title: Taylor Hawkins

text: Hawkins said that his drumming was chiefly influenced by classic rock drummers, including Phil Collins of Genesis, Stewart Copeland of The Police whom he idolized and eventually befriended, Budgie of Siouxsie and the Banshees, Roger Taylor of Queen, Stephen Perkins of Jane's Addiction, Ringo Starr of The Beatles, Jim Gordon of Derek and the Dominos, Neil Peart of Rush, and Alex Van Halen of Van Halen.


title: Taylor Hawkins

text: Hawkins was once guest editor of "Rhythm" and interviewed Collins, Copeland, Taylor, and Perkins for the issue.


title: Taylor Hawkins

text: It was thanks to Rush that I got into Genesis. After listening to Neil, I bought the "Seconds Out" live album, which was released in 1977. It's just amazing. Not only hearing Phil Collins playing the drums, but also singing. He gets a really bad rap from some people for 'daring' to take over after Peter Gabriel quit, but you just hear the way he sounds here. The beauty of this album is that it's got songs from ..."Trick of the Tail", the first album Genesis did without Gabriel. Collins is an incredible drummer. Anyone who wants to be good on the drums should check him out – the man is a master.


title: Taylor Hawkins

text: Hawkins's first two major inspirations were Roger Taylor and Stewart Copeland. He reported that listening to these two drummers' different styles showed him a wide spectrum of drumming style. He also mentioned that he would play along with songs on the radio or records, like Queen's "News of the World", to enhance his skills when he was young.


title: Taylor Hawkins

text: Hawkins and his wife, Alison, married in 2005. Together they have three children. They lived in Hidden Hills, California, after moving from Topanga Canyon in 2012.


title: Taylor Hawkins

text: Hawkins overdosed on heroin in August 2001, which left him in a coma for two weeks. Hawkins' bandmate and best friend, Dave Grohl, was beside his hospital bed in London for two weeks until he woke up. Grohl said he was ready to quit music while Hawkins was in the hospital. He also revealed in the 2011 documentary "", that he wrote the song "On the Mend" from the band's 2005's album "In Your Honor", about Hawkins while he was in a coma. Speaking to Beats 1 host Matt Wilkinson in 2018 about the incident, Hawkins said: "I was partying a lot. I wasn't a junkie, per se, but I was partying. There was a year where the partying just got a little too heavy. Thank God on some level this guy gave me the wrong line with the wrong thing one night and I woke up going, 'What the fuck happened?' That was a real changing point for me." In the same interview, Hawkins also said he was sober.


title: Taylor Hawkins

text: Hawkins suffered from stage fright. Speaking about his health in a June 2021 interview with "Rolling Stone" magazine, Hawkins said; "I'm healthy. I'm good... I get sinus infections really bad. And I just found out from my doctor, got all my blood tests and my heart everything checked and he goes, 'Dude, you're in amazing shape. Your heart's big, because you exercise a lot. It's like a runner's heart.' And that's fine. The only thing is, he said, 'I think you have sleep apnea.' And my wife's always saying you snore and you fucking make weird noises while you're sleeping and stuff."


title: Taylor Hawkins

text: On March 25, 2022, emergency services were called to the Four Seasons Casa Medina hotel in Bogotá, Colombia, where Hawkins was suffering from chest pain in his hotel room. Health personnel arrived and found Hawkins unresponsive; they performed CPR, but he was declared dead at the scene, at the age of 50. No cause of death was given.


title: Taylor Hawkins

text: The following day, Colombian authorities announced that a preliminary urine toxicology test indicated that Hawkins had ten substances in his system at the time of his death, including opioids, benzodiazepines, tricyclic antidepressants, and THC (a psychoactive drug found in cannabis). Colombia's National Institute of Forensic Medicine stated that they would "continue the medical studies to achieve total clarification of the events that led to the death of Taylor Hawkins", and the attorney general's office would continue to investigate his cause of death in a "timely manner".


title: Taylor Hawkins

text: A May 2022 article from Andy Green and Kory Grow in "Rolling Stone" made the case that Hawkins was suffering from extreme exhaustion in the period leading up to his death, and allegedly asked that no more dates be added to the scheduled tour. However, two of the interviewees, Chad Smith and Matt Cameron, have since disavowed the article, stating that they were quoted out of context and that their comments were misrepresented. Noting that official autopsy results had not, at that time, been made public, the Green-Grow article went on to state that "his friends believe[d] he wasn't using hard drugs recreationally at the time of his death" and that the post-mortem forensics performed indicated that the mass of Hawkins' heart was "about double... normal size", raising the possibility that "it could have collapsed" apart from any contribution of drugs in his system.


title: Taylor Hawkins

text: The Foo Fighters announced the death of Hawkins in a Twitter statement on March 25, writing that "His musical spirit and infectious laughter will live on with all of us forever." On March 29, the band cancelled all their upcoming shows.


title: Taylor Hawkins

text: On the night of his death, Hawkins was scheduled to perform with the Foo Fighters at the Estéreo Picnic Festival in Bogotá as part of their ongoing South American tour. The festival stage was turned into a candlelight vigil for Hawkins.


title: Taylor Hawkins

text: Miley Cyrus, who had worked with Hawkins on her 2020 album, "Plastic Hearts", dedicated her Brazil's Lollapalooza performance to him. Liam Gallagher, a longtime friend of Hawkins, dedicated "Live Forever" to him at his London concert for Teenage Cancer Trust the day after his death, describing him as a "brother" and "a fucking joy". Elton John, who had worked with Hawkins on his album, "The Lockdown Sessions", dedicated a performance of "Don't Let the Sun Go Down on Me" to him. At his investiture ceremony at Windsor Castle, Queen drummer Roger Taylor dedicated his OBE to Hawkins, describing him as "sunshine in human form" and adding that he had been a mentor to his own son Rufus. At the New Orleans Jazz & Heritage Festival the Red Hot Chili Peppers drummer, Chad Smith, used a drum kit with "Taylor" written on it. At the end of their set, Smith made a speech dedicating their performance to Hawkins. On April 3, 2022, the 64th Grammy Awards paid tribute to Hawkins with a brief montage of blended interviews and backstage home videos with live footage, ending in an audience singalong of the Foo Fighters' 1998 single "My Hero". Billie Eilish paid tribute during her Grammys performance wearing an oversized T-shirt with a picture of Hawkins' face on the front and his last name in capital letters printed on the back.


title: Taylor Hawkins

text: On June 8, 2022, the Foo Fighters and the family of Hawkins announced two shows in tribute to Hawkins. The first show, played at Wembley Stadium in London, took place on September 3, 2022. The Foo Fighters were joined by Travis Barker, Nandi Bushell, Martin Chambers, Chris Chaney, Dave Chappelle, Stewart Copeland, Josh Freese, Liam Gallagher, Omar Hakim, Justin Hawkins, Joshua Homme, Chrissie Hynde, The James Gang, Alain Johannes, Brian Johnson, John Paul Jones, Greg Kurstin, Geddy Lee, Alex Lifeson, Brian May, Paul McCartney, Chris Rock, Nile Rodgers, Mark Ronson, Luke Spiller, Supergrass, Roger Taylor, Rufus Taylor, Lars Ulrich, and Wolfgang Van Halen, plus members of Hawkins's cover band, Chevy Metal. His son Oliver Shane Hawkins paid tribute to his father with the band during a performance of "My Hero" by taking on his father's role as drummer. The concert at Wembley Stadium was streamed live on YouTube and Paramount+, with a rerun being broadcast on CBS.


title: Taylor Hawkins

text: The guest list reunited, in whole or in part, several of Hawkins' favorite bands including Queen and Rush. Some had not played together for a long time, including Dave Grohl's supergroup Them Crooked Vultures (12 years) and the James Gang (16 years).


title: Taylor Hawkins

text: The second tribute concert, at the Kia Forum in Inglewood, California, took place on September 27, 2022, and featured Hawkins' bandmates as well as numerous guest artists. Returning from the London show were Barker, Copeland, Hakim, Homme, Johannes, Jones, Kurstin, Lee, Lifeson, May, Novoselic, Ronson, Smith, Spiller, Roger Taylor, Rufus Taylor, Ulrich, Van Halen, the James Gang, and Chevy Metal. Also appearing were Jack Black, Geezer Butler, Danny Carey, Phil Collen, Miley Cyrus, Joe Elliott, Joan Jett, Mark King, Tommy Lee, Alanis Morissette, Pink, Nikki Sixx, Chad Smith, Jon Theodore, Brad Wilk, Nancy Wilson, and Patrick Wilson.


title: Pi

text: The number (; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number appears in many formulas across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as formula_1 are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers. The transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of appear to be randomly distributed, but no proof of this conjecture has been found.


title: Pi

text: For thousands of years, mathematicians have attempted to extend their understanding of , sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of for practical computations. Around 250BC, the Greek mathematician Archimedes created an algorithm to approximate with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for , based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706.


title: Pi

text: The invention of calculus soon led to the calculation of hundreds of digits of , enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test supercomputers.


title: Pi

text: Because its definition relates to the circle, is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. In modern mathematical analysis, it is often instead defined without any reference to geometry; therefore, it also appears in areas having little to do with geometry, such as number theory and statistics. The ubiquity of makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to have been published, and record-setting calculations of the digits of often result in news headlines.


title: Pi

text: The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter, sometimes spelled out as "pi." In English, is pronounced as "pie" ( ). In mathematical use, the lowercase letter is distinguished from its capitalized and enlarged counterpart , which denotes a product of a sequence, analogous to how denotes summation.


title: Pi

text: The ratio is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio . This definition of implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula .


title: Pi

text: Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits—a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation , as the integral:


title: Pi

text: An integral such as this was adopted as the definition of by Karl Weierstrass, who defined it directly as an integral in 1841.


title: Pi

text: Integration is no longer commonly used in a first analytical definition because, as explains, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of that does not rely on the latter. One such definition, due to Richard Baltzer and popularized by Edmund Landau, is the following: is twice the smallest positive number at which the cosine function equals 0. is also the smallest positive number at which the sine function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a power series, or as the solution of a differential equation.


title: Pi

text: In a similar spirit, can be defined using properties of the complex exponential, , of a complex variable . Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which is equal to one is then an (imaginary) arithmetic progression of the form:


title: Pi

text: A variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: there is a unique (up to automorphism) continuous isomorphism from the group R/Z of real numbers under addition modulo integers (the circle group), onto the multiplicative group of complex numbers of absolute value one. The number is then defined as half the magnitude of the derivative of this homomorphism.


title: Pi

text: is an irrational number, meaning that it cannot be written as the ratio of two integers. Fractions such as and are commonly used to approximate , but no common fraction (ratio of whole numbers) can be its exact value. Because is irrational, it has an infinite number of digits in its decimal representation, and does not settle into an infinitely repeating pattern of digits. There are several proofs that is irrational; they generally require calculus and rely on the "reductio ad absurdum" technique. The degree to which can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of or but smaller than the measure of Liouville numbers.


title: Pi

text: The digits of have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that is normal has not been proven or disproven.


title: Pi

text: Since the advent of computers, a large number of digits of have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of , and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found. Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem. Thus, because the sequence of 's digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of . This is also called the "Feynman point" in mathematical folklore, after Richard Feynman, although no connection to Feynman is known.


title: Pi

text: In addition to being irrational, is also a transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as .


title: Pi

text: The transcendence of has two important consequences: First, cannot be expressed using any finite combination of rational numbers and square roots or "n"-th roots (such as or ). Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle. Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.


title: Pi

text: Like all irrational numbers, cannot be represented as a common fraction (also known as a simple or vulgar fraction), by the very definition of irrational number (i.e., not a rational number). But every irrational number, including , can be represented by an infinite series of nested fractions, called a continued fraction:


title: Pi

text: Truncating the continued fraction at any point yields a rational approximation for ; the first four of these are , , , and . These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to than any other fraction with the same or a smaller denominator. Because is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, cannot have a periodic continued fraction. Although the simple continued fraction for (shown above) also does not exhibit any other obvious pattern, mathematicians have discovered several generalized continued fractions that do, such as:


title: Pi

text: Any complex number, say , can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or "r") is used to represent 's distance from the origin of the complex plane, and the other (angle or ) the counter-clockwise rotation from the positive real line:


title: Pi

text: where is the imaginary unit satisfying = −1. The frequent appearance of in complex analysis can be related to the behaviour of the exponential function of a complex variable, described by Euler's formula:


title: Pi

text: where the constant is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of and points on the unit circle centred at the origin of the complex plane. Setting = in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing five important mathematical constants:


title: Pi

text: There are different complex numbers satisfying , and these are called the "-th roots of unity" and are given by the formula:


title: Pi

text: The best-known approximations to dating before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics in particular by the mid-first millennium, to an accuracy of seven decimal places.


title: Pi

text: The earliest written approximations of are found in Babylon and Egypt, both within one percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats as  = 3.125. In Egypt, the Rhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats as 3.16. Although some pyramidologists such as Flinders Petrie have theorized that the Great Pyramid of Giza was built with proportions related to , this theory is not widely accepted by scholars.


title: Pi

text: In the Shulba Sutras of Indian mathematics, dating to an oral tradition from the first or second millennium BC, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.


title: Pi

text: The first recorded algorithm for rigorously calculating the value of was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. This polygonal algorithm dominated for over 1,000 years, and as a result is sometimes referred to as Archimedes's constant. Archimedes computed upper and lower bounds of by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that (that is ). Archimedes' upper bound of may have led to a widespread popular belief that is equal to . Around 150 AD, Greek-Roman scientist Ptolemy, in his "Almagest", gave a value for of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.


title: Pi

text: In ancient China, values for included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and (3rd century, approximately 3.1556). Around 265 AD, the Wei Kingdom mathematician Liu Hui created a polygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of of 3.1416. Liu later invented a faster method of calculating and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4. The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that and suggested the approximations = 3.14159292035... and = 3.142857142857..., which he termed the "Milü" (<nowiki>"</nowiki>close ratio") and "Yuelü" ("approximate ratio"), respectively, using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value remained the most accurate approximation of available for the next 800 years.


title: Pi

text: The Indian astronomer Aryabhata used a value of 3.1416 in his "Āryabhaṭīya" (499 AD). Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. Italian author Dante apparently employed the value .


title: Pi

text: The Persian astronomer Jamshīd al-Kāshī produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×2 sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×2 sides. Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, was called the "Ludolphian number" in Germany until the early 20th century). Dutch scientist Willebrord Snellius reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 10 sides. Christiaan Huygens was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to Richardson extrapolation.


title: Pi

text: The calculation of was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence. Infinite series allowed mathematicians to compute with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach also appeared in the Kerala school sometime between 1400 and 1500 AD. Around 1500 AD, a written description of an infinite series that could be used to compute was laid out in Sanskrit verse in "Tantrasamgraha" by Nilakantha Somayaji. The series are presented without proof, but proofs are presented in a later work, "Yuktibhāṣā", from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 – c. 1425. Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory–Leibniz series. Madhava used infinite series to estimate to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.


title: Pi

text: In 1593, François Viète published what is now known as Viète's formula, an infinite product (rather than an infinite sum, which is more typically used in calculations):


title: Pi

text: In the 1660s, the English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz discovered calculus, which led to the development of many infinite series for approximating . Newton himself used an arcsin series to compute a 15-digit approximation of in 1665 or 1666, writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."


title: Pi

text: In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for formula_14 to compute to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm. The Gregory–Leibniz series for formula_15 is simple, but converges very slowly (that is, approaches the answer gradually), so it is not used in modern calculations.


title: Pi

text: In 1706, John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster:


title: Pi

text: Machin reached 100 digits of with this formula. Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of . Machin-like formulae remained the best-known method for calculating well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device.


title: Pi

text: In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of in his head at the behest of German mathematician Carl Friedrich Gauss.


title: Pi

text: In 1853, British mathematician William Shanks calculated to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.


title: Pi

text: Some infinite series for converge faster than others. Given the choice of two infinite series for , mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate to any given accuracy. A simple infinite series for is the Gregory–Leibniz series:


title: Pi

text: As individual terms of this infinite series are added to the sum, the total gradually gets closer to , and – with a sufficient number of terms – can get as close to as desired. It converges quite slowly, though – after 500,000 terms, it produces only five correct decimal digits of .


title: Pi

text: An infinite series for (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:


title: Pi

text: After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of , whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of . Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term.


title: Pi

text: Not all mathematical advances relating to were aimed at increasing the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between and the prime numbers that later contributed to the development and study of the Riemann zeta function:


title: Pi

text: Swiss scientist Johann Heinrich Lambert in 1768 proved that is irrational, meaning it is not equal to the quotient of any two integers. Lambert's proof exploited a continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that is transcendental, confirming a conjecture made by both Legendre and Euler. Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".


title: Pi

text: In the earliest usages, the Greek letter was used to denote the semiperimeter ("semiperipheria" in Latin) of a circle. and was combined in ratios with δ (for diameter or semidiameter) or ρ (for radius) to form circle constants. (Before then, mathematicians sometimes used letters such as "c" or "p" instead.) The first recorded use is Oughtred's , to express the ratio of periphery and diameter in the 1647 and later editions of . Barrow likewise used "formula_20" to represent the constant 3.14..., while Gregory instead used "formula_21" to represent 6.28... .


title: Pi

text: The earliest known use of the Greek letter alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work "; or, a New Introduction to the Mathematics". The Greek letter first appears there in the phrase "1/2 Periphery ()" in the discussion of a circle with radius one. However, he writes that his equations for are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones. Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.


title: Pi

text: Euler started using the single-letter form beginning with his 1727 "Essay Explaining the Properties of Air", though he used , the ratio of periphery to radius, in this and some later writing. Euler first used in his 1736 work "Mechanica", and continued in his widely-read 1748 work (he wrote: "for the sake of brevity we will write this number as ; thus is equal to half the circumference of a circle of radius 1"). Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the Western world, though the definition still varied between 3.14... and 6.28... as late as 1761.


title: Pi

text: The development of computers in the mid-20th century again revolutionized the hunt for digits of . Mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator. Using an inverse tangent (arctan) infinite series, a team led by George Reitwiesner and John von Neumann that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIAC computer. The record, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973.


title: Pi

text: Two additional developments around 1980 once again accelerated the ability to compute . First, the discovery of new iterative algorithms for computing , which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly. Such algorithms are particularly important in modern computations because most of the computer's time is devoted to multiplication. They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods.


title: Pi

text: The iterative algorithms were independently published in 1975–1976 by physicist Eugene Salamin and scientist Richard Brent. These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic–geometric mean method (AGM method) or Gauss–Legendre algorithm. As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.


title: Pi

text: The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally "multiply" the number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step. Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing between 1995 and 2002. This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.


title: Pi

text: For most numerical calculations involving , a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the circumference of the observable universe with a precision of one atom. Accounting for additional digits needed to compensate for computational round-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application. Despite this, people have worked strenuously to compute to thousands and millions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with often make headlines around the world. They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of .


title: Pi

text: Modern calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive. The fast iterative algorithms were anticipated in 1914, when Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for , remarkable for their elegance, mathematical depth and rapid convergence. One of his formulae, based on modular equations, is


title: Pi

text: This series converges much more rapidly than most arctan series, including Machin's formula. Bill Gosper was the first to use it for advances in the calculation of , setting a record of 17 million digits in 1985. Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers (Jonathan and Peter) and the Chudnovsky brothers. The Chudnovsky formula developed in 1987 is


title: Pi

text: It produces about 14 digits of per term, and has been used for several record-setting calculations, including the first to surpass 1 billion (10) digits in 1989 by the Chudnovsky brothers, 10 trillion (10) digits in 2011 by Alexander Yee and Shigeru Kondo, and 100 trillion digits by Emma Haruka Iwao in 2022. For similar formulas, see also the Ramanujan–Sato series.


title: Pi

text: In 2006, mathematician Simon Plouffe used the PSLQ integer relation algorithm to generate several new formulas for , conforming to the following template:


title: Pi

text: where is (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.


title: Pi

text: Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of . Buffon's needle is one such technique: If a needle of length is dropped times on a surface on which parallel lines are drawn units apart, and if of those times it comes to rest crossing a line ( > 0), then one may approximate based on the counts:


title: Pi

text: Another Monte Carlo method for computing is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equal .


title: Pi

text: Another way to calculate using probability is to start with a random walk, generated by a sequence of (fair) coin tosses: independent random variables such that with equal probabilities. The associated random walk is


title: Pi

text: so that, for each , is drawn from a shifted and scaled binomial distribution. As varies, defines a (discrete) stochastic process. Then can be calculated by


title: Pi

text: This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem, discussed below.


title: Pi

text: These Monte Carlo methods for approximating are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained. Thus they are never used to approximate when speed or accuracy is desired.


title: Pi

text: Two algorithms were discovered in 1995 that opened up new avenues of research into . They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of that are not reused after they are calculated. This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.


title: Pi

text: Mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995. Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.


title: Pi

text: Another spigot algorithm, the BBP digit extraction algorithm, was discovered in 1995 by Simon Plouffe:


title: Pi

text: This formula, unlike others before it, can produce any individual hexadecimal digit of without calculating all the preceding digits. Individual binary digits may be extracted from individual hexadecimal digits, and octal digits can be extracted from one or two hexadecimal digits. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found that rapidly produces decimal digits. An important application of digit extraction algorithms is to validate new claims of record computations: After a new record is claimed, the decimal result is converted to hexadecimal, and then a digit extraction algorithm is used to calculate several random hexadecimal digits near the end; if they match, this provides a measure of confidence that the entire computation is correct.


title: Pi

text: Between 1998 and 2000, the distributed computing project PiHex used Bellard's formula (a modification of the BBP algorithm) to compute the quadrillionth (10th) bit of , which turned out to be 0. In September 2010, a Yahoo! employee used the company's Hadoop application on one thousand computers over a 23-day period to compute 256 bits of at the two-quadrillionth (2×10th) bit, which also happens to be zero.


title: Pi

text: Because is closely related to the circle, it is found in many formulae from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics, Fourier analysis, and number theory, also include in some of their important formulae.


title: Pi

text: appears in formulae for areas and volumes of geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. Below are some of the more common formulae that involve .


title: Pi

text: Some of the formulae above are special cases of the volume of the "n"-dimensional ball and the surface area of its boundary, the ("n"−1)-dimensional sphere, given below.


title: Pi

text: Apart from circles, there are other curves of constant width. By Barbier's theorem, every curve of constant width has perimeter times its width. The Reuleaux triangle (formed by the intersection of three circles with the sides of an equilateral triangle as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular smooth and even algebraic curves of constant width.


title: Pi

text: Definite integrals that describe circumference, area, or volume of shapes generated by circles typically have values that involve . For example, an integral that specifies half the area of a circle of radius one is given by:


title: Pi

text: In that integral the function represents the height over the formula_30-axis of a semicircle (the square root is a consequence of the Pythagorean theorem), and the integral computes the area below the semicircle.


title: Pi

text: The trigonometric functions rely on angles, and mathematicians generally use radians as units of measurement. plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2 radians. The angle measure of 180° is equal to radians, and 1° = /180 radians.


title: Pi

text: Common trigonometric functions have periods that are multiples of ; for example, sine and cosine have period 2, so for any angle and any integer ,


title: Pi

text: Many of the appearances of in the formulas of mathematics and the sciences have to do with its close relationship with geometry. However, also appears in many natural situations having apparently nothing to do with geometry.


title: Pi

text: In many applications, it plays a distinguished role as an eigenvalue. For example, an idealized vibrating string can be modelled as the graph of a function on the unit interval , with fixed ends . The modes of vibration of the string are solutions of the differential equation formula_32, or formula_33. Thus is an eigenvalue of the second derivative operator formula_34, and is constrained by Sturm–Liouville theory to take on only certain specific values. It must be positive, since the operator is negative definite, so it is convenient to write , where is called the wavenumber. Then satisfies the boundary conditions and the differential equation with .


title: Pi

text: The value is, in fact, the "least" such value of the wavenumber, and is associated with the fundamental mode of vibration of the string. One way to show this is by estimating the energy, which satisfies Wirtinger's inequality: for a function formula_35 with and , both square integrable, we have:


title: Pi

text: with equality precisely when is a multiple of . Here appears as an optimal constant in Wirtinger's inequality, and it follows that it is the smallest wavenumber, using the variational characterization of the eigenvalue. As a consequence, is the smallest singular value of the derivative operator on the space of functions on vanishing at both endpoints (the Sobolev space formula_37).


title: Pi

text: The number serves appears in similar eigenvalue problems in higher-dimensional analysis. As mentioned above, it can be characterized via its role as the best constant in the isoperimetric inequality: the area enclosed by a plane Jordan curve of perimeter satisfies the inequality


title: Pi

text: Ultimately, as a consequence of the isoperimetric inequality, appears in the optimal constant for the critical Sobolev inequality in "n" dimensions, which thus characterizes the role of in many physical phenomena as well, for example those of classical potential theory. In two dimensions, the critical Sobolev inequality is


title: Pi

text: for "f" a smooth function with compact support in , formula_40 is the gradient of "f", and formula_41 and formula_42 refer respectively to the and -norm. The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.


title: Pi

text: Wirtinger's inequality also generalizes to higher-dimensional Poincaré inequalities that provide best constants for the Dirichlet energy of an "n"-dimensional membrane. Specifically, is the greatest constant such that


title: Pi

text: for all convex subsets of of diameter 1, and square-integrable functions "u" on of mean zero. Just as Wirtinger's inequality is the variational form of the Dirichlet eigenvalue problem in one dimension, the Poincaré inequality is the variational form of the Neumann eigenvalue problem, in any dimension.


title: Pi

text: The constant also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function on the real line to the function defined as:


title: Pi

text: Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve "somewhere". The above is the most canonical definition, however, giving the unique unitary operator on that is also an algebra homomorphism of to .


title: Pi

text: The Heisenberg uncertainty principle also contains the number . The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,


title: Pi

text: The physical consequence, about the uncertainty in simultaneous position and momentum observations of a quantum mechanical system, is discussed below. The appearance of in the formulae of Fourier analysis is ultimately a consequence of the Stone–von Neumann theorem, asserting the uniqueness of the Schrödinger representation of the Heisenberg group.


title: Pi

text: The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution. The Gaussian function, which is the probability density function of the normal distribution with mean and standard deviation , naturally contains :


title: Pi

text: The factor of formula_47 makes the area under the graph of equal to one, as is required for a probability distribution. This follows from a change of variables in the Gaussian integral:


title: Pi

text: The central limit theorem explains the central role of normal distributions, and thus of , in probability and statistics. This theorem is ultimately connected with the spectral characterization of as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function. Equivalently, is the unique constant making the Gaussian normal distribution equal to its own Fourier transform. Indeed, according to , the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.


title: Pi

text: Let be the set of all twice differentiable real functions formula_49 that satisfy the ordinary differential equation formula_50. Then is a two-dimensional real vector space, with two parameters corresponding to a pair of initial conditions for the differential equation. For any formula_51, let formula_52 be the evaluation functional, which associates to each formula_53 the value formula_54 of the function at the real point . Then, for each "t", the kernel of formula_55 is a one-dimensional linear subspace of . Hence formula_56 defines a function from formula_57 from the real line to the real projective line. This function is periodic, and the quantity can be characterized as the period of this map.


title: Pi

text: The constant appears in the Gauss–Bonnet formula which relates the differential geometry of surfaces to their topology. Specifically, if a compact surface has Gauss curvature "K", then


title: Pi

text: where is the Euler characteristic, which is an integer. An example is the surface area of a sphere "S" of curvature 1 (so that its radius of curvature, which coincides with its radius, is also 1.) The Euler characteristic of a sphere can be computed from its homology groups and is found to be equal to two. Thus we have


title: Pi

text: The constant appears in many other integral formulae in topology, in particular, those involving characteristic classes via the Chern–Weil homomorphism.


title: Pi

text: Vector calculus is a branch of calculus that is concerned with the properties of vector fields, and has many physical applications such as to electricity and magnetism. The Newtonian potential for a point source situated at the origin of a three-dimensional Cartesian coordinate system is


title: Pi

text: which represents the potential energy of a unit mass (or charge) placed a distance from the source, and is a dimensional constant. The field, denoted here by , which may be the (Newtonian) gravitational field or the (Coulomb) electric field, is the negative gradient of the potential:


title: Pi

text: Special cases include Coulomb's law and Newton's law of universal gravitation. Gauss' law states that the outward flux of the field through any smooth, simple, closed, orientable surface containing the origin is equal to :


title: Pi

text: It is standard to absorb this factor of into the constant , in which case it appears in the numerator of the equation for the potential. This argument shows why it must appear "somewhere". Furthermore, is the surface area of the unit sphere, but we have not assumed that is the sphere. However, as a consequence of the divergence theorem, because the region away from the origin is vacuum (source-free) it is only the homology class of the surface in that matters in computing the integral, so it can be replaced by any convenient surface in the same homology class, in particular, a sphere, where spherical coordinates can be used to calculate the integral.


title: Pi

text: A consequence of the Gauss law is that the negative Laplacian of the potential is equal to times the Dirac delta function:


title: Pi

text: More general distributions of matter (or charge) are obtained from this by convolution, giving the Poisson equation


title: Pi

text: The constant also plays an analogous role in four-dimensional potentials associated with Einstein's equations, a fundamental formula which forms the basis of the general theory of relativity and describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy:


title: Pi

text: where is the Ricci curvature tensor, is the scalar curvature, is the metric tensor, is the cosmological constant, is Newton's gravitational constant, is the speed of light in vacuum, and is the stress–energy tensor. The left-hand side of Einstein's equation is a non-linear analogue of the Laplacian of the metric tensor, and reduces to that in the weak field limit, with the formula_65 term playing the role of a Lagrange multiplier, and the right-hand side is the analogue of the distribution function, times .


title: Pi

text: One of the key tools in complex analysis is contour integration of a function over a positively oriented (rectifiable) Jordan curve . A form of Cauchy's integral formula states that if a point is interior to , then


title: Pi

text: Although the curve is not a circle, and hence does not have any obvious connection to the constant , a standard proof of this result uses Morera's theorem, which implies that the integral is invariant under homotopy of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates. More generally, it is true that if a rectifiable closed curve does not contain , then the above integral is times the winding number of the curve.


title: Pi

text: The general form of Cauchy's integral formula establishes the relationship between the values of a complex analytic function on the Jordan curve and the value of at any interior point of :


title: Pi

text: provided is analytic in the region enclosed by and extends continuously to . Cauchy's integral formula is a special case of the residue theorem, that if is a meromorphic function the region enclosed by and is continuous in a neighbourhood of , then


title: Pi

text: The factorial function formula_69 is the product of all of the positive integers through . The gamma function extends the concept of factorial (normally defined only for non-negative integers) to all complex numbers, except the negative real integers, with the identity formula_70. When the gamma function is evaluated at half-integers, the result contains . For example, formula_71 and formula_72.


title: Pi

text: where is the Euler–Mascheroni constant. Evaluated at and squared, the equation reduces to the Wallis product formula. The gamma function is also connected to the Riemann zeta function and identities for the functional determinant, in which the constant plays an important role.


title: Pi

text: The gamma function is used to calculate the volume of the "n"-dimensional ball of radius "r" in Euclidean "n"-dimensional space, and the surface area of its boundary, the ("n"−1)-dimensional sphere:


title: Pi

text: The gamma function can be used to create a simple approximation to the factorial function for large : formula_77 which is known as Stirling's approximation. Equivalently,


title: Pi

text: As a geometrical application of Stirling's approximation, let denote the standard simplex in "n"-dimensional Euclidean space, and denote the simplex having all of its sides scaled up by a factor of . Then


title: Pi

text: Ehrhart's volume conjecture is that this is the (optimal) upper bound on the volume of a convex body containing only one lattice point.


title: Pi

text: The Riemann zeta function is used in many areas of mathematics. When evaluated at it can be written as


title: Pi

text: Finding a simple solution for this infinite series was a famous problem in mathematics called the Basel problem. Leonhard Euler solved it in 1735 when he showed it was equal to . Euler's result leads to the number theory result that the probability of two random numbers being relatively prime (that is, having no shared factors) is equal to . This probability is based on the observation that the probability that any number is divisible by a prime is (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is , and the probability that at least one of them is not is . For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:


title: Pi

text: This probability can be used in conjunction with a random number generator to approximate using a Monte Carlo approach.


title: Pi

text: The solution to the Basel problem implies that the geometrically derived quantity is connected in a deep way to the distribution of prime numbers. This is a special case of Weil's conjecture on Tamagawa numbers, which asserts the equality of similar such infinite products of "arithmetic" quantities, localized at each prime "p", and a "geometrical" quantity: the reciprocal of the volume of a certain locally symmetric space. In the case of the Basel problem, it is the hyperbolic 3-manifold .


title: Pi

text: The zeta function also satisfies Riemann's functional equation, which involves as well as the gamma function:


title: Pi

text: A consequence is that can be obtained from the functional determinant of the harmonic oscillator. This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula. The calculation can be recast in quantum mechanics, specifically the variational approach to the spectrum of the hydrogen atom.


title: Pi

text: The constant also appears naturally in Fourier series of periodic functions. Periodic functions are functions on the group of fractional parts of real numbers. The Fourier decomposition shows that a complex-valued function on can be written as an infinite linear superposition of unitary characters of . That is, continuous group homomorphisms from to the circle group of unit modulus complex numbers. It is a theorem that every character of is one of the complex exponentials formula_84.


title: Pi

text: There is a unique character on , up to complex conjugation, that is a group isomorphism. Using the Haar measure on the circle group, the constant is half the magnitude of the Radon–Nikodym derivative of this character. The other characters have derivatives whose magnitudes are positive integral multiples of 2. As a result, the constant is the unique number such that the group T, equipped with its Haar measure, is Pontrjagin dual to the lattice of integral multiples of 2. This is a version of the one-dimensional Poisson summation formula.


title: Pi

text: The constant is connected in a deep way with the theory of modular forms and theta functions. For example, the Chudnovsky algorithm involves in an essential way the j-invariant of an elliptic curve.


title: Pi

text: Modular forms are holomorphic functions in the upper half plane characterized by their transformation properties under the modular group formula_85 (or its various subgroups), a lattice in the group formula_86. An example is the Jacobi theta function


title: Pi

text: which is a kind of modular form called a Jacobi form. This is sometimes written in terms of the nome formula_88.


title: Pi

text: The constant is the unique constant making the Jacobi theta function an automorphic form, which means that it transforms in a specific way. Certain identities hold for all automorphic forms. An example is


title: Pi

text: which implies that transforms as a representation under the discrete Heisenberg group. General modular forms and other theta functions also involve , once again because of the Stone–von Neumann theorem.


title: Pi

text: The Cauchy distribution plays an important role in potential theory because it is the simplest Furstenberg measure, the classical Poisson kernel associated with a Brownian motion in a half-plane. Conjugate harmonic functions and so also the Hilbert transform are associated with the asymptotics of the Poisson kernel. The Hilbert transform "H" is the integral transform given by the Cauchy principal value of the singular integral


title: Pi

text: The constant is the unique (positive) normalizing factor such that "H" defines a linear complex structure on the Hilbert space of square-integrable real-valued functions on the real line. The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space : up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line. The constant is the unique normalizing factor that makes this transformation unitary.


title: Pi

text: An occurrence of in the fractal called the Mandelbrot set was discovered by David Boll in 1991. He examined the behaviour of the Mandelbrot set near the "neck" at . When the number of iterations until divergence for the point is multiplied by , the result approaches as approaches zero. The point at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root of tends to .


title: Pi

text: Although not a physical constant, appears routinely in equations describing fundamental principles of the universe, often because of 's relationship to the circle and to spherical coordinate systems. A simple formula from the field of classical mechanics gives the approximate period of a simple pendulum of length , swinging with a small amplitude ( is the earth's gravitational acceleration):


title: Pi

text: One of the key formulae of quantum mechanics is Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Δ) and momentum (Δ) cannot both be arbitrarily small at the same time (where is Planck's constant):


title: Pi

text: The fact that is approximately equal to 3 plays a role in the relatively long lifetime of orthopositronium. The inverse lifetime to lowest order in the fine-structure constant is


title: Pi

text: The field of fluid dynamics contains in Stokes' law, which approximates the frictional force exerted on small, spherical objects of radius , moving with velocity in a fluid with dynamic viscosity :


title: Pi

text: In electromagnetics, the vacuum permeability constant "μ" appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation. Before 20 May 2019, it was defined as exactly


title: Pi

text: A relation for the speed of light in vacuum, can be derived from Maxwell's equations in the medium of classical vacuum using a relationship between "μ" and the electric constant (vacuum permittivity), in SI units:


title: Pi

text: Under ideal conditions (uniform gentle slope on a homogeneously erodible substrate), the sinuosity of a meandering river approaches . The sinuosity is the ratio between the actual length and the straight-line distance from source to mouth. Faster currents along the outside edges of a river's bends cause more erosion than along the inside edges, thus pushing the bends even farther out, and increasing the overall loopiness of the river. However, that loopiness eventually causes the river to double back on itself in places and "short-circuit", creating an ox-bow lake in the process. The balance between these two opposing factors leads to an average ratio of between the actual length and the direct distance between source and mouth.


title: Pi

text: Piphilology is the practice of memorizing large numbers of digits of , and world-records are kept by the "Guinness World Records". The record for memorizing digits of , certified by Guinness World Records, is 70,000 digits, recited in India by Rajveer Meena in 9 hours and 27 minutes on 21 March 2015. In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.


title: Pi

text: One common technique is to memorize a story or poem in which the word lengths represent the digits of : The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. Such memorization aids are called mnemonics. An early example of a mnemonic for pi, originally devised by English scientist James Jeans, is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics." When a poem is used, it is sometimes referred to as a "piem". Poems for memorizing have been composed in several languages in addition to English. Record-setting memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and the method of loci.


title: Pi

text: A few authors have used the digits of to establish a new form of constrained writing, where the word lengths are required to represent the digits of . The "Cadaeic Cadenza" contains the first 3835 digits of in this manner, and the full-length book "Not a Wake" contains 10,000 words, each representing one digit of .


title: Pi

text: Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae, has been represented in popular culture more than other mathematical constructs.


title: Pi

text: In the 2008 Open University and BBC documentary co-production, "The Story of Maths", aired in October 2008 on BBC Four, British mathematician Marcus du Sautoy shows a visualization of the – historically first exact – formula for calculating when visiting India and exploring its contributions to trigonometry.


title: Pi

text: In the Palais de la Découverte (a science museum in Paris) there is a circular room known as the "pi room". On its wall are inscribed 707 digits of . The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1873 calculation by English mathematician William Shanks, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.


title: Pi

text: In Carl Sagan's 1985 novel "Contact" it is suggested that the creator of the universe buried a message deep within the digits of . The digits of have also been incorporated into the lyrics of the song "Pi" from the 2005 album "Aerial" by Kate Bush. In the 1967 "" episode "Wolf in the Fold", an out-of-control computer is contained by being instructed to "Compute to the last digit the value of ".


title: Pi

text: In the United States, Pi Day falls on 14 March (written 3/14 in the US style), and is popular among students. and its digital representation are often used by self-described "math geeks" for inside jokes among mathematically and technologically minded groups. A college cheer variously attributed to the Massachusetts Institute of Technology or the Rensselaer Polytechnic Institute includes "3.14159". Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi. In parts of the world where dates are commonly noted in day/month/year format, 22 July represents "Pi Approximation Day", as 22/7 = 3.142857.


title: Pi

text: During the 2011 auction for Nortel's portfolio of valuable technology patents, Google made a series of unusually specific bids based on mathematical and scientific constants, including .


title: Pi

text: In 1958 Albert Eagle proposed replacing by (tau), where , to simplify formulas, but this use of is otherwise unknown. Some propose , arguing that , as the number of radians in one turn or the ratio of a circle's circumference to its radius, is more natural than and simplifies many formulas. This use of has not made its way into mainstream mathematics, but has been was added to several programming languages as a predefined constant.


title: Pi

text: In 1897, an amateur mathematician attempted to persuade the Indiana legislature to pass the Indiana Pi Bill, which described a method to square the circle and contained text that implied various incorrect values for , including 3.2. The bill is notorious as an attempt to establish a value of mathematical constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate, meaning it did not become a law.


title: Pi

text: In contemporary internet culture, individuals and organizations frequently pay homage to the number . For instance, the computer scientist Donald Knuth let the version numbers of his program TeX approach . The versions are 3, 3.1, 3.14, and so forth.


title: Mckenna Grace

text: Mckenna Grace (born June 25, 2006) is an American actress and singer. She began acting professionally at the age of six, with her earliest roles including Jasmine Bernstein in the Disney XD sitcom "Crash & Bernstein" (2012–2014) and Faith Newman in the soap opera "The Young and the Restless" (2013–2015). In 2018 and 2019, "The Hollywood Reporter" named her one of the top 30 stars under age 18. She is known for her roles as Mary Adler in the 2017 film "Gifted", Abigail Stone in the 2021 film "Spirit Untamed" and Phoebe Spengler in the 2021 film "".


title: Mckenna Grace

text: She released her first song in 2021 titled "Haunted House" with the release of "Ghostbusters: Afterlife". Following this release, she began her music career.


title: Mckenna Grace

text: In 2013, Grace made her acting debut playing Sydney in the television pilot "Joe, Joe & Jane". Later she recurred on the Disney XD series "Crash & Bernstein". She had a recurring role as Faith Newman on "The Young and the Restless" between 2013 and 2015. In 2015, she had a recurring role on the CBS series "". Grace played the role of Daisy in the science fiction thriller ""; the film, directed by Roland Emmerich, was released on June 24, 2016. Other early film roles include the drama film "Mr. Church" (2016), and the comedy film "How to Be a Latin Lover" (2017).


title: Mckenna Grace

text: In 2017, Grace played a lead role in the Marc Webb drama film "Gifted" as child prodigy Mary Adler, starring alongside Chris Evans. She gained recognition for her performance in the role, being nominated for the Critics' Choice Movie Award for Best Young Performer.


title: Mckenna Grace

text: She learned to ice skate to portray the younger version of Tonya Harding in the biopic "I, Tonya", describing it as the "most challenging role" she'd ever undertaken. Grace also played the significant role of Juliet in the horror film "" along with Bella Thorne; Franck Khalfoun directed the film.


title: Mckenna Grace

text: Grace has occasionally appeared in "Young Sheldon" from 2018 to 2021 as Paige Swanson, a child prodigy and a rival to Sheldon Cooper.


title: Mckenna Grace

text: Grace starred as Judy Warren, daughter of paranormal investigators Ed and Lorraine Warren, in the 2019 horror film "Annabelle Comes Home". Also that year, she appeared in the Marvel Studios film "Captain Marvel" as a younger version of the titular character.


title: Mckenna Grace

text: Grace appeared as the younger version of Sabrina Spellman on the first season of Netflix's "Chilling Adventures of Sabrina".


title: Mckenna Grace

text: She voiced the younger Daphne Blake in the 2020 animated film "Scoob!", and was set to reprise the role in the follow-up "Scoob!: Holiday Haunt", which was set for release in late 2022 on HBO Max, but it was canceled in August 2022.


title: Mckenna Grace

text: Grace's career has encompassed portraying the "younger version" of various characters. On this notion she stated, "It's really funny. I don't know how that ended up happening. But I'm really glad that it did, because I definitely got to learn from all of the actresses that I played younger versions of."


title: Mckenna Grace

text: In 2021, Grace played Phoebe in "", plus the role of Esther Keyes in the fourth season of the Hulu adaptation of Margaret Atwood's "The Handmaid's Tale" and earned a nomination for Outstanding Guest Actress in a Drama Series at the 73rd Primetime Creative Arts Emmy Awards. In August, she joined Thomasin McKenzie and Olivia Wilde in Searchlight Pictures' feature "Perfect".


title: Mckenna Grace

text: In 2020, Grace signed with Photo Finish Records, and in November 2021 she released her debut single "Haunted House" alongside an accompanying music video. While not originally written for "," the song was eventually added to the film's soundtrack and used for the closing credits. When asked, Grace explained that the song was written during a rough period of her life in the midst of the COVID-19 pandemic, and that one "could take it as a breakup song, but it could also be about a friend or a family member or any kind of relationship that's ended."


title: Mckenna Grace

text: On February 13, 2022, Grace released her follow up single "Do All My Friends Hate Me?" along with a music video. On April 21, Grace released her third single "You Ruined Nirvana", also with a music video. On July 8, Grace released her fourth single "Post Party Trauma", but with a lyric video instead.


title: Mckenna Grace

text: Grace is a vegetarian and has directed her affection for animals towards work with Farm Sanctuary and PETA, including appearing in a campaign reminding people not to leave their dogs in hot cars.


title: Mckenna Grace

text: She was diagnosed with scoliosis at age 12, and on October 9th, 2022, she underwent spinal surgery to correct the curvature of her spine, which was over 45 degrees. Post-op, her spinal curvature is at only 6 degrees.


title: Boris Yeltsin

text: Boris Nikolayevich Yeltsin (; 1 February 1931 – 23 April 2007) was a Soviet and Russian politician who served as the first president of the Russian Federation from 1991 to 1999. He was a member of the Communist Party of the Soviet Union from 1961 to 1990. He later stood as a political independent, during which time he was viewed as being ideologically aligned with liberalism and Russian nationalism.


title: Boris Yeltsin

text: Yeltsin was born in Butka, Ural Oblast. He grew up in Kazan and Berezniki. After studying at the Ural State Technical University, he worked in construction. After joining the Communist Party, he rose through its ranks, and in 1976 he became First Secretary of the party's Sverdlovsk Oblast committee. Yeltsin was initially a supporter of the "perestroika" reforms of Soviet leader Mikhail Gorbachev. He later criticized the reforms as being too moderate, and called for a transition to a multi-party representative democracy. In 1987 he was the first person to resign from the Politburo of the Communist Party of the Soviet Union, which established his popularity as an anti-establishment figure. In 1990, he was elected chair of the Russian Supreme Soviet and in 1991 was elected president of the Russian Soviet Federative Socialist Republic (RSFSR). Yeltsin allied with various non-Russian nationalist leaders, and was instrumental in the formal dissolution of the Soviet Union in December of that year. With the dissolution of the Soviet Union, the RSFSR became the Russian Federation, an independent state. Through that transition, Yeltsin remained in office as president. He was later reelected in the 1996 election, which was claimed by critics to be pervasively corrupt.


title: Boris Yeltsin

text: Yeltsin transformed Russia's command economy into a capitalist market economy by implementing economic shock therapy, market exchange rate of the ruble, nationwide privatization, and lifting of price controls. Economic volatility and inflation ensued. Amid the economic shift, a small number of oligarchs obtained a majority of the national property and wealth, while international monopolies came to dominate the market. A constitutional crisis emerged in 1993 after Yeltsin ordered the unconstitutional dissolution of the Russian parliament, leading parliament to impeach him. The crisis ended after troops loyal to Yeltsin stormed the parliament building and stopped an armed uprising; he then introduced a new constitution which significantly expanded the powers of the president. Secessionist sentiment in the Russian Caucasus led to the First Chechen War, War of Dagestan, and Second Chechen War between 1994 and 1999. Internationally, Yeltsin promoted renewed collaboration with Europe and signed arms control agreements with the United States. Amid growing internal pressure, he resigned by the end of 1999 and was succeeded by his chosen successor, Vladimir Putin, whom he had appointed prime minister a few months earlier. He kept a low profile after leaving office and was accorded a state funeral upon his death in 2007.


title: Boris Yeltsin

text: Domestically, he was highly popular in the late 1980s and early 1990s, although his reputation was damaged by the economic and political crises of his presidency, and he left office widely unpopular with the Russian population. He received praise and criticism for his role in dismantling the Soviet Union, transforming Russia into a representative democracy, and introducing new political, economic, and cultural freedoms to the country. Conversely, he was accused of economic mismanagement, overseeing a massive growth in inequality and corruption, and sometimes of undermining Russia's standing as a major world power.


title: Boris Yeltsin

text: Boris Yeltsin was born on 1 February 1931 in the village of Butka, Talitsky District, Sverdlovsk Oblast, then in the Russian Soviet Federative Socialist Republic, one of the republics of the Soviet Union. His family, who were ethnic Russians, had lived in this area of the Urals since at least the eighteenth century. His father, Nikolai Yeltsin, had married his mother, Klavdiya Vasilyevna Starygina, in 1928. Yeltsin always remained closer to his mother than to his father; the latter beat his wife and children on various occasions.


title: Boris Yeltsin

text: The Soviet Union was then under the dictatorship of Joseph Stalin, who led the one-party state governed by the Communist Party of the Soviet Union. Seeking to convert the country into a socialist society according to Marxist–Leninist doctrine, in the late 1920s Stalin's government had initiated a project of mass rural collectivisation coupled with dekulakization. As a prosperous farmer, Yeltsin's paternal grandfather, Ignatii, was accused of being a "kulak" in 1930. His farm, which was in Basmanovo (also known as Basmanovskoye), was confiscated; and he and his family were forced to reside in a cottage in nearby Butka. There, Nikolai and Ignatii's other children were allowed to join the local kolkhoz (collective farm), but Ignatii himself was not; he and his wife, Anna, were exiled in 1934 to Nadezhdinsk, where he died two years later.


title: Boris Yeltsin

text: As an infant, Yeltsin was christened in the Russian Orthodox Church; his mother was devout, and his father unobservant. In the years after his birth, the area was hit by the famine of 1932–1933; throughout his childhood, Yeltsin was often hungry. In 1932, Yeltsin's parents moved to Kazan, where Yeltsin attended kindergarten. There, in 1934, the OGPU state security services arrested Nikolai, accused him of anti-Soviet agitation, and sentenced him to three years in the Dmitrov labor camp. Yeltsin and his mother then were ejected from their residence and were taken in by friends; Klavdiya worked at a garment factory in her husband's absence. In October 1936, Nikolai returned; in July 1937, the couple's second child, Mikhail, was born. That month, they moved to Berezniki, in Perm Krai, where Nikolai got work on a potash combine project. There, in July 1944, they had a third child, the daughter, Valentina.


title: Boris Yeltsin

text: Between 1939 and 1945, Yeltsin received a primary education at Berezniki's Railway School Number 95. Academically, he did well at primary school and was repeatedly elected class monitor by fellow pupils. There, he also took part in activities organized by the Komsomol and Vladimir Lenin All-Union Pioneer Organization. This overlapped with Soviet involvement in the Second World War, during which Yeltsin's paternal uncle, Andrian, served in the Red Army and was killed.


title: Boris Yeltsin

text: From 1945 to 1949, Yeltsin studied at the municipal secondary school number 1, also known as Pushkin High School. Yeltsin did well at secondary school, and there took an increasing interest in sport, becoming captain of the school's volleyball squad. He enjoyed playing pranks and in one instance played with a grenade, which blew off the thumb and index finger of his left hand. With friends, he would go on summer walking expeditions in the adjacent taiga, sometimes for many weeks.


title: Boris Yeltsin

text: In September 1949, Yeltsin was admitted to the Ural Polytechnic Institute (UPI) in Sverdlovsk. He took the stream in industrial and civil engineering, which included courses in maths, physics, materials and soil science, and draftsmanship. He was also required to study Marxist-Leninist doctrine and choose a language course, for which he selected German, although never became adept at it. Tuition was free and he was provided a small stipend to live on, which he supplemented by unloading railway trucks for a small wage. Academically, he achieved high grades, although temporarily dropped out in 1952 when afflicted with tonsillitis and rheumatic fever. He devoted much time to athletics, and joined the UPI volleyball team. He avoided any involvement in political organizations while there. During the summer 1953 break, he traveled across the Soviet Union, touring the Volga, central Russia, Belarus, Ukraine, and Georgia; much of the travel was achieved by hitchhiking on freight trains. It was at UPI that he began a relationship with Naina Iosifovna Girina, a fellow student who would later become his wife. Yeltsin completed his studies in June 1955.


title: Boris Yeltsin

text: Leaving the Ural Polytechnic Institute, Yeltsin was assigned to work with the Lower Iset Construction Directorate in Sverdlovsk; at his request, he served the first year as a trainee in various building trades. He quickly rose through the organization's ranks. In June 1956 he was promoted to foreman ("master"), and in June 1957 was promoted again, to the position of work superintendent ("prorab"). In these positions, he confronted a widespread alcoholism and a lack of motivation among construction workers, an irregular supply of materials, and the regular theft or vandalism of materials that were available. He soon imposed fines for those who damaged or stole materials or engaged in absenteeism, and closely monitored productivity. His work on the construction of a textile factory, for which he oversaw 1000 workers, brought him wider recognition. In June 1958 he became a senior work superintendent ("starshii prorab") and in January 1960 was made head engineer ("glavni inzhener") of Construction Directorate Number 13.


title: Boris Yeltsin

text: At the same time, Yeltsin's family was growing; in September 1956, he married Girina. She soon got work at a scientific research institute, where she remained for 29 years. In August 1957, their daughter Yelena was born, followed by a second daughter, Tatyana, in January 1960. During this period, they moved through a succession of apartments. On family holidays, Yeltsin took his family to a lake in northern Russia and to the Black Sea coast.


title: Boris Yeltsin

text: In March 1960, Yeltsin became a probationary member of the governing Communist Party and a full member in March 1961. In his later autobiography, he stated that his original reasons for joining were "sincere" and rooted in a genuine belief in the party's socialist ideals. In other interviews he instead stated that he joined because membership was a necessity for career advancement. His career continued to progress during the early 1960s; in February 1962 he was promoted chief ("nachal'nik") of the construction directorate. In June 1963, Yeltsin was reassigned to the Sverdlovsk House-Building Combine as its head engineer, and in December 1965 became the combine's director. During this period he was largely involved in building residential housing, the expansion of which was a major priority for the government. He gained a reputation within the construction industry as a hard worker who was punctual and effective and who was used to meeting the targets set forth by the state apparatus. There had been plans to award him the Order of Lenin for his work, although this was scrapped after a five-story building he was constructing collapsed in March 1966. An official investigation found that Yeltsin was not culpable for the accident.


title: Boris Yeltsin

text: Within the local Communist Party, Yeltsin gained a patron in , who became the first secretary of the party "gorkom" in 1963. In April 1968, Ryabov decided to recruit Yeltsin into the regional party apparatus, proposing him for a vacancy in the "obkom" department for construction. Ryabov ensured that Yeltsin got the job despite objections that he was not a longstanding party member. That year, Yeltsin and his family moved into a four-room apartment on Mamin-Sibiryak Street, downtown Sverdlovsk. Yeltsin then received his second Order of the Red Banner of Labor for his work completing a cold-rolling mill at the Upper Iset Works, a project for which he had overseen the actions of 15,000 laborers. In the late 1960s, Yeltsin was permitted to visit the West for the first time as he was sent on a trip to France. In 1975, Yeltsin was then made one of the five "obkom" secretaries in the Sverdlovsk Oblast, a position that gave him responsibility not only for construction in the region but also for the forest and the pulp-and-paper industries. Also in 1975, his family relocated to a flat in the House of Old Bolsheviks on March Street.


title: Boris Yeltsin

text: In October 1976, Ryabov was promoted to a new position in Moscow. He recommended that Yeltsin replace him as the First Secretary of the Party Committee in Sverdlovsk Oblast. Leonid Brezhnev, who then led the Soviet Union as General Secretary of the party's Central Committee, interviewed Yeltsin personally to determine his suitability and agreed with Ryabov's assessment. At the Central Committee's recommendation, the Sverdlovsk obkom then unanimously voted to appoint Yeltsin as its first secretary. This made him one of the youngest provincial first secretaries in the RSFSR, and gave him significant power within the province.


title: Boris Yeltsin

text: Where possible, Yeltsin tried to improve consumer welfare in the province, arguing that it would make for more productive workers. Under his provincial leadership, work started on various construction and infrastructure projects in the city of Sverdlovsk, including a subway system, the replacement of its barracks housing, new theaters and a circus, the refurbishment of its 1912 opera house, and youth housing projects to build new homes for young families. In September 1977, Yeltsin carried out orders to demolish the Ipatiev House, the location where the Romanov royal family had been killed in 1918, over the government's fears that it was attracting growing foreign and domestic attention. He was also responsible for punishing those living in the province who wrote or published material that the Soviet government considered to be seditious or damaging to the established order.


title: Boris Yeltsin

text: Yeltsin sat on the civil-military collegium of the Urals Military District and attended its field exercises. In October 1978, the Ministry of Defence gave him the rank of colonel. Also in 1978, Yeltsin was elected without opposition to the Supreme Soviet. In 1979 Yeltsin and his family moved into a five-room apartment at the Working Youth Embankment in Sverdlovsk. In February 1981, Yeltsin gave a speech to the 25th CPSU Congress and on the final day of the Congress was selected to join the Communist Party Central Committee.


title: Boris Yeltsin

text: Yeltsin's reports to party meetings reflected the ideological conformity that was expected within the authoritarian state. Yeltsin played along with the personality cult surrounding Brezhnev, but he was contemptuous of what he saw as the Soviet's leader's vanity and sloth. He later claimed to have quashed plans for a Brezhnev museum in Sverdlovsk. While First Secretary, his world-view began to shift, influenced by his reading; he kept up with a wide range of journals published in the country and also claimed to have read an illegally-printed "samizdat" copy of Aleksandr Solzhenitsyn's "The Gulag Archipelago". Many of his concerns about the Soviet system were prosaic rather than ideological, as he believed that the system was losing effectiveness and beginning to decay. He was increasingly faced with the problem of Russia's place within the Soviet Union; unlike other republics in the country, the RSFSR lacked the same levels of autonomy from the central government in Moscow. In the early 1980s, he and Yurii Petrov privately devised a tripartite scheme for reforming the Soviet Union that would involve strengthening the Russian government, but it was never presented publicly.


title: Boris Yeltsin

text: By 1980, Yeltsin had developed the habit of appearing unannounced in factories, shops, and public transport to get a closer look at the realities of Soviet life. In May 1981, he held a question-and-answer session with college students at the Sverdlovsk Youth Palace, where he was unusually frank in his discussion of the country's problems. In December 1982 he then gave a television broadcast for the region in which he responded to various letters. This personalised approach to interacting with the public brought disapproval from some Communist Party figures, such as First Secretary of Tyumen Oblast, Gennadii Bogomyakov, although the Central Committee showed no concern. In 1981, he was awarded the Order of Lenin for his work. The following year, Brezhnev died and was succeeded by Yuri Andropov, who in turn ruled for 15 months before his own death; Yeltsin spoke positively about Andropov. Andropov was succeeded by another short-lived leader, Konstantin Chernenko. After his death, Yeltsin took part in the Central Committee plenum which appointed Mikhail Gorbachev the new General Secretary of the Communist Party of the Soviet Union, and thus "de facto" Soviet leader, in March 1985.


title: Boris Yeltsin

text: Gorbachev was interested in reforming the Soviet Union and, at the urging of Yegor Ligachyov, the organisational secretary of the Central Committee, soon summoned Yeltsin to meet with him as a potential ally in his efforts. Yeltsin had some reservations about Gorbachev as a leader, deeming him controlling and patronising, but committed himself to the latter's project of reform. In April 1985, Gorbachev appointed Yeltsin as the Head of the Construction Department of the Party's Central Committee. Although it entailed moving to the capital city, Yeltsin was unhappy with what he regarded as a demotion. There, he was issued a nomenklatura flat at 54 Second Tverskaya-Yamskaya Street, where his daughter Tatyana and her son and husband soon joined him and his wife. Gorbachev soon promoted Yeltsin to secretary of the Central Committee for construction and capital investment, a position within the powerful CPSU Central Committee Secretariat, a move approved by the Central Committee plenum in July 1985.


title: Boris Yeltsin

text: With Gorbachev's support, in December 1985, Yeltsin was installed as the first secretary of the Moscow gorkom of the CPSU. He was now responsible for managing the Soviet capital city, which had a population of 8.7 million. In February 1986, Yeltsin became a candidate (non-voting) member of the Politburo. At that point he formally left the Secretariat to concentrate on his role in Moscow. Over the coming year he removed many of the old secretaries of the gorkom, replacing them with younger individuals, particularly with backgrounds in factory management. In August 1986, Yeltsin gave a two-hour report to the party conference in which he talked about Moscow's problems, including issues that had previously not been spoken about publicly. Gorbachev described the speech as a "strong fresh wind" for the party. Yeltsin expressed a similar message at the 22nd Congress of the CPSU in February 1986 and then in a speech at the House of Political Enlightenment in April.


title: Boris Yeltsin

text: On 10 September 1987, after a lecture from hard-liner Yegor Ligachyov at the Politburo for allowing two small unsanctioned demonstrations on Moscow streets, Yeltsin wrote a letter of resignation to Gorbachev who was holidaying on the Black Sea. When Gorbachev received the letter he was stunned – nobody in Soviet history had voluntarily resigned from the ranks of the Politburo. Gorbachev phoned Yeltsin and asked him to reconsider.


title: Boris Yeltsin

text: On 27 October 1987 at the plenary meeting of the Central Committee of the CPSU, Yeltsin, frustrated that Gorbachev had not addressed any of the issues outlined in his resignation letter, asked to speak. He expressed his discontent with the slow pace of reform in society, the servility shown to the general secretary, and opposition to him from Ligachyov making his position untenable, before requesting to resign from the Politburo, adding that the City Committee would decide whether he should resign from the post of First Secretary of the Moscow Communist Party. Aside from the fact that no one had ever quit the Politburo before, no one in the party had addressed a leader of the party in such a manner in front of the Central Committee since Leon Trotsky in the 1920s. In his reply, Gorbachev accused Yeltsin of "political immaturity" and "absolute irresponsibility". Nobody in the Central Committee backed Yeltsin.


title: Boris Yeltsin

text: Within days, news of Yeltsin's actions leaked and rumours of his "secret speech" at the Central Committee spread throughout Moscow. Soon fabricated "samizdat" versions began to circulate – this was the beginning of Yeltsin's rise as a rebel and growth in popularity as an anti-establishment figure. Gorbachev called a meeting of the Moscow City Party Committee for 11 November 1987 to launch another crushing attack on Yeltsin and confirm his dismissal. On 9 November 1987, Yeltsin apparently tried to kill himself and was rushed to the hospital bleeding profusely from self-inflicted cuts to his chest. Gorbachev ordered the injured Yeltsin from his hospital bed to the Moscow party plenum two days later where he was ritually denounced by the party faithful in what was reminiscent of a Stalinist show trial before he was fired from the post of First Secretary of the Moscow Communist Party. Yeltsin said he would never forgive Gorbachev for this "immoral and inhuman" treatment.


title: Boris Yeltsin

text: Yeltsin was demoted to the position of First Deputy Commissioner for the State Committee for Construction. At the next meeting of the Central Committee on 24 February 1988, Yeltsin was removed from his position as a Candidate member of the Politburo. He was perturbed and humiliated but began plotting his revenge. His opportunity came with Gorbachev's establishment of the Congress of People's Deputies. Yeltsin recovered, and started intensively criticizing Gorbachev, highlighting the slow pace of reform in the Soviet Union as his major argument.


title: Boris Yeltsin

text: Yeltsin's criticism of the Politburo and Gorbachev led to a smear campaign against him, in which examples of Yeltsin's awkward behavior were used against him. Speaking at the CPSU conference in 1988, Yegor Ligachyov stated, "Boris, you are wrong". An article in "Pravda" described Yeltsin as drunk at a lecture during his visit to the United States in September 1989, an allegation which appeared to be confirmed by a TV account of his speech; however, popular dissatisfaction with the regime was very strong, and these attempts to smear Yeltsin only added to his popularity. In another incident, Yeltsin fell from a bridge. Commenting on this event, Yeltsin hinted that he was helped to fall by the enemies of perestroika, but his opponents suggested that he was simply drunk.


title: Boris Yeltsin

text: On 26 March 1989, Yeltsin was elected to the Congress of People's Deputies of the Soviet Union as the delegate from Moscow district with a decisive 92% of the vote, and on 29 May 1989, he was elected by the Congress of People's Deputies to a seat on the Supreme Soviet of the Soviet Union. On 19 July 1989, Yeltsin announced the formation of the radical pro-reform faction in the Congress of People's Deputies, the Inter-Regional Group of Deputies, and on 29 July 1989 was elected one of the five co-Chairmen of the Inter-Regional Group.


title: Boris Yeltsin

text: On 16 September 1989, Yeltsin toured a medium-sized grocery store (Randalls) in Texas. Leon Aron, quoting a Yeltsin associate, wrote in his 2000 biography, "Yeltsin, A Revolutionary Life" (St. Martin's Press): "For a long time, on the plane to Miami, he sat motionless, his head in his hands. 'What have they done to our poor people?' he said after a long silence." He added, "On his return to Moscow, Yeltsin would confess the pain he had felt after the Houston excursion: the 'pain for all of us, for our country so rich, so talented and so exhausted by incessant experiments'." He wrote that Mr. Yeltsin added, "I think we have committed a crime against our people by making their standard of living so incomparably lower than that of the Americans." An aide, Lev Sukhanov, was reported to have said that it was at that moment that "the last vestige of Bolshevism collapsed" inside his boss. In his autobiography, "Against the Grain: An Autobiography", written and published in 1990, Yeltsin hinted in a small passage that after his tour, he made plans to open his own line of grocery stores and planned to fill it with government subsidized goods in order to alleviate the country's problems.


title: Boris Yeltsin

text: On 4 March 1990, Yeltsin was elected to the Congress of People's Deputies of Russia representing Sverdlovsk with 72% of the vote. On 29 May 1990, he was elected chairman of the Supreme Soviet of the Russian Soviet Federative Socialist Republic (RSFSR), in spite of the fact that Gorbachev personally pleaded with the Russian deputies not to select Yeltsin. He was supported by both democratic and conservative members of the Supreme Soviet, which sought power in the developing political situation in the country.


title: Boris Yeltsin

text: A part of this power struggle was the opposition between power structures of the Soviet Union and the RSFSR. In an attempt to gain more power, on 12 June 1990, the Congress of People's Deputies of the RSFSR adopted a declaration of sovereignty. On 12 July 1990, Yeltsin resigned from the CPSU in a dramatic speech before party members at the 28th Congress of the Communist Party of the Soviet Union, some of whom responded by shouting "Shame!"