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Book: The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
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  1. Mario Livio _ The Golden Ratio: The Story of

    Even though it is almost impossible to attribute with certainty any specific mathematical achievements either to Pythagoras himself or to his followers, there is no question that they have been responsible for a mingling of mathematics, philosophy of life, and religion unparalleled in history. In this respect it is perhaps interesting to note the historical coincidence that Pythagoras was a contemporary of Buddha and Confucius.
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  2. Mario Livio _ The Golden Ratio: The Story of

    There were also many cases of feedback between physics and mathematics, where a physical phenomenon inspired a mathematical model that later proved to be the explanation of an entirely different physical phenomenon. An excellent example is provided by the phenomenon known as Brownian motion. In 1827, British botanist Robert Brown {1773-1858} observed that wen pollen particles are suspended in water, they get into a state of agitated motion. This effect was explained by Einstein in 1905 as resulting from the collisions that the colloidal particles experience with the molecules of the surrounding fluid. Each single collision has a negligible effect, because the pollen grains are millions of times more massive than the water molecules, but the persistent bombardment has a cumulative effect. Amazingly, the same model was found to apply to the motions of stars in star clusters. There the Brownian motion is produced by the cumulative effect of many stars passing by any given star, with each passage altering the motion {through gravitational interaction} by a tiny amount.
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  3. Mario Livio _ The Golden Ratio: The Story of

    Nature loves logarithmic spirals. From sunflowers, seashells, and whirlpools, to hurricanes and giant spiral galaxies, it seems that nature chose this marvelous shape as its favorite "ornament." The constant shape of the logarithmic spiral on all size scales reveals itself beautifully in nature in the shapes of minuscule fossils or unicellular organisms known as foraminifera. Although the spiral shells in this care are composite structures {and not one continuous tube}, X-ray images of the internal structure of these fossils show that the shape of the logarithmic spiral remained essentially unchanged for millions of years.
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  4. Mario Livio _ The Golden Ratio: The Story of

    Jacques was so impressed with the beauty of the curve known as a logarithmic spiral {Figure 37; the name was derived from the way in which the radius grows as we move around the curve clockwise} that he asked that this shape, and the motto he assigned to it: "Eadem mutato resurgo" {although changed, I rise again the same}, be engraved on his tombstone.The motto describes a fundamental property unique to the logarithmic spiral-it does not alter its shape as its size increases. This feature is known as self-similarity. Fascinated by this property, Jacques wrote that the logarithmic spiral "may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self.
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  5. Mario Livio _ The Golden Ratio: The Story of

    The curriculum for the education of statesmen at the time of Plato included arithmetic, geometry, solid geometry, astronomy, and music-all of which, the Pythagorean Archytas tells us, fell under the general definition of "mathematics." According to legend, when Alexander the Great asked his teacher Menaechmus {who is reputed to have discovered the curves of the ellipse, the parabola, and the hyperbola} for a shortcut to geometry, he got the reply: "O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry there is one road for all.
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  6. Mario Livio _ The Golden Ratio: The Story of

    Supporters of the "modified Platonic view" of mathematics like to point out that, over the centuries, mathematicians have produced {or "discovered"} numerous objects of pure mathematics with absolutely no application in mind. Decades later, these mathematical constructs and models were found to provide solutions to problems in physics. Penrose tilings and non-Euclidean geometries are beautiful testimonies to this process of mathematics unexpectedly feeding into physics, but there are many more."
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