 Golden Ratio The Idea Behind It Have a try yourself (use the slider): Beauty This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it? Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape. Do you think it is the "most pleasing rectangle"? Maybe you do or don't, that is up to you! Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way. The Actual Value The Golden Ratio is equal to: 1.61803398874989484820... The digits just keep on going, with no pattern. Calculating It You can calculate it yourself by starting with any number and following these steps: A) divide 1 by your number (=1/number) B) add 1 C) that is your new number, start again at A With a calculator, just keep pressing "1/x", "+", "1", "=", around and around. It is getting closer and closer! Drawing It Here is one way to draw a rectangle with the Golden Ratio: Pentagram

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Adolf Zeising Adolf Zeising (24 September 1810 – 27 April 1876) was a German psychologist, whose main interests were mathematics and philosophy. Among his discoveries, Zeising found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law, the universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.

Nature, The Golden Ratio and Fibonacci Numbers Plants can grow new cells in spirals, such as the pattern of seeds in this beautiful sunflower. The spiral happens naturally because each new cell is formed after a turn. "New cell, then turn, then another cell, then turn, ..." How Far to Turn? So, if you were a plant, how much of a turn would you have in between new cells? Imaginary time The relationship of real and imaginary time can be visualised as perpendicular axes of direction. Imaginary time is a concept derived from quantum mechanics and is essential in connecting quantum mechanics with statistical mechanics. Imaginary time can be difficult to visualize. If we imagine "regular time" as a horizontal line running between "past" in one direction and "future" in the other, then imaginary time would run perpendicular to this line as the imaginary numbers run perpendicular to the real numbers in the complex plane.

Fibonacci Sequence The Fibonacci Sequence is the series of numbers: The next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1) Similarly, the 3 is found by adding the two numbers before it (1+2), And the 5 is (2+3), and so on! Example: the next number in the sequence above is 21+34 = 55 It is that simple! Minkowski space In theoretical physics, Minkowski space is often contrasted with Euclidean space. While a Euclidean space has only spacelike dimensions, a Minkowski space also has one timelike dimension. The isometry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincaré group. History In 1905 (published 1906) it was noted by Henri Poincaré that, by taking time to be the imaginary part of the fourth spacetime coordinate √−1 ct, a Lorentz transformation can be regarded as a rotation of coordinates in a four-dimensional Euclidean space with three real coordinates representing space, and one imaginary coordinate, representing time, as the fourth dimension.

Fractal Foundation Online Course - Chapter 11 - FIBONACCI FRACTALS The Golden Ratio The Fibonacci Sequence appears in many seemingly unrelated areas. In this section we'll see how the Fibonacci Sequence generates the Golden Ratio, a relationship so special it has even been called "the Divine Proportion." Remember that the next number in the Fibonacci Sequence is just made by adding the current number in the sequence to the number before it.

Spacetime symmetries Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field equations of general relativity. Physical motivation Physical problems are often investigated and solved by noticing features which have some form of symmetry. For example, in the Schwarzschild solution, the role of spherical symmetry is important in deriving the Schwarzschild solution and deducing the physical consequences of this symmetry (such as the non-existence of gravitational radiation in a spherically pulsating star). In cosmological problems, symmetry finds a role to play in the cosmological principle which restricts the type of universes that are consistent with large-scale observations (e.g. the Friedmann-Lemaître-Robertson-Walker (FLRW) metric).

Golden ratio Line segments in the golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, or Spacetime In non-relativistic classical mechanics, the use of Euclidean space instead of spacetime is appropriate, as time is treated as universal and constant, being independent of the state of motion of an observer.[disambiguation needed] In relativistic contexts, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer and also on the strength of gravitational fields, which can slow the passage of time for an object as seen by an observer outside the field. Until the beginning of the 20th century, time was believed to be independent of motion, progressing at a fixed rate in all reference frames; however, later experiments revealed that time slows at higher speeds of the reference frame relative to another reference frame. Such slowing, called time dilation, is explained in special relativity theory.

List of fractals by Hausdorff dimension Deterministic fractals Random and natural fractals See also Notes and references Time dilation Time dilation explains why two working clocks will report different times after different accelerations. For example, ISS astronauts return from missions having aged slightly less than they would have been if they had remained on Earth, and GPS satellites work because they adjust for similar bending of spacetime to coordinate with systems on Earth. An accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer's own equally accurate clocks. This effect arises neither from technical aspects of the clocks nor from the fact that signals need time to propagate, but from the nature of spacetime itself. Overview In theory, and to make a clearer example, time dilation could affect planned meetings for astronauts with advanced technologies and greater travel speeds.

Julia set Three-dimensional slices through the (four-dimensional) Julia set of a function on the quaternions. The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Formal definition Let f(z) be a complex rational function from the plane into itself, that is, Higgs boson The Higgs boson is named after Peter Higgs, one of six physicists who, in 1964, proposed the mechanism that suggested the existence of such a particle. Although Higgs's name has come to be associated with this theory, several researchers between about 1960 and 1972 each independently developed different parts of it. In mainstream media the Higgs boson has often been called the "God particle", from a 1993 book on the topic; the nickname is strongly disliked by many physicists, including Higgs, who regard it as inappropriate sensationalism. In 2013 two of the original researchers, Peter Higgs and François Englert, were awarded the Nobel Prize in Physics for their work and prediction (Englert's co-researcher Robert Brout had died in 2011). A non-technical summary "Higgs" terminology

Koch snowflake The first seven iterations in animation Zooming into the Koch curve The Koch snowflake (also known as the Koch star and Koch island) is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire) by the Swedish mathematician Helge von Koch.

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