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 After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are se ...

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Golden ratio, Golden ratio - Definition, Golden ratio - History, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Alternate forms, Golden ratio - Mathematical uses, Golden ratio - Aesthetic uses, Golden ratio - Decimal expansion

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Golden ratio, Golden ratio - Definition, Golden ratio - History, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Alternate forms, Golden ratio - Mathematical uses, Golden ratio - Aesthetic uses, Golden ratio - Decimal expansion

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The formula can be expanded recursively to obtain a continued fraction for the golden ratio: and its reciprocal: Note that the successive convergents of these continued fractions are ratios of Fibonacci numbers. 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 ...

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Golden ratio, Golden ratio - Definition, Golden ratio - History, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Alternate forms, Golden ratio - Mathematical uses, Golden ratio - Aesthetic uses, Golden ratio - Decimal expansion

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See also. Meta-Golden Rule Stone Rule Other related archivesBritish Rule, Fermi's Golden Rule, Golden Rule, Golden Rule (ethics), Golden Rule savings rate, Golden ratio, HM Treasury, Meta-Golden Rule, Solow growth model, UK, economics, ethic of reciprocity, ethical, ethics, fiscal policy, philosophy, quantum mechanics, religion

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Golden mean. In philosophy (especially that of Aristotle), the golden mean is the felicitous middle between two extremes, one of excess and the other of deficiency; for this meaning, see golden mean (philosophy). Golden mean is also sometimes used as a synonym for the golden ratio (also called golden section, golden number, or divine proportion), the irrational number approximately 1.61803..., which has applications in several fields inc

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Macrocosm and microcosm is an ancient Greek schema of seeing the same patterns reproduced in all levels of reality. It may have begun with Democritus in the fifth century B.C. or with Pythagoras and is a philosophical conception that runs through Socrates, and Plato and through to the Renaissance. With Pythagoras, the discovery of the golden ratio and its philosophical conception called the Golden mean, the Greeks saw that this golden ratio is repeated in all parts of the ordered universe both large and small. The Greeks were very con ...

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• Macrocosm and microcosm - Ancient thought
• Macrocosm and microcosm - Medieval and modern thought
• Macrocosm and microcosm - Bibliography

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In geometry, a pentagon is any five-sided polygon. However, the term is commonly used to mean a regular pentagon, where all sides are equal and all angles are equal (to 108°). Its Schläfli symbol is {5}. The area of a regular pentagon with side length a is given by A pentagram can be formed from a regular pentagon either by extending its sides or by drawing its diagonals. The two differ by a linear scale factor φ + 1, or conversely 2 - φ, where φ = (1+√5)/2, the golden ratio. The resulting figure contains also various other lengths ...

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• Pentagon - Constructing a pentagon
• Pentagon - Some relevant trigonometric values

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Book design is the art of incorporating the content, style, format, design, and sequence of the various components of a book into a coherent whole. In the words of Jan Tschichold, book design "[...] though largely forgotten today, methods and rules upon which it is impossible to improve have been developed over centuries. To produce perfect books these rules have to be brought back to life and applied."^{» Book design: Encyclopedia - Book design}

Phi (upper case Φ, lower case φ or φ) is the 21st letter of the Greek alphabet. It is pronounced fee by modern Greeks, or fie (depending on context and, often, personal inclination), representing the phoneme 'f'. In Ancient Greek it represented a strongly aspirated 'p'. In the system of Greek numerals it has a value of 500. The lower-case letter φ (or often its variant, » Phi letter: Encyclopedia - Phi letter

Sacred geometry is geometry that is sacred to the observer or discoverer of the geometry. This meaning is sometimes described as being the language of the God of the religion of the people who discovered or used it. Sacred geometry can be described as attributing a religious or cultural value to the graphical representation of the mathematical relationships and the design of the man-made objects that sy ...

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• Sacred geometry - External link

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Beauty is the phenomenon of the experience of pleasure, through the perception of balance and proportion of stimulus. It involves the cognition of a balanced form and structure that elicits attraction and appeal towards a person, animal, inanimate object, scene, music, or idea. The opposite of beauty is ugliness, the experience of displeasure at some stimulus. Beauty - Beauty and aesthetics. Understanding the nature and meaning of beauty is one of the key themes in the philosophical discipline known as aest ...

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• Beauty - Beauty and aesthetics
• Beauty - Theories of beauty
• Beauty - Mathematical Beauty
• Beauty - Effects of beauty in human society

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Every rational number can be represented as a recurring base-φ expansion, as can any element of the field Q[√5] = Q + √5Q, the field generated by the rational numbers and √5. Conversely any recurring (or terminating) base-φ expansion is an element of Q[√5]. Some examples (with spaces added for emphasis): 1/2 = 0.010 010 010 ... φ 1/3 = 0.00101000 00101000 00101000... φ √5 = 10.100000φ 2+(1/13)√5 = 10.010 10001000101010 ...

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Golden ratio base, Golden ratio base - Examples, Golden ratio base - Writing golden ratio base numbers in standard form, Golden ratio base - Representing integers as golden ratio base numbers, Golden ratio base - Non-uniqueness, Golden ratio base - Representing rational numbers as golden ratio base numbers, Golden ratio base - Addition subtraction and multiplication, Golden ratio base - Calculate then convert to standard form, Golden ratio base - Avoid digits other than 0 and 1, Golden ratio base - Division, Golden ratio base - Fibonacci coding: a close relation

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211.01φ is not a standard base-φ numeral, since it contains a "11" and a "2", which isn't a "0" or "1", and contains a 1=-1, which isn't a "0" or "1" either. To "standardize" a numeral, we can use the following substitutions: 011φ = 100φ, 0200φ = 1001φ and 010φ = 101φ. We can apply the substitutions in any order we like, as the result is the ...

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Golden ratio base, Golden ratio base - Examples, Golden ratio base - Writing golden ratio base numbers in standard form, Golden ratio base - Representing integers as golden ratio base numbers, Golden ratio base - Non-uniqueness, Golden ratio base - Representing rational numbers as golden ratio base numbers, Golden ratio base - Addition subtraction and multiplication, Golden ratio base - Calculate then convert to standard form, Golden ratio base - Avoid digits other than 0 and 1, Golden ratio base - Division, Golden ratio base - Fibonacci coding: a close relation

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We can either consider our integer to be the (only) digit of a nonstandard base-φ numeral, and standardize it, or do the following: 1×1 = 1, φ × φ = 1 + φ and 1/φ = -1 + φ. Therefore, we can compute (a + bφ) + (c + dφ) = ((a + c) + (b + d)φ), (a + bφ) - (c + dφ) = ((a - c) + (b - d)φ) and (a + bφ) × (c + dφ) = ((a × c + b × d) + (a × d + b × c + b × d)φ). So, using integer values only, we can add, subtract and multiply numbers of the form (a + bφ), and even r ...

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Golden ratio base, Golden ratio base - Examples, Golden ratio base - Writing golden ratio base numbers in standard form, Golden ratio base - Representing integers as golden ratio base numbers, Golden ratio base - Non-uniqueness, Golden ratio base - Representing rational numbers as golden ratio base numbers, Golden ratio base - Addition subtraction and multiplication, Golden ratio base - Calculate then convert to standard form, Golden ratio base - Avoid digits other than 0 and 1, Golden ratio base - Division, Golden ratio base - Fibonacci coding: a close relation

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Note the use of words such as "times", "parts", "number", etc. Because two objects are being compared using the same measure, ratios are unitless; the units cancel out of the ratio. For example, the ingredients in a recipe that required 500 grams and 300 grams of each, would be in the ratio of 5:3, with no units. Note also the difference between ratios and vulgar fractions. For example, if there are three raspberry candies and five blackcurrant candies, then the ratio of raspberry candies to blackcurrant candies is 3:5. This indicates ...

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Ratio, Ratio - Examples

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The Parthenon was built at the initiative of Pericles, the leading Athenian politician of the 5th century BC. It was built under the general supervision of the sculptor Phidias, who also had charge of the sculptural decoration. The architects were Iktinos and Kallikrates. Construction began in 447 BC, and the building was substantially completed by 438 BC, but work on the decorations continued until at least 433 BC. Some of the financial accounts for the Parthenon survive and show that the largest single expense was transporting the stone fr ...

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Parthenon, Parthenon - Design and construction, Parthenon - Decorations, Parthenon - Later history, Parthenon - Recent events, Parthenon - Reconstruction, Parthenon - Treasury or temple?, Parthenon - Sources and further reading

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It is possible to adapt all the standard algorithms of base-10 arithmetic to base-φ arithmetic. There are two approaches to this: Golden ratio base - Calculate then convert to standard form. For addition of two base-φ numbers, add each pair of digits, without carry, and then convert the numeral to standard form. For subtraction, subtract each pair of digits without borrow (borrow is a negative amount of carry), and then convert the numeral to standard form. For multiplication, multiply in the typi ...

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Golden ratio base, Golden ratio base - Examples, Golden ratio base - Writing golden ratio base numbers in standard form, Golden ratio base - Representing integers as golden ratio base numbers, Golden ratio base - Non-uniqueness, Golden ratio base - Representing rational numbers as golden ratio base numbers, Golden ratio base - Addition subtraction and multiplication, Golden ratio base - Calculate then convert to standard form, Golden ratio base - Avoid digits other than 0 and 1, Golden ratio base - Division, Golden ratio base - Fibonacci coding: a close relation

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As was pointed out by Johannes Kepler, the ratio of consecutive Fibonacci numbers, that is: , converges to the golden ratio φ (phi) defined as the positive solution of the equation: or equivalently Proof: Like every series defined by linear recursion, the Fibonacci numbers have a closed-form solution. It has become known as Binet's formula: ...

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Fibonacci number, Fibonacci number - Origins, Fibonacci number - The bee ancestry code, Fibonacci number - Relation to the golden ratio, Fibonacci number - Matrix form, Fibonacci number - Computation, Fibonacci number - Applications, Fibonacci number - Fibonacci numbers in nature, Fibonacci number - Identities, Fibonacci number - Common factors, Fibonacci number - Power series, Fibonacci number - Reciprocal sum constant, Fibonacci number - Generalizations, Fibonacci number - Vector space, Fibonacci number - Similar integer sequences, Fibonacci number - Other generalizations, Fibonacci number - Fibonacci primes, Fibonacci number - Fibonacci strings, Fibonacci number - Fiction, Fibonacci number - Journals

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One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the negation and showing that that leads to a contradiction, which means that the proposition must be true. Assume that √2 is a rational number. This would mean that there exist integers a and b such that a / b = √2. Then √2< ...

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Irrational number, Irrational number - History, Irrational number - The square root of 2, Irrational number - Another proof, Irrational number - The golden ratio, Irrational number - Transcendental and algebraic irrationals, Irrational number - Logarithms, Irrational number - Decimal expansions, Irrational number - Open questions, Irrational number - The set of all irrationals, Irrational number - Another irrational number

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Almost all irrational numbers are transcendental and all transcendental numbers are irrational: the article on transcendental numbers lists several examples. er and πr are irrational if r ≠ 0 is rational; eπ is also irrational. Another way to construct irrational numbers is as irrational algebraic numbers, i.e. as zeros of polynomials with integer coefficients: start with a polynomial equation p(x) = an xn + an-1 xn−1 ...

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Irrational number, Irrational number - History, Irrational number - The square root of 2, Irrational number - Another proof, Irrational number - The golden ratio, Irrational number - Transcendental and algebraic irrationals, Irrational number - Logarithms, Irrational number - Decimal expansions, Irrational number - Open questions, Irrational number - The set of all irrationals, Irrational number - Another irrational number

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