five-pointed star, such as those appearing on the U. S. flag, may be considered to be composed of a central pentagon and five isosceles triangles of equal size conjoined so that the base (shorter side) of each triangle coincides with a side of the pentagon. The ratio of the length of each of the two equal sides of one of the isosceles triangles to its base is . 2. If we divide a line segment into two pieces so that the ratio of the length of the whole segment to the length of the longer piece is the same as the ratio of the length of the longer piece to the length of the shorter piece, then this ratio is as shown in the picture below. x 1 x , or 1 x 1 5 , and equivalently x 2 x 1 0 , a quadratic equation whose solutions are x 2 2 1 5 1 x . Two interesting properties of the number are apparent from this 2 2 equation: a) the square of may be found by adding 1 to , and b) the reciprocal of may be found by subtracting 1 from . This may be expressed mathematically in the form of the equation 3. If we construct a regular (equal-sided) pentagon, then construct an isosceles triangle by extending two line segments from one vertex of the pentagon to the two opposing vertices, we obtain what is called a Golden Triangle. It is called this because the ratio of the length of either of the two equal sides to the base is : aba a 1 . If we let b = 1, then this equation is equivalent to the ab b 2 cos72 quadratic equation in the preceding paragraph. 4. is the simplest continued square root, and also the simplest continued fraction. I.e., 1 1 1 1 1 .... , and 1 1 1 . Either of these two identities may 1 1 .... be verified by using the algebraic formulae from paragraph 2 (try it). 1 5. If we construct a rectangle for which the ratio of the longer side to the shorter side is , as shown in the figure below, and then partition the rectangle into a square and another rectangle, the new rectangle will also have the property that the ratio of the 1 longer side to the shorter side is , since . 1 6. The Golden Section is also involved in the characteristics of two of the Platonic solids (convex regular polyhedrons). There are exactly five Platonic solids: 1) the regular tetradhedron, 2) the hexahedron (cube), 3) the regular octahedron, 4) the regular dodecahedron, and 5) the regular icosahedron. 1“The five Platonic solids are distinguished by the following properties: they are the only existing solids in which all the faces (of a given solid) are identical and equilateral, and each of the solids can be circumscribed by a sphere, (with all of its vertices lying on the sphere).” As a consequence of these characteristics, it is possible to construct a fair die based on each Platonic solid; for the regular tetrahedron, the die would have four faces, labeled 1, 2, 3, 4; for the cube, the die would be the ordinary die used at gaming tables; for the octahedron, the die would have eight sides, labeled 1, 2, 3, 4, 5, 6, 7, 8; for the dodecahedron, the die would have 12 sides, labeled 1, 2, 3, 4, …, 11, 12; and for the icosahedron, the die would have 20 sides, labeled 1, 2, 3, 4, …, 19, 20. Pictures of the Platonic solids are given below. 1) The tetrahedron: 2) The hexahedron: 3) The octahedron: 4) The dodecahedron: 5) The icosahedron: If a dodecahedron has an edge length of 1 unit, then its total surface area is 15 3 , and its volume is 5 5 5 3 . An icosahedron with unit edge length has volume . 6 6 2 7. The Fibonacci sequence is a sequence of natural numbers in which each number, starting at the third place in the sequence, is the sum of the two preceding numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …. This sequence occurs in nature in some surprising places. 1 “Suppose we have two glass plates made of slightly different types of glass (different light refraction properties, or „indices of refraction‟) mounted face to face. If we shine light through the plates, the light rays can (in principle) reflect internally at four reflective surfaces before emerging. More specifically, they can either pass through without reflecting at all, or they can undergo one internal reflection, or two internal reflections, three internal reflections, and so on, potentially an infinite number of internal reflections before re-emerging. All of these are paths allowed by the laws of optics. Now count the number of beams that emerge from this two-plate system. There is only one emerging beam in the case of no reflections at all. There are two emerging beams when all the possibilities for the rays to undergo precisely one internal reflection are considered, because there are two paths the ray can follow. There are three emerging beams for all the possibilities of two internal reflections, five beams for three internal reflections, eight paths if the ray is reflected four times, thirteen paths for five reflections, and so on. The number of emerging beams – 1, 2, 3, 5, 8, 13, … – form a Fibonacci sequence.” If we form the sequence of ratios of pairs of successive Fibonacci numbers, we obtain 1/1 1.000000000 2/1 2.000000000 3/2 1.500000000 5/3 1.666666666 8/5 1.600000000 13/8 1.625000000 21/13 1.615384615 34/21 1.619047619 55/34 1.617647059 89/55 1.618181818 144/89 1.617977528 233/144 1.618055556 377/233 1.618025751 610/377 1.618037135 987/610 1.618032787 Etc. The limit of the (oscillating) sequence of ratios of successive Fibonacci numbers is . 1 The Golden Ratio, by Mario Livio, 2002, Broadway Books. 8. Consider the sequence of powers of a real number x: 1, x, x2, x3, x4, …. There are exactly two such numbers that satisfy the condition that the sequence has the Fibonacci property, i.e., that each number in the sequence, starting at the third, is the sum of the two preceding numbers. The two numbers that satisfy this condition are ϕ and -1/ϕ. 9. The Fibonacci sequence (and thus the Golden Mean) also occurs in many other natural systems: The distribution of seeds in a sunflower is spiral. The seeds of the sunflower spiral outwards in both clockwise and counterclockwise directions from the center of the flower. The number of clockwise and counterclockwise spirals are two consecutive numbers in the Fibonacci sequence. Pine cones are one of the well-known examples of the Fibonacci sequence. All cones grow in spirals, starting from the base where the stalk is, and going round and round the sides until they reach the top. Another notable example is the human body. In the human body, the ratio of the length of forearm to the length of the hand is equal to 1.618, that is, Golden Ratio. Other wellknown examples for the human body are: 1. The ratio between the length and width of face 2. Ratio of the distance between the lips and where the eyebrows meet to the length of nose 3. Ratio of the length of mouth to the width of nose 4. Ratio of the distance between the shoulder line and the top of the head to the head length 5. Ratio of the distance between the navel and knee to the distance between the knee and the end of the foot 6. Ratio of the distance between the finger tip and the elbow to the distance between the wrist and the elbow