_The Golden Ratio: The Story of Phi, the World's Most Astonishing Number_ by Mario Livio
NY: Random House, 2002
ISBN 978-0-7679-0815-3
(6) Roger Herz-Fischler, _A Mathematical History of the Golden Number_
(26) Pythagoras emphasized the importance of learning above all other activities, because, in his words, "most men and women, by birth or nature, lack the means to advance in wealth and power, but all have the ability to advance in knowledge."
(67) The key figure and driving force behind the geometrical theorems concerning the Golden Ratio was probably Theaetetus (ca. 417 BC - ca. 369 BC), who according to the Byzantine collection _Suidas_ "was the first to construct the five so-called solids."
NB: Which actually go back to at least the Neolithic
(79) In other words, in a regular pentagon, the ratio of the diagonal to the side is equal to ø. This fact illustrates that the ability to construct a line divided in a Golden Ratio provides at the same time a simple means of constructing the regular pentagon. The construction of the pentagon was the main reason for the Greek interest in the Golden Ratio. The triangle in the middle of Figure 25a, with a ratio of side to base of ø, is known as a Golden Triangle; the two triangles on the sides, with a ratio of side to base of 1/ø, are sometimes called Golden Gnomons.
(81) The Golden Ratio has the unique properties that we produce its square by simply adding the number 1 and its reciprocal by subtracting the number 1. Incidentally, the negative solution of the equation x sub 2=(1-√5/2) is equal precisely to the negative of 1/ø.
(85) The Golden Rectangle is the _only_ rectangle with the property that cutting a square from it produces a similar rectangle.
NB: Successive cutting results in the ability to trace a logarithmic spiral converging on one point, the so-called Eye of God
(101) As we go farther and farther down the Fibonacci sequence, the ratio of two successive Fibonacci numbers oscillates about (being alternatively greater and smaller) but comes closer and closer to the Golden Ratio.
(111-112) One of the discoveries of the Bravais brothers in 1837 was that new leaves advance roughly by the same angle around the circle and that this angle (known as the divergence angle) is usually close to 137.5 degrees. Are you shocked to hear that this value is determined by the the Golden Ratio? The angle that divides a complete turn in a Golden Ration is 360º/ø=222.5 degrees. Since this is more than half a circle (180 degrees), we should measure it going in the opposite direction around the circle. In other words, we should subtract 222.5 from 360, giving us the observed angle of 137.5 degrees (sometimes called the Golden Angle).
(126) Three of Piero's [della Francesca] mathematical works have survived: _De Prospective pingendi (On perspective in painting), _Libellus de quinque Corporibus Regularibus (Short book on the five regular solids), and _Trattato d'Abaco (Treatise on the abacus).
(140) Dürer's polyhedron in Melancolia I: http://en.wikipedia.org/wiki/Melencolia_I
(146-147) Kepler: The Earth's sphere is the measure of all other orbits. Circumscribe a dodecahedron around it. The sphere surrounding it will be that of Mars. Circumscribe a tetrahedron around Mars. The sphere surrounding it will be that of Jupiter. Circumscribe a cube around Jupiter. The surrounding sphere will be that of Saturn. Now, inscribe an icosahedron inside the orbit of the Earth. The sphere inscribed in it will be that of Venus. Inscribe an octahedron inside Venus. The sphere inscribed in it will be that of Mercury. There you have the basis for the number of the planets.
(155) Kepler's songs of the planets: http://www.guitar-theory-in-depth.com/musical-scales.html and in modern notation athttp://soundpossibilities.wordpress.com/2011/08/07/education-music-physics-science-math-and-so-on-ad-infitum/
(171) Another art theorist who had great interest in the Golden Ratio at the beginning of the twentieth century was the American Jay Hambidge (1867-1924). In a series of articles and books, Hambidge defined two types of symmetry in classical and modern art. One, which he called "static symmetry," was based on regular figures like the square and equilateral triangle, and was supposed to produce lifeless art. The other, which he dubbed "dynamic symmetry," had the Golden Ratio and the logarithmic spiral in leading roles. Hambidge's basic thesis was that the use of "dynamic symmetry" in design leads to vibrant and moving art. Few today take his ideas seriously.
(193) [Joseph] Schillinger was a great believer in the mathematical basis of music, and, in particular, he developed a System of Musical Composition in which successive notes in the melody followed Fibonacci intervals when counted in units of half-steps.
(205) One of the most startling properties of any Penrose kite-dart tiling design is that the number of kites is about 1.618 times the number of darts. That is, if we denote by Nkites the number of kites and Ndarts the number of darts, then Nkites/Ndarts approaches ø the larger the area we take in....
Another pair of Penrose tiles that can fill the entire plane (nonperiodically) is composed of two diamonds (rhombi), one fat (obtuse) and one thin (acute). As in the kite-dart pair, each of the rhombi is composed of two Golden Triangles or Golden Gnomons, and special matching rules have to be obeyed (in this case described by decorating the appropriate sides or angles of the rhombi) to obtain a plane-filling pattern. Again, in large areas there are 1.618 times more fat rhombi than thin ones, Nfat/Nthin=ø.
(206) Penrose's work on tiling has been expanded to three dimensions. In the same way that two-dimensional tiles can be used to fill the plane, three-dimensional "blocks" can be used to fill up space. IN 1976, mathematician Robert Ammann discovered a pair of "cubes", one "squashed" and one "stretched," known as rhombohedra, that can fill up space with no gaps. Ammann was further abel to show that given a set of face-matching rules, the pattern that emerges is nonperiodic and has the symmetry properties of the icosahedron; this is the equivalent of fivefold symmetry in three dimensions, since five symmetric edges meet at every vertex). Not surprisingly, the two rhombohedra are Golden Rhombohedra - their faces actually are identical to the rhombi of the Penrose tiles.
(212) Consider the following simple algorithm for the creation of a sequence known as the Golden Sequence. Start with the number 1, and then replace 1 by 10. From then on, replace each 1 by 10 and each 0 by 1.
1
10
101
10110
10110101
1011010110110
101101011011010110101
NB: What is this in binary?
181 181 21
181 22
181
22
5
2
1
(216) These musings have turned into the by now-famous question: "How long is the coast of Britain?" Mandelbrot's surprising answer is that the length of the coastline depends on the length of your ruler.
NB: Zeno's paradox
(219) Clearly, for many systems (eg, a drainage system or a blood circulatory system), we may be interested in finding out at what reduction factor precisely do the branches just touch and start to overlap. Surprisingly (or maybe not, by now), this happens for a reduction factor that is equal precisely to _one over the Golden Ratio_, 1/ø=0.618... This is known as a Golden Tree, and its fractal dimension turns out to be about 1.4404.
NB: Constructal theory
(224) Elliot's basic idea was relatively simple. He claimed that market variations can be characterized by a fundamental pattern consisting of five waves during an upward ("optimistic") trend and three waves during a downward ("pessimistic") trend.
(233) Newcomb... came up with an actual formula that was supposed to give the probability that a random number begins with a particular digit. That formula gives for 1 a probability of 30%; for 2, about 17.6%; for 3, about 12.5%; for 4, about 9.7%; for 5, about 8 %; for 6, about 6.7%; for 7, about 5.8%; for 8, about 5%; and for 9, about 4.6%.
NY: Random House, 2002
ISBN 978-0-7679-0815-3
(6) Roger Herz-Fischler, _A Mathematical History of the Golden Number_
(26) Pythagoras emphasized the importance of learning above all other activities, because, in his words, "most men and women, by birth or nature, lack the means to advance in wealth and power, but all have the ability to advance in knowledge."
(67) The key figure and driving force behind the geometrical theorems concerning the Golden Ratio was probably Theaetetus (ca. 417 BC - ca. 369 BC), who according to the Byzantine collection _Suidas_ "was the first to construct the five so-called solids."
NB: Which actually go back to at least the Neolithic
(79) In other words, in a regular pentagon, the ratio of the diagonal to the side is equal to ø. This fact illustrates that the ability to construct a line divided in a Golden Ratio provides at the same time a simple means of constructing the regular pentagon. The construction of the pentagon was the main reason for the Greek interest in the Golden Ratio. The triangle in the middle of Figure 25a, with a ratio of side to base of ø, is known as a Golden Triangle; the two triangles on the sides, with a ratio of side to base of 1/ø, are sometimes called Golden Gnomons.
(81) The Golden Ratio has the unique properties that we produce its square by simply adding the number 1 and its reciprocal by subtracting the number 1. Incidentally, the negative solution of the equation x sub 2=(1-√5/2) is equal precisely to the negative of 1/ø.
(85) The Golden Rectangle is the _only_ rectangle with the property that cutting a square from it produces a similar rectangle.
NB: Successive cutting results in the ability to trace a logarithmic spiral converging on one point, the so-called Eye of God
(101) As we go farther and farther down the Fibonacci sequence, the ratio of two successive Fibonacci numbers oscillates about (being alternatively greater and smaller) but comes closer and closer to the Golden Ratio.
(111-112) One of the discoveries of the Bravais brothers in 1837 was that new leaves advance roughly by the same angle around the circle and that this angle (known as the divergence angle) is usually close to 137.5 degrees. Are you shocked to hear that this value is determined by the the Golden Ratio? The angle that divides a complete turn in a Golden Ration is 360º/ø=222.5 degrees. Since this is more than half a circle (180 degrees), we should measure it going in the opposite direction around the circle. In other words, we should subtract 222.5 from 360, giving us the observed angle of 137.5 degrees (sometimes called the Golden Angle).
(126) Three of Piero's [della Francesca] mathematical works have survived: _De Prospective pingendi (On perspective in painting), _Libellus de quinque Corporibus Regularibus (Short book on the five regular solids), and _Trattato d'Abaco (Treatise on the abacus).
(140) Dürer's polyhedron in Melancolia I: http://en.wikipedia.org/wiki/Melencolia_I
(146-147) Kepler: The Earth's sphere is the measure of all other orbits. Circumscribe a dodecahedron around it. The sphere surrounding it will be that of Mars. Circumscribe a tetrahedron around Mars. The sphere surrounding it will be that of Jupiter. Circumscribe a cube around Jupiter. The surrounding sphere will be that of Saturn. Now, inscribe an icosahedron inside the orbit of the Earth. The sphere inscribed in it will be that of Venus. Inscribe an octahedron inside Venus. The sphere inscribed in it will be that of Mercury. There you have the basis for the number of the planets.
(155) Kepler's songs of the planets: http://www.guitar-theory-in-depth.com/musical-scales.html and in modern notation athttp://soundpossibilities.wordpress.com/2011/08/07/education-music-physics-science-math-and-so-on-ad-infitum/
(171) Another art theorist who had great interest in the Golden Ratio at the beginning of the twentieth century was the American Jay Hambidge (1867-1924). In a series of articles and books, Hambidge defined two types of symmetry in classical and modern art. One, which he called "static symmetry," was based on regular figures like the square and equilateral triangle, and was supposed to produce lifeless art. The other, which he dubbed "dynamic symmetry," had the Golden Ratio and the logarithmic spiral in leading roles. Hambidge's basic thesis was that the use of "dynamic symmetry" in design leads to vibrant and moving art. Few today take his ideas seriously.
(193) [Joseph] Schillinger was a great believer in the mathematical basis of music, and, in particular, he developed a System of Musical Composition in which successive notes in the melody followed Fibonacci intervals when counted in units of half-steps.
(205) One of the most startling properties of any Penrose kite-dart tiling design is that the number of kites is about 1.618 times the number of darts. That is, if we denote by Nkites the number of kites and Ndarts the number of darts, then Nkites/Ndarts approaches ø the larger the area we take in....
Another pair of Penrose tiles that can fill the entire plane (nonperiodically) is composed of two diamonds (rhombi), one fat (obtuse) and one thin (acute). As in the kite-dart pair, each of the rhombi is composed of two Golden Triangles or Golden Gnomons, and special matching rules have to be obeyed (in this case described by decorating the appropriate sides or angles of the rhombi) to obtain a plane-filling pattern. Again, in large areas there are 1.618 times more fat rhombi than thin ones, Nfat/Nthin=ø.
(206) Penrose's work on tiling has been expanded to three dimensions. In the same way that two-dimensional tiles can be used to fill the plane, three-dimensional "blocks" can be used to fill up space. IN 1976, mathematician Robert Ammann discovered a pair of "cubes", one "squashed" and one "stretched," known as rhombohedra, that can fill up space with no gaps. Ammann was further abel to show that given a set of face-matching rules, the pattern that emerges is nonperiodic and has the symmetry properties of the icosahedron; this is the equivalent of fivefold symmetry in three dimensions, since five symmetric edges meet at every vertex). Not surprisingly, the two rhombohedra are Golden Rhombohedra - their faces actually are identical to the rhombi of the Penrose tiles.
(212) Consider the following simple algorithm for the creation of a sequence known as the Golden Sequence. Start with the number 1, and then replace 1 by 10. From then on, replace each 1 by 10 and each 0 by 1.
1
10
101
10110
10110101
1011010110110
101101011011010110101
NB: What is this in binary?
181 181 21
181 22
181
22
5
2
1
(216) These musings have turned into the by now-famous question: "How long is the coast of Britain?" Mandelbrot's surprising answer is that the length of the coastline depends on the length of your ruler.
NB: Zeno's paradox
(219) Clearly, for many systems (eg, a drainage system or a blood circulatory system), we may be interested in finding out at what reduction factor precisely do the branches just touch and start to overlap. Surprisingly (or maybe not, by now), this happens for a reduction factor that is equal precisely to _one over the Golden Ratio_, 1/ø=0.618... This is known as a Golden Tree, and its fractal dimension turns out to be about 1.4404.
NB: Constructal theory
(224) Elliot's basic idea was relatively simple. He claimed that market variations can be characterized by a fundamental pattern consisting of five waves during an upward ("optimistic") trend and three waves during a downward ("pessimistic") trend.
(233) Newcomb... came up with an actual formula that was supposed to give the probability that a random number begins with a particular digit. That formula gives for 1 a probability of 30%; for 2, about 17.6%; for 3, about 12.5%; for 4, about 9.7%; for 5, about 8 %; for 6, about 6.7%; for 7, about 5.8%; for 8, about 5%; and for 9, about 4.6%.
