Encyclopedia - Golden Ratio: Encyclopedia Ii - Golden Ratio - Definition
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 par...   » Read the article

Encyclopedia - Golden Ratio: Encyclopedia Ii - Golden Ratio - Aesthetic Uses
It has been claimed that the ancient Egyptians knew the golden ratio because ratios close to the golden ratio may be found in the positio...   » Read the article

Encyclopedia - Golden Ratio: Encyclopedia Ii - Golden Ratio - Alternate Forms
The formula can be expanded recursively to obtain a continued fraction for the golden ratio: and its reciprocal: Note that the suc...   » Read the article

Encyclopedia - Golden Rule: Encyclopedia - Golden Rule
See also. Meta-Golden Rule Stone Rule Other related archivesBritish Rule, Fermi's Golden Rule, Golden Rule, Golden Rule (ethics), Go...   » Read the article

Encyclopedia - Golden Mean: Encyclopedia - Golden Mean
Golden mean. In philosophy (especially that of Aristotle), the golden mean is the felicitous middle between two extremes, one of excess...   » Read the article

Encyclopedia - Macrocosm And Microcosm: Encyclopedia - Macrocosm And Microcosm
Macrocosm and microcosm is an ancient Greek schema of seeing the same patterns reproduced in all levels of reality. It may have begun wit...   » Read the article

Encyclopedia - Pentagon: Encyclopedia - Pentagon
In geometry, a pentagon is any five-sided polygon. However, the term is commonly used to mean a regular pentagon, where all sides are equ...   » Read the article

Encyclopedia - Book Design: Encyclopedia - Book Design
Book design is the art of incorporating the content, style, format, design, and sequence of the various components of a book into a coher...   » Read the article

Encyclopedia - Phi Letter: Encyclopedia - Phi Letter
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 (dependi...   » Read the article

Encyclopedia - Sacred Geometry: Encyclopedia - Sacred Geometry
Sacred geometry is geometry that is sacred to the observer or discoverer of the geometry. This meaning is sometimes described as being th...   » Read the article

Encyclopedia - Beauty: Encyclopedia - Beauty
Beauty is the phenomenon of the experience of pleasure, through the perception of balance and proportion of stimulus. It involves the cog...   » Read the article

Encyclopedia - Golden Ratio Base: Encyclopedia Ii - Golden Ratio Base - Representing Rational Numbers As Golden Ratio Base Numbers
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...   » Read the article

Encyclopedia - Golden Ratio Base: Encyclopedia Ii - Golden Ratio Base - Writing Golden Ratio Base Numbers In Standard Form
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'...   » Read the article

Encyclopedia - Golden Ratio Base: Encyclopedia Ii - Golden Ratio Base - Representing Integers As Golden Ratio Base Numbers
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...   » Read the article

Encyclopedia - Ratio: Encyclopedia Ii - Ratio - Examples
Note the use of words such as "times", "parts", "number", etc. Because two objects are being compared using the same measure, ratios are ...   » Read the article

Encyclopedia - Parthenon: Encyclopedia Ii - Parthenon - Design And Construction
The Parthenon was built at the initiative of Pericles, the leading Athenian politician of the 5th century BC. It was built under the gene...   » Read the article

Encyclopedia - Golden Ratio Base: Encyclopedia Ii - Golden Ratio Base - Addition Subtraction And Multiplication
It is possible to adapt all the standard algorithms of base-10 arithmetic to base-φ arithmetic. There are two approaches to this: Golde...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Relation To The Golden Ratio
As was pointed out by Johannes Kepler, the ratio of consecutive Fibonacci numbers, that is: , converges to the golden ratio φ (phi) de...   » Read the article

Encyclopedia - Irrational Number: Encyclopedia Ii - Irrational Number - The Square Root Of 2
One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the n...   » Read the article

Encyclopedia - Irrational Number: Encyclopedia Ii - Irrational Number - Transcendental And Algebraic Irrationals
Almost all irrational numbers are transcendental and all transcendental numbers are irrational: the article on transcendental numbers lis...   » Read the article

Encyclopedia - Irrational Number: Encyclopedia Ii - Irrational Number - Logarithms
Perhaps the numbers most easily proved to be irrational are certain logarithms. Here is a proof by reductio ad absurdum that log23 is irr...   » Read the article

Encyclopedia - Irrational Number: Encyclopedia Ii - Irrational Number - Decimal Expansions
It is often erroneously assumed that mathematicians define "irrational number" in terms of decimal expansions, calling a number irrationa...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Matrix Form
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is or The eigenvalues of the matrix A a...   » Read the article

Encyclopedia - Parthenon: Encyclopedia Ii - Parthenon - Decorations
The Parthenon was elaborately decorated with marble sculptures both internally and externally. They survive only in part, but there are g...   » Read the article

Encyclopedia - Parthenon: Encyclopedia Ii - Parthenon - Recent Events
By the late eighteenth century, many more Europeans were visiting Athens, and the picturesque ruins of the Parthenon were much drawn and ...   » Read the article

Encyclopedia - Irrational Number: Encyclopedia Ii - Irrational Number - History
The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC. The first proof of irrational nu...   » Read the article

Encyclopedia - Parthenon: Encyclopedia Ii - Parthenon - Reconstruction
In 1975, the Greek government began a concerted effort to restore the Parthenon and other Acropolis structures. The project later attract...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Reciprocal Sum Constant
The sum of the reciprocals of the positive Fibonacci numbers converges: A079586 Convergence is easy to show with the ratio test. This va...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Computation
Computing Fibonacci numbers by computing powers of the golden mean is not very practical except for small values of n, since rounding err...   » Read the article

Encyclopedia - Parthenon: Encyclopedia Ii - Parthenon - Later History
The Parthenon survived as the most important temple of the Ancient Greek religion for close to a thousand years. It was certainly still i...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Power Series
The power series has a simple and interesting closed-form solution for x < 1/φ: This function is therefore the generating funct...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Fibonacci Strings
In analogy to its numerical counterpart, a Fibonacci string is defined by: , where + denotes the concatenation of two strings. The sequ...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Generalizations
Fibonacci number - Vector space. The term Fibonacci sequence is also applied more generally to any function g where g(n + 2) = g(n) + g...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Applications
The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two intege...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Applications
The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two intege...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Origins
The earliest known reference to Fibonacci numbers is contained in a book on meters by an Indian mathematician named Pingala called Chhand...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - The Bee Ancestry Code
Fibonacci is also stated as having described the sequence "encoded in the ancestry of a male bee." This turns out to be the Fibonacci seq...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Identities
These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using c...   » Read the article

Encyclopedia - Golden Angle: Encyclopedia Ii - Golden Angle - Derivation
The golden ratio is defined as given the conditions above. This provides an interesting relationship. Let f be the fraction of the circu...   » Read the article

Encyclopedia - Golden Rule Savings Rate: Encyclopedia Ii - Golden Rule Savings Rate - Derivation Of The Golden Rule Savings Rate
If the other parameters of the Solow model are taken as 'given' (beyond the policy-maker's control), but the savings rate can be set exog...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - J Examples
All of the following J examples generate . Fibonacci number program - Double recursion. f0a and f0b use the basic identity . f0c uses...   » Read the article

Encyclopedia - Pi Film: Encyclopedia Ii - Pi Film - Mathematics And π Explain The Ratio Problem ?
While the film's characters make several mathematical "goofs", such as saying * the ratio of a/b is the same as the ratio of a/(a + b) in...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Common Lisp
Fibonacci number program - Calculating fibonacci through Lucas' formula. (defun fib (n) (cond ((= n 0) 0) ((or (= n 1) (= n 2)...   » Read the article

Encyclopedia - 137 Number: Encyclopedia Ii - 137 Number - In Mathematics
One hundred [and] thirty-seven is the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen pr...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Haskell Examples
Fibonacci number program - Lazy infinite list. module Main where import System.Environment fibo = 1 : 1 : zipWith (+) fibo (tail fibo...   » Read the article

Encyclopedia - Beauty: Encyclopedia Ii - Beauty - Theories Of Beauty
The earliest theory of beauty can be found in the works of early Greek philosophers from the pre-Socratic period, such as Pythagoras. The...   » Read the article

Encyclopedia - Proportion Architecture: Encyclopedia Ii - Proportion Architecture - Sacred Proportions
Among the Cistercians, Gothic, Renaissance, Egyptian, Semitic, Babylonian, Arab, Greek and Roman traditions; the harmonic proportions, hu...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Postscript Example
Fibonacci number program - Iterative. 20 % how many Fibonacci numbers to print ...   » Read the article

Encyclopedia - Numeral System: Encyclopedia Ii - Numeral System - Types Of Numeral Systems
The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbol...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Ada Example
Fibonacci number program - Recursive snippet. function fib(n : integer) return integer is begin if n < 2 then return n; els...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Matlab Example
Fibonacci number program - Recursive snippet. function F = fibonacci_recursive(n) if n < 2 F = n; else F = fibonacci_recur...   » Read the article

Encyclopedia - Numeral System: Encyclopedia Ii - Numeral System - Positional Systems In Detail
Also see Positional notation. In a positional base-b numeral system (with b a positive natural number known as the radix), b basic symbol...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Php Scripting Language Example
Fibonacci number program - Contained snippet. function generate_fibonacci_sequence( $length, $method = 0 ) { # ---- ...   » Read the article

Encyclopedia - Numeral System: Encyclopedia Ii - Numeral System - Change Of Radix
A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give...   » Read the article

Encyclopedia - Numeral System: Encyclopedia Ii - Numeral System - Generalized Variable-length Integers
More general is using a notation (here written little-endian) like a0a1a2 for a0 + a1b1 + a2b1b2, etc. This is used in punycode, one aspe...   » Read the article

Encyclopedia - Numeral System: Encyclopedia Ii - Numeral System - History
Tallies carved from wood and stone have been used since prehistoric times. Stone age cultures, including ancient American Indian groups, ...   » Read the article

Encyclopedia - Numeral System: Encyclopedia Ii - Numeral System - Bases Used
The base-10 system is the one most commonly used today. It is assumed to have originated because humans have ten fingers. A base-eight sy...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Perl Examples
Fibonacci number program - One example. #! /usr/bin/perl use bigint; my ($a, $b) = (0, 1); for (;;) { print "$a\n"; ($a, $b) ...   » Read the article

Encyclopedia - Recurrence Relation: Encyclopedia Ii - Recurrence Relation - Example: Fibonacci Numbers
The Fibonacci numbers are defined using a linear recurrence relation: and has solution (letting be the golden ratio) The initial...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Python Examples
Fibonacci number program - Recursion. def fib(n): if n < 2: return n else: return fib(n - 1) + fib(n - 2) ...   » Read the article

Encyclopedia - Recurrence Relation: Encyclopedia Ii - Recurrence Relation - Example: Fibonacci Numbers
The Fibonacci numbers are defined using the linear recurrence relation whose solution is where denotes the golden ratio. There...   » Read the article

Encyclopedia - Viswanath's Constant: Encyclopedia Ii - Viswanath's Constant - Explication
The constant was discovered by Divakar Viswanath in 1999 (see references). His work uses the theory of random matrix product developed by...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - Scheme Examples
Fibonacci number program - Binary recursion snippet. (define fibo (lambda (x) (if (< x 2) x (+ (fibo (- x 1)) (fibo (...   » Read the article

Encyclopedia - Fibonacci Number Program: Encyclopedia Ii - Fibonacci Number Program - C/c++/java Example
Fibonacci number program - Recursive snippet. int fib(int n) { if (n < 2) return n; else return fib(n-1) + fib(n-2); } ...   » Read the article

Encyclopedia - Pentagram: Encyclopedia Ii - Pentagram - Geometry
A regular pentagram is the {5/2} star polygon. It is most easily drawn by drawing a regular pentagon, joining the corners with lines and ...   » Read the article

Encyclopedia - Continued Fraction: Encyclopedia Ii - Continued Fraction - Other Continued Fraction Expansions
While one cannot discern any pattern in the infinite continued fraction expansion of Ï€, this is not true for e, the base of the natural ...   » Read the article

Encyclopedia - Cinema Of The United States: Encyclopedia Ii - Cinema Of The United States - Independent Film
Studios supplement these movies with independent productions, made with small budgets and often independently of the studio corporation. ...   » Read the article

Encyclopedia - Cinema Of The United States: Encyclopedia Ii - Cinema Of The United States - Independent Film
Studios supplement these movies with independent productions, made with small budgets and often independently of the studio corporation. ...   » Read the article

Dictionary - Metrology: Spiritual - Theosophy Dictionary On Metrology
Metrology One key of the ancient symbol-language, which concealed and revealed certain aspects of the esoteric teachings. It is seen i...   » Read the article

Encyclopedia - Golden Mean Philosophy: Encyclopedia Ii - Golden Mean Philosophy - History Of The Golden Mean In Philosophy
Golden mean philosophy - Crete. The earliest representation of this idea in culture is probably in the mythological Cretan tale of Daed...   » Read the article

Dictionary - Pentagram, Pentalpha: Spiritual - Theosophy Dictionary On Pentagram, Pentalpha
Pentagram, Pentalpha The five-pointed star, or star pentagon, called pentalpha by Pythagoreans because its corners are like five (pent...   » Read the article

Encyclopedia - Individual Visit Scheme: Encyclopedia Ii - Individual Visit Scheme - Economic Impact
According to the statistics released by the Immigration Department, about 800,000 to 900,000 Mainland visitors travel to Hong Kong each m...   » Read the article

Dictionary - Temple Of Solomon: Spiritual - Theosophy Dictionary On Temple Of Solomon
Temple of Solomon The building of this temple, according to the Bible, was first projected by King David, but on command of the Lord w...   » Read the article

Encyclopedia - Silver Ratio: Encyclopedia Ii - Silver Ratio - Definition
Silver ratio - Definition as . The silver ratio (δS) is defined as the irrational number formed from the sum of 1 and the square root ...   » Read the article

Encyclopedia - Silver Ratio: Encyclopedia Ii - Silver Ratio - Properties
In diophantine approximation, the sequence of fractional parts of xn, n = 1, 2, 3, ... is shown to be equidistributed mod 1, for almost...   » Read the article

Encyclopedia - Nautilus: Encyclopedia Ii - Nautilus - Camouflage
A nautilus shell when viewed from above. The same shell viewed from underneath. The shell of the nautilus makes it as invisible as possib...   » Read the article

Encyclopedia - Padovan Sequence: Encyclopedia Ii - Padovan Sequence - Binet-like Formula
The Padovan sequence numbers can be written in terms of powers of the roots of the equation This equation has 3 roots; one real root p...   » Read the article

Encyclopedia - Perrin Number: Encyclopedia Ii - Perrin Number - Binet Like Formula
The Perrin sequence numbers can be written in terms of powers of the roots of the equation x3 − x − 1 = 0. This equation has 3 root...   » Read the article

Encyclopedia - Icosidodecahedron: Encyclopedia Ii - Icosidodecahedron - Related Polyhedra
The icosidodecahedron is a rectified dodecahedron and also a rectified icosahedron. Compare: ... See also:Icosidodecahedron, Icosidode...   » Read the article

Encyclopedia - Rhombicosidodecahedron: Encyclopedia Ii - Rhombicosidodecahedron - Geometric Relations
If you blow up an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the ...   » Read the article

Encyclopedia - Macrocosm And Microcosm: Encyclopedia Ii - Macrocosm And Microcosm - Medieval And Modern Thought
The English physician and alchemist Robert Fludd (1574-1637) expicitly based his work Utriusque Cosmi Historia (The history of the two wo...   » Read the article

Encyclopedia - Polylogarithm: Encyclopedia Ii - Polylogarithm - Particular Values
See also the "Relationship to other functions" section below. For integer values of s, we have the following explicit expressions: ...   » Read the article

Encyclopedia - Proportion Architecture: Encyclopedia Ii - Proportion Architecture - Classical Orders
The Classical Orders are largely known through the writings of Vitruvius, particularly De Archetura (The Ten Books of Architecture) and s...   » Read the article

Encyclopedia - Padovan Sequence: Encyclopedia Ii - Padovan Sequence - Generating Function
The generating function of the Padovan sequence is This can be used to prove identities involving products of the Padovan sequence wit...   » Read the article

Encyclopedia - Khinchin's Constant: Encyclopedia Ii - Khinchin's Constant - Series Expressions
Khinchin's constant may be expressed as a rational zeta series in the form or, by peeling off terms in the series, where N is an in...   » Read the article

Encyclopedia - Khinchin's Constant: Encyclopedia Ii - Khinchin's Constant - Hölder Means
The Khinchin constant can be viewed as the first in a series of the Hölder means of the terms of continued fractions. Given an arbitrary...   » Read the article

Encyclopedia - Generating Function: Encyclopedia Ii - Generating Function - Examples
Generating functions for the sequence of square numbers an = n2 are: Generating function - Ordinary generating function. Generating...   » Read the article

Encyclopedia - Generating Function: Encyclopedia Ii - Generating Function - Another Example
Generating functions can be created by extending simpler generating functions. For example, starting with and replacing x with 2x, we ...   » Read the article

Encyclopedia - Truncated Icosahedron: Encyclopedia Ii - Truncated Icosahedron - Uses
A football (soccer ball) comprises the same pattern of regular pentagons and regular hexagons, but is more spherical due to the pressure ...   » Read the article

Encyclopedia - Polylogarithm: Encyclopedia Ii - Polylogarithm - The Dilogarithm
The dilogarithm is just the polylogarithm with s = 2. An alternate integral expression for the dilogarithm is: The Abel identity for t...   » Read the article

Encyclopedia - 137 Number: Encyclopedia Ii - 137 Number - In Physics
The numerical value of the fine structure constant α, a dimensionless physical constant, is approximately 1/137. The physicist Arthur Ed...   » Read the article

Encyclopedia - E Mathematical Constant: Encyclopedia Ii - E Mathematical Constant - Properties
The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivati...   » Read the article

Encyclopedia - Facial Symmetry: Encyclopedia Ii - Facial Symmetry - Health And Physical Attractiveness
It has been proposed that facial bilateral symmetry is a universal determinant of health and therefore of beauty. Facial asymmetries and ...   » Read the article

Encyclopedia - Generating Function: Encyclopedia Ii - Generating Function - More Detailed Example — Fibonacci Numbers
Consider the problem of finding a closed formula for the Fibonacci numbers Fn defined by F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2 for n â...   » Read the article

Encyclopedia - Trigonometric Identity: Encyclopedia Ii - Trigonometric Identity - Identities Without Variables
Richard Feynman is reputed to have learned as a boy, and always remembered, the following curious identity: However, this is a special...   » Read the article

Encyclopedia - Golden Ratio: Encyclopedia Ii - Golden Ratio - Alternate Forms
The formula can be expanded recursively to obtain a continued fraction for the golden ratio: and its reciprocal: Note that the suc...   » Read the article

Encyclopedia - Golden Ratio: Encyclopedia Ii - Golden Ratio - Mathematical Uses
The number φ turns up frequently in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regula...   » Read the article

Encyclopedia - Golden Ratio: Encyclopedia Ii - Golden Ratio - Decimal Expansion
(sequence A001622 in OEIS) 1.6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244 9707207204 189391...   » Read the article

Encyclopedia - Golden Ratio: Encyclopedia Ii - Golden Ratio - A Startlingly Quick Proof Of Irrationality
Recall that we denoted the "larger part" by a and the "smaller part" by b, and concluded that This gives a startlingly quick proof tha...   » Read the article

Article - Popular Pages Sitemap V - G
This is a sitemap for Popular Pages V - G . Click on a link and you will find multiple definitions and articles related to the word. &...   » Read the article