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'''Thealogy''' is literally the study of the [[Goddess]] ([[Greek language|Greek]] θεά, ''thea'', "goddess" + λόγος, ''logos'', "study"). In [[1993]], [[Charlotte Caron]]'s definition of '''thealogy''' as "reflection on the divine in feminine and feminist terms" appeared, but the term actually originates in the writings of [[Isaac Bonewits]] in [[1974]].
{{otheruses|pie}}
==First uses==
[[Image:Greek pi.png|right|thumb|110px|The [[minuscule]], or lower-case, pi]]
===First(?) usages===
The [[mathematical constant]] '''π''' is the [[ratio]] of a [[circle]]'s [[circumference]] ([[Greek language|Greek]]<!-- It would be nice if <nobr> worked: --> <u>'''π'''</u>εριφέρεια, periphery) to its [[diameter]] and is commonly used in [[mathematics]], [[physics]], and [[engineering]]. The name of the [[Greek alphabet|Greek letter]] [[Pi (letter)|π]] is '''pi''' (pronounced ''pie''), and this spelling can be used in typographical contexts where the Greek letter is not available. π is also known as '''[[Archimedes]]' constant''' (not to be confused with [[Archimedes number|Archimedes' number]]) and '''[[Ludolph van Ceulen|Ludolph]]'s number'''.
In "The Druid Chronicles (Evolved)," privately published in [[1976]], Isaac Bonewits used "thealogian" to refer to a Wiccan author ([[Aidan Kelly]], aka "C. Taliesin Edwards," who may have given him the term or vice versa) and "theilogy" (defined as "the study of more than one God"). Bonewits also used "theilogy" (and possibly "thealogy," since he thinks he coined them at the same time) in the pages of the widely-distributed "Gnostica" magazine he edited in 1974 and [[1975]].
In [[plane geometry|Euclidean plane geometry]], π may be defined either as the [[ratio]] of a [[circle]]'s [[circumference]] to its [[diameter]], or as the ratio of a circle's [[area]] to the area of a square whose side is the radius. Advanced textbooks define π [[mathematical analysis|analytically]] using [[trigonometric function]]s, for example as the smallest positive ''x'' for which [[trigonometric function|sin]](''x'') = 0, or as twice the smallest positive ''x'' for which [[trigonometric function|cos]](''x'') = 0.
All these definitions are equivalent.
"The Druid Chronicles (Evolved)" were a three-year project starting in 1974 and finished (published) in 1976. The article referred to within "The Druid Chronicles (Evolved)" is dated to the summer of 1976. Moreover, this is almost certainly not the first usage; the context of "thealogian" is in citing a work by C. Taliesin Edwards, "Essays towards a Meta''thealogy'' of the Goddess." [stress added] There is, however, a possibility that Bonewits altered the name of the work to fit with his terminology. He is attempting to track this down. Kelley himself has said to Bonewits that he can't remember which of the two of them said "thealogy" to the other first.
The numerical value of π rounded to 50 [[decimal|decimal places]] {{OEIS|id=A000796}} is:
In [[1976]], [[Valerie Saiving]], ending her "[[Androcentrism]] in Religious Studies" made a much quoted invocation that yearns towards something as yet undefined-
:3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511
:''it is just possible that the unheard testimony of that half of the human species which has for so long been rendered inarticulate may have something to tell us about the holy which we have not known - something which can finally make us whole.''
Although this precision is more than sufficient for use in [[engineering]] and [[science]], much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, in addition to [[supercomputer]] calculations that have determined over 1 trillion digits of π, no pattern in the digits has ever been found. Digits of π are available from multiple resources on the Internet, and a regular [[personal computer]] can be used to compute billions of digits.
::(Saiving 1976:197)
===Second(?) Propertiesusage ===
In "The Changing of the Gods" 1979:96, [[Naomi Goldenberg]] selfconsciously introduces the term as a half whimsical possibility, an inspirational comment, not a prelude to exegesis. She does not go on to define what thealogy might be, other than the implicit femininity of the coinage. This lack was perhaps because at that time the very assertion of a serious feminist analysis of religion was virtually unheard of, and the introduction of the concept was an excitingly powerful, but vague, possibility.
π is an [[irrational number]]; that is, it cannot be written as the ratio of two [[integer]]s, as was proven in [[1761]] by [[Johann Heinrich Lambert]].
This is not to say that both Goldenberg and Saiving do not both offer extremely solid chunks of thealogy, but they do not give an overview of something to which they were midwives.
π is also [[transcendental number|transcendental]], as was proven by [[Ferdinand von Lindemann]] in [[1882]]. This means that there is no [[polynomial]] with [[rational number|rational]] coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not [[constructible number|constructible]]. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to [[squaring the circle|square the circle]], that is, it is impossible to construct, using [[ruler-and-compass construction|ruler and compass]] alone, a square whose area is equal to the area of a given circle.
===Bonewits again===
== Formulae involving π ==
Also in [[1979]], in the first revised edition of "Real Magic," Bonewits defined "thealogy" in his Glossary this way: "Intellectual speculations concerning the nature of the Goddess and Her relations to the world in general and humans in particular; rational explanations of religious doctrines, practices and beliefs, which may or may not bear any connection to any religion as actually conceived and practiced by the majority of its members." While the last clause was his editorializing, the majority of the definition was adapted by removing sexist assumptions from a dictionary then in his library. Also in the same glossary, he defined "theology" and "theoilogy" (spelled correctly this time) with nearly identical words, changing the pronouns appropriately. He has since dropped the use of "theoilogy" in favor of "polytheology," also first published by him in the 1974 "Druid Chronicles."
===Geometry===
<math>\pi</math> appears in many formulae in [[geometry]] involving [[circle]]s and [[sphere]]s.
In [[2003]] he pointed out that "thealogy" is an obvious coinage that may have been invented many times, and that feminist scholars are unlikely to have been familiar with his writings.
{| border="1" cellspacing="4" cellpadding="4" style="border-collapse: collapse;"
!Geometrical shape
!Formula
|-
|[[Circumference]] of circle of [[radius]] ''r'' and [[diameter]] ''d''
|<math>C = \pi d = 2 \pi r \,\!</math>
|-
|[[area (geometry)|Area]] of circle of radius ''r''
|<math>A = \pi r^2 \,\!</math>
|-
|Area of [[ellipse]] with semiaxes ''a'' and ''b''
|<math>A = \pi a b \,\!</math>
|-
|[[Volume]] of sphere of radius ''r'' and diameter ''d''
|<math>V = \frac{4}{3} \pi r^3 = \frac{1}{6} \pi d^3 \,\!</math>
|-
|[[Surface area]] of sphere of radius ''r''
|<math>A = 4 \pi r^2 \,\!</math>
|-
|Volume of [[cylinder]] of height ''h'' and radius ''r''
|<math>V = \pi r^2 h \,\!</math>
|-
|Surface area of cylinder of height ''h'' and radius ''r''
|<math>A = 2 ( \pi r^2 ) + ( 2 \pi r ) h = 2 \pi r (r + h) \,\!</math>
|-
|Volume of [[cone]] of height ''h'' and radius ''r''
|<math>V = \frac{1}{3} \pi r^2 h \,\!</math>
|-
|Surface area of cone of height ''h'' and radius ''r''
|<math>A = \pi r \sqrt{r^2 + h^2} + \pi r^2 = \pi r (r + \sqrt{r^2 + h^2}) \,\!</math>
|}
=== Growing usage by Carol Christ and Ursula King ===
(All of these are a consequence of the first one, as the area of a circle can be written as
''A'' = ∫(2''πr'')d''r'' ("sum of [[annulus|annuli]] of infinitesimal width"), and others concern a surface or [[solid of revolution]].)
[[Carol Christ]] used the term more substantially in "Laughter of Aphrodite" [[1987]].
Also, the [[angle]] measure of 180° ([[Degree (angle)|degrees]]) is equal to π [[radian]]s.
In [[1989]] [[Ursula King]] notes its growing usage as a fundamental departure from traditional male-oriented theology, characterised by its privileging of symbols over rational explanation. She chronicles sympathetically that-
===Analysis===
Many formulae in [[Mathematical analysis|analysis]] contain π, including [[infinite series]] (and [[infinite product]]) representations, [[integral]]s, and so-called [[List of mathematical functions|special functions]].
:''most writing on the Goddess, when not historical, is either inspirational or devotional, and a systematically ordered body of thought, even with reference to symbols, is only slowly coming into existence.''
*[[François Viète]], [[1593]] ([[Proof of Viète formula|proof]]):
::(1989:126-127)
:<math>\frac2\pi=
\frac{\sqrt2}2
\frac{\sqrt{2+\sqrt2}}2
\frac{\sqrt{2+\sqrt{2+\sqrt2}}}2\ldots</math>
== Further expansion of thealogy by Starr* Saffa ==
*[[Gottfried Leibniz|Leibniz]]' formula ([[Proof of Leibniz formula|proof]]):
:<math>\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \frac{\pi}{4}</math>
:This commonly cited infinite series is usually written as above, but is more technically expressed as:
:<math>\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} = \frac{\pi}{4}</math>
Tahirih Thealogy
*[[Wallis product|Wallis's product]] (see that article for a proof):
:<math> \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \frac{\pi}{2} </math>
: <math>\prod_{n=1}^{\infty} \frac{(2n)^2}{(2n)^2-1} = \prod_{n=1}^{\infty} \frac{2n}{2n-1} \cdot \frac{2n}{2n+1} = \frac{\pi}{2}</math>
The basic Definition of TheAlogy as opposed to Theology means viewing the world incorporating the Female lens which to a great extent in the past has been omitted in Theology.
*1995 Bailey-Borwein-Plouffe algorithm
:<math>\pi=\sum_{k=0}^\infty\frac{1}{16^k}\left [ \frac {4}{8k+1} - \frac {2}{8k+4} - \frac {1}{8k+5} - \frac {1}{8k+6}\right ]</math>
Tahirih TheAlogy is religion beyond religion, politics beyond politics, and spiritual feminism beyond feminism in that it recognizes the Cosmic Christ Spirit in every individual and sets out the pattern of balance for the Sixth Cycle of humanity based on magnetic attraction vs. force and patriarchal constructs.
*An [[integral]] formula from [[calculus]] (see also [[Error function]] and [[Normal distribution]]):
:<math>\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}</math>
During the later part of 2004 Starr* Saffa introduced Tahirih Thealogy and the Tahirih Path in her book entitled “Tahirih Thealogy: Female Christ Spirit of the Age” based on the figure of the 19th Century Iranian born Prophet-Poetess Tahirih who was also known as Qurratu’l-ayn, and the return of Fatima.
*[[Basel problem]], first solved by [[Leonhard Euler|Euler]] (see also [[Riemann zeta function]]):
:<math>\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}</math>
:<math>\zeta(4)= \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots = \frac{\pi^4}{90}</math>
:and generally, <math>\zeta(2n)</math> is a rational multiple of <math>\pi^{2n}</math> for positive integer n
*[[Gamma function]] evaluated at 1/2:
:<math>\Gamma\left({1 \over 2}\right)=\sqrt{\pi}</math>
Tahirih taught that inner knowledge is trumps and Starr* Saffa says Tahirih TheAlogy has the potential to unite East and West where everyone can be living Tahirih’s in this day through the continuous flow of Spirit.
*[[Stirling's approximation]]:
:<math>n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math>
== Definition by Charlotte Caron ==
*[[Euler's identity]] (called by [[Richard Feynman]] "the most remarkable formula in mathematics"):
:<math>e^{i \pi} + 1 = 0\;</math>
In [[1993]] Charlotte Caron's definition of thealogy as "reflection on the divine in feminine and feminist terms" appeared in "To Make and Make Again" (quoted from Russell & Clarkson 1996). By this time the concept had gained considerable (though conventionally marginal) status, broadly analogous to Ruether's view of radical feminist theology as opposed to reformist [[feminist theology]].
*Property of [[Euler's totient function]] (see also [[Farey sequence]]):
:<math>\sum_{k=1}^{n} \phi (k) \sim 3 n^2 / \pi^2</math>
=== Melissa Raphael's view ===
*Area of one quarter of the unit circle:
:<math>\int_0^1 \sqrt{1-x^2}\,dx = {\pi \over 4}</math>
In [[1997]] [[Melissa Raphael]] wrote "Thealogy & Embodiment" which put the usage firmly on the map, and which she sustained in her subsequent "Thealogy: Discourse on the Goddess" ([[1999]]?). Together with Carol Christ's "Rebirth of the Goddess" 1997 Raphael's work provides a start for the "systematically ordered body of thought" King found lacking in 1989.
*An application of the [[residue theorem]]
:<math>\oint\frac{dz}{z}=2\pi i ,</math>
:where the path of integration is a circle around the origin, traversed in the standard (anti-clockwise) direction.
== Three interpretations of thealogy ==
===Continued fractions===
π has many [[continued fraction]]s representations, including:
There are perhaps three distinct interpretations of thealogy, and they are evident in the briefing above.
:<math> \frac{4}{\pi} = 1 + \frac{1}{3 + \frac{4}{5 + \frac{9}{7 + \frac{16}{9 + \frac{25}{11 + \frac{36}{13 + ...}}}}}} </math>
*Christ, King and Raphael focus thealogy specifically on [[Goddess]] spirituality.
(Other representations are available at [http://functions.wolfram.com/Constants/Pi/10/ The Wolfram Functions Site].)
*Caron defines a broader field of a female worldview of the [[sacred]].
*Goldenberg's neologism as a political stance that marks the [[androcentrism]] of historical [[theology]] permeates the other two and raises its own issues.
=== Thealogy as Goddess spirituality ===
===Number theory===
Some results from [[number theory]]:
*The [[probability]] that two [[random]]ly chosen integers are [[coprime]] is 6/π<sup>2</sup>.
Taking the Goddess variant first, and it seems the commonest to the point where thealogy is typically assumed to be purely Goddess based, a linguistic derivation from the Greek "thea"
*The probability that a randomly chosen integer is [[square-free]] is 6/π<sup>2</sup>.
(goddess). Goddess systematics inevitably face the question of "god in a skirt" or not, a subtly [[sexism|sexist]] tag that nonetheless carries a genuine issue. This can be viewed as sexist because "in a skirt" defines a subject norm as altered, trivialised, and definitely derivative, much as some have considered the female to have been historically defined in relation to the male. Thealogy specifically aims to counter what its proponents perceive as the massive [[dualism|dualistic]] sexism in the field of religion, by asserting a female [[worldview]] that is not merely reformist or derivative, so its proponents would see this quip as especially destructive.
=== Broad interpretation of thealogy (Caron) ===
*The [[mean|average]] number of ways to write a positive integer as the sum of two [[perfect square]]s (order matters) is π/4.
Caron's definition "Reflection on the divine in feminine and feminist terms" holds a caution for feminist theologians and thealogians alike that the female sacred extends beyond the feminist agenda. Often theology or feminist thealogy writes as if the Goddess is a feminist discovery. The "womenspirit" Goddess is a highly selected deity who for thealogians such as Christ has nothing to do with goddess practices such as violent sacrifice, or validating a male conqueror. However, this can be seen to be as inauthentic as the habit of some Christians of disowning the [[Inquisition]] as "not done by real Christians" (see the "[[no true Scotsman]]" [[logical fallacy]]).
* The [[product]] of (1-1/p<sup>2</sup>) over the primes, ''p'', is 6/π<sup>2</sup>.<math> \prod_{p\in\mathbb{P}} \left(1-\frac {1} {p^2} \right) = \frac {6} {\pi^2} </math>
Nor is it a matter only of past history: many members of a huge international organisation like the [[Fellowship of Isis]] would not identify as feminist, nor would a great many [[Pagan]]s. Outside the goddessing of western [[New religious movement|NRMs]] thealogy can recognise and give due respect to the world millions in village and tribal religions who look to goddesses in ways that may or may
Here, "probability", "average", and "random" are taken in a limiting sense, e.g. we consider the probability for the set of integers {1, 2, 3,..., ''N''}, and then take the [[limit (mathematics)|limit]] as ''N'' approaches infinity.
not be feminist, and Caron's definition allows thealogy to be this widely inclusive.
This broader view accords well with the kind of fluid systematics profiled by [[Cynthia Eller]] when she reports her respondent [[Margaret Keane]] as saying:
The remarkable fact that
: <math>e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007...</math>
or equivalently,
: <math>e^{\pi \sqrt{163}} = 640320^3+743.99999999999925007...</math>
can be explained by the theory of [[complex multiplication]].
:''I don't make those kind of distinctions that you hear about, they don't make any sense to me. You can say it's the Great Goddess, and that's the one Goddess, but she's also all of the many goddesses, and that's true. And she's everywhere. She's immanent in everything, in the sparkle of the sun on the sea, and even in an animistic concept. I think certain objects can embody that force and power. So I worship the Great Goddess, and I'm polytheistic and pantheistic and monotheistic too. And I also have a feeling for nature spirits...''
===Dynamical systems and ergodic theory===
::(1993 :132-133)
Consider the [[recurrence relation]]
:<math>x_{i+1} = 4 x_i (1 - x_i) \,</math>
Then for [[almost everywhere|almost every]] initial value ''x''<sub>0</sub> in the [[unit interval]] [0,1],
:<math> \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} \sqrt{x_i} = \frac{2}{\pi} </math>
This recurrence relation is the [[logistic map]] with parameter ''r'' = 4, known from [[dynamical system]]s theory. See also: [[ergodic theory]].
This broader view has most recently been labelled by [[Michael York]] as "polymorphic thealogy." He also raises the issue of whether thealogy venerates one Goddess or many, which some thealogicians consider a non-question since it arises from a monotheist worldview that they do not hold.
===Physics===
In [[physics]], appearance of π in formulae is usually only a matter of convention and normalization. For example, by using the reduced [[Planck's constant]] <math> \hbar = \frac{h}{2\pi} </math> one can avoid writing π explicitly in many formulae of quantum mechanics. In fact, the reduced version is the more fundamental, and presence of factor ''1/2π'' in formulas using ''h'' can be considered an artifact of the conventional definition of Planck's constant.
However Caron's definition falls short of explicitly allowing for male positions in thealogy.
*[[Uncertainty principle|Heisenberg's uncertainty principle]]:
:<math> \Delta x \Delta p \ge \frac{h}{4\pi} </math>
*[[Einstein's field equation]] of [[general relativity]]:
:<math> R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik} </math>
*[[Coulomb's law]] for the [[electric force]]:
:<math> F = \frac{\left|q_1q_2\right|}{4 \pi \epsilon_0 r^2} </math>
*[[Permeability (electromagnetism)|Magnetic permeability of free space]]:
:<math> \mu_0 = 4 \pi \times 10^{-7}\,\mathrm{H/m}\,</math>
=== A challenge to androcentrism ===
===Probability and statistics===
In [[probability]] and [[statistics]], there are many [[probability distribution|distributions]] whose formulae contain π, including:
*[[probability density function]] (pdf) for the [[normal distribution]] with [[mean]] μ and [[standard deviation]] σ:
The third interpretation of thealogy as an assertion of female sacred worldviews is clearly political. The notes above touch on how this usage aims to counter the deeply established dualistic relegation of female as derivative, making the male the norm: as [[Mary Daly]] put it "If God is male, then the male is God."
:<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}</math>
*pdf for the (standard) [[Cauchy distribution]]:
Thealogy has been criticised as [[essentialism|essentialist]] by [[queer theory|queer theorists]] and others.
:<math>f(x) = \frac{1}{\pi (1 + x^2)}</math>
To a thealogian it is important to explore the female worldview (not only but notably of the sacred) and not be compelled to take off female spectacles when looking at themes beyond female [[psychobiology]]. A speaker may choose to adopt a kind of gender neutral stance insofar as she can, or she may try to empathise with a male worldview, and a male speaker vice versa.
Note that since <math>\int_{-\infty}^{\infty} f(x)\,dx = 1</math>, for any pdf ''f''(''x''), the above formulae can be used to produce other integral formulas for π.
== Linguistic twiddling ==
An interesting empirical approximation of π is based on [[Buffon's needle]] problem. Consider dropping a needle of length ''L'' repeatedly on a surface containing parallel lines drawn ''S'' units apart (with ''S'' > ''L''). If the needle is dropped ''n'' times and ''x'' of those times it comes to rest crossing a line (''x'' > 0), then one may approximate π using:
:<math>\pi \approx \frac{2nL}{xS}</math>
Many scholars find the term "thealogy" exasperating, a linguistic twiddling, including some feminist theologians. But the position of women operating within the male worldview of theology, as in most of [[feminist theology]], is more marginal than in the general run of professional occupations these days. The rigidly entrenched sexism in the contemporary academy perceived by some thealogs recalls situations of general Women's Liberation in 1972, rather than society 30 years later (see recent research studies Ofsted UK).
== History of π ==
==See also==
''Main article: [[History of Pi]]''.
*[[God and gender]]
*[[feminist theology]]
π has been known in some form since antiquity. References to measurements of a circular basin in the [[Bible]] give a corresponding value of 3 for π: "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." — [[1 Kings]] 7:23; KJV.
*[[goddess]]
*[[goddess worship]]
Nehemiah, a [[Late Antiquity|late antique]] Jewish rabbi and mathematician explained this apparent lack of precision in π, by considering the thickness of the basin, and assuming that the thirty cubits was the inner circumference, while the ten cubits was the diameter of the outside of the basin.
== Numerical approximations of π ==
Due to the transcendental nature of π, there are no closed expressions for the number in terms of algebraic numbers and functions. Therefore numerical calculations must use [[approximation]]s of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 [[significant figures]]) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple [[continued fraction]] expansion of π.
An Egyptian scribe named [[Ahmes]] wrote the oldest known text to give an approximate value for π. The [[Moscow and Rhind Mathematical Papyri|Rhind Mathematical Papyrus]] dates from the [[Ancient Egypt|Egyptian]] [[Second Intermediate Period]]—though Ahmes stated that he copied a [[Middle Kingdom of Egypt|Middle Kingdom]] [[papyrus]]—and describes the value in such a way that the result obtained comes out to 256 divided by 81 or 3.160.
The Chinese mathematician [[Liu Hui]] computed π to 3.141014 (good to three decimal places) in AD [[263]] and suggested that 3.14 was a good approximation.
The Indian mathematician and astronomer [[Aryabhata]] gave an accurate approximation for π. He wrote "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words (4+100)×8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of π = 62832/20000 = 3.1416, correct when rounded off to four decimal places.
The Chinese mathematician and astronomer [[Zu Chongzhi]] computed π to 3.1415926 to 3.1415927 and gave two approximations of π 355/113 and 22/7 in the [[5th century]].
The Iranian mathematician and astronomer, [[Ghyath ad-din Jamshid Kashani]], 1350-1439, computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:
:2 π = 6.2831853071795865
The German mathematician [[Ludolph van Ceulen]] (''circa'' [[1600]]) computed the first 35 decimals. He was so proud of this accomplishment that he had them inscribed on his [[tomb stone|tombstone]].
The Slovene mathematician [[Jurij Vega]] in [[1789]] calculated the first 140 decimal places for π of which the first 137 were correct and held the world record for 52 years until [[1841]], when [[William Rutherford]] calculated 208 decimal places of which the first 152 were correct. Vega improved [[John Machin]]'s formula from [[1706]] and his method is still mentioned today.
None of the formulas given above can serve as an efficient way of approximating π. For fast calculations, one may use formulas such as [[John_Machin|Machin's]]:
: <math>\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} </math>
together with the [[Taylor series]] expansion of the function [[arctan]](''x''). This formula is most easily verified using [[polar coordinates]] of [[complex number]]s, starting with
:<math>(5+i)^4\cdot(-239+i)=-114244-114244i.</math>
Formulas of this kind are known as ''[[Machin-like formula]]s''.
Extremely long decimal expansions of π are typically computed with the [[Gauss-Legendre algorithm]] and [[Borwein's algorithm]]; the [[Salamin-Brent algorithm]] which was invented in [[1976]] has also been used in the past.
The first one million digits of π and 1/π are available from [[Project Gutenberg]] (see external links below).
The current record (December [[2002]]) by [[Yasumasa Kanada]] of [[Tokyo University]] stands at 1,241,100,000,000 digits, which were computed in September [[2002]] on a 64-node [[Hitachi (company)|Hitachi]] [[supercomputer]] with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulas were used for this:
:<math> \frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}</math>
:K. Takano ([[1982]]).
: <math> \frac{\pi}{4} = 44 \arctan\frac{1}{57} + 7 \arctan\frac{1}{239} - 12 \arctan\frac{1}{682} + 24 \arctan\frac{1}{12943}</math>
:F. C. W. Störmer ([[1896]]).
These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers and (obviously) for establishing new π calculation records.
In [[1996]], [[David H. Bailey]], [[Peter Borwein]] and [[Simon Plouffe]] published a paper on a new formula for π as an [[infinite series]]:
: <math>\pi = \sum_{k = 0}^{\infty} \frac{1}{16^k}
\left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right)</math>
This formula permits one to easily compute the ''k''<sup>th</sup> [[Binary numeral system|binary]] or [[hexadecimal]] digit of π, without
having to compute the preceding ''k'' − 1 digits. [http://www.nersc.gov/~dhbailey/ Bailey's website] contains the derivation as well as implementations in various [[programming language|programming languages]]. The [[PiHex]] project computed 64-bits around the [[quadrillion]]th bit of π (which turns out to be 0).
Other formulas that have been used to compute estimates of π include:
:<math>
\frac{\pi}{2}=
\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}=
1+\frac{1}{3}\left(1+\frac{2}{5}\left(1+\frac{3}{7}\left(1+\frac{4}{9}(1+...)\right)\right)\right)
</math>
:[[Isaac Newton|Newton]].
:<math> \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} </math>
:[[Ramanujan]].
This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π.
:<math> \frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}} </math>
:[[David Chudnovsky (mathematician)|David Chudnovsky]] and [[Gregory Chudnovsky]].
: <math>{\pi} = 20 \arctan\frac{1}{7} + 8 \arctan\frac{3}{79} </math>
:[[Euler]].
On computers running [[Microsoft Windows]] [[Operating_system|OS]], the program [[PiFast]] can be used to quickly calculate a large amount of digits. The largest number of digits of π calculated on a home computer, 25,000,000,000, was calculated with PiFast in 17 days.
===Miscellaneous formulas===
In [[radix|base]] 60, π can be approximated to eight significant figures as
:<math> 3 + \frac{8}{60} + \frac{29}{60^2} + \frac{44}{60^3}</math>
In addition, the following expressions can be used to estimate π
* accurate to 9 digits:
:<math>(63/25)((17+15\sqrt 5)/(7+15\sqrt5))</math>
* accurate to 17 digits:
:<math>3 + \frac{48178703}{340262731}</math>
* accurate to 3 digits:
:<math>\sqrt{2} + \sqrt{3}</math>
:[[Karl Popper]] conjectured that [[Plato]] knew this expression; that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of [[right triangle]]s which are either [[isosceles]] or halves of [[equilateral]] triangles.
===Less accurate approximations===
In 1897, a physician and amateur mathematician from [[Indiana]] named [[Edward J. Goodwin]] believed that the [[transcendental number|transcendental]] value of π was wrong. He proposed a bill to Indiana Representative [[T. I. Record]] which expressed the "new mathematical truth" in several ways:
:''The ratio of the diameter of a circle to its circumference is 5/4 to 4.'' (π = 3.2)
:''The ratio of the length of a 90 degree arc to the length of a segment connecting the arc's two endpoints is 8 to 7.'' (π ≈ 3.23...)
:''The area of a circle equals the area of a square whose side is 1/4 the circumference of the circle.'' (π = 4)
:''It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side.'' (π ≈ 9.24 if ''rectangle'' is emended to ''triangle''; if not, as above.)
The bill also recites Goodwin's previous accomplishments: "his solutions of the [[trisection of the angle]], [[doubling the cube]] [and the value of π] having been already accepted as contributions to science by the [[American Mathematical Monthly]]....And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man's ability to comprehend." These false claims are typical of a mathematical [[crank (person)|crank]]. The claims trisection of an angle and the doubling of the cube are particularly widespread in crank literature.
The Indiana [[Assembly]] referred the bill to the Committee on Swamp Lands, which [[Petr Beckmann]] has seen as symbolic. It was transferred to the Committee on Education, which reported favorably, and the bill passed unanimously. One argument used was that Goodwin had copyrighted his discovery, and proposed to let the State use it in the public schools for free. As this debate concluded, Professor [[C. A. Waldo]] arrived in [[Indianapolis]] to secure the annual appropriation for the [[Indiana Academy of Sciences]]. An assemblyman handed him the bill, offering to introduce him to the genius who wrote it. He declined, saying that he already knew as many crazy people as he cared to.
The Indiana Senate had not yet finally passed the bill (which they had referred to the Committee on Temperance), and Professor Waldo coached enough Senators overnight that they postponed the bill indefinitely. [http://faqs.jmas.co.jp/FAQs/sci-math-faq/indianabill source]
== Open questions ==
The most pressing open question about π is whether it is a [[normal number]] -- whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly". This must be true in any base, not just in base 10. Current knowledge in this direction is very weak; e.g., it is not even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of π.
Bailey and Crandall showed in [[2000]] that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can be reduced to a plausible [[conjecture]] of [[chaos theory]]. See Bailey's above mentioned web site for details.
It is also unknown whether π and [[E (mathematical constant)|''e'']] are [[algebraically independent]], i.e. whether there is a polynomial relation between π and ''e'' with rational coefficients.
[[John Harrison]], (1693-1776) (of Longitude fame), devised a [[meantone temperament]] musical tuning system derived from π. This [[Lucy Tuning]] system (due to the unique mathematical properties of π), can map all musical intervals, harmony and harmonics. This suggests that musical harmonics beat, and that using π could provide a more precise model for the analysis of both musical and other harmonics in vibrating systems.
== The nature of π ==
In [[non-Euclidean geometry]] the sum of the angles of a [[triangle (geometry)|triangle]] may be more or less than π [[radians]], and the ratio of a circle's circumference to its diameter may also differ from π. This does not change the definition of π, but it does affect many formulae in which π appears. So, in particular, π is not affected by the [[shape of the universe]]; it is not a [[physical constant]] but a mathematical constant defined independently of any physical measurements. The reason it occurs so often in physics is simply because it's convenient in many physical models.
For example, consider [[Coulomb's law]]
:<math> F = \frac{1}{ 4 \pi \epsilon_0} \frac{\left|q_1 q_2\right|}{r^2} </math>.
Here, 4''πr''<sup>2</sup> is just the surface area of sphere of radius ''r''. In this form, it is a convenient way of describing the inverse square relationship of the force at a distance ''r'' from a point source. It would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient. If [[Planck charge]] is used, it can be written as
:<math> F = \frac{q_1 q_2}{r^2} </math>
and thus eliminate the need for π.
== Fictional references ==
* ''[[Contact (novel)|Contact]]'' -- [[Carl Sagan|Carl Sagan's]] [[science fiction]] work. Sagan contemplates the possibility of finding a signature embedded in the [[Positional notation|base-11]] expansion of Pi by the creators of the universe.
* ''[[Pi (film)|π (film)]]'' -- On the relationship between numbers and nature: finding one without being a [[numerologist]].
* ''[[Time's Eye]]'' -- [[science fiction]] by [[Arthur C. Clarke]] and [[Stephen Baxter]]. In a world restructured by alien forces, a spherical device is observed whose circumference to diameter ratio appears to be an exact integer 3 across all planes. T
* ''[[The Simpsons]]'' -- "Pi is exactly 3!" was an announcement used by [[Professor Frink]] to gain the full attention of a hall full of scientists.
== π culture ==
There is an entire field of humorous yet serious study that involves the use of [[mnemonic technique]]s to remember the digits of π, which is known as [[piphilology]]. See [[:q:English_mnemonics#Pi|Pi mnemonics]] for examples.
[[March 14]] (3/14 in [[US]] date format) marks [[Pi Day]] which is celebrated by many lovers of π.
On [[July 22]], [[Pi Approximation Day]] is celebrated (22/7 - in European date format - is a popular approximation of π).
In the early hours of Saturday [[2 July]], [[2005]], a Japanese mental health counsellor, Akira Haraguchi, 59, managed to recite π's first 83,431 decimal places from [[memory]], thus breaking the standing world record [http://news.bbc.co.uk/1/hi/world/asia-pacific/4644103.stm].
355/113 (~3.1415929) is sometimes jokingly referred to as "not π, but an incredible simulation!"
== Related articles ==
*[[List of topics related to pi]]
*[[Pi (letter)|Greek letter pi]]
*[[Calculus]]
*[[Geometry]]
*[[Trigonometric function]]
*[[Pi through experiment]]
*[[Lindemann-Weierstrass theorem|Proof that π is transcendental]]
*[[Proof that 22/7 exceeds pi|A simple proof that 22/7 exceeds π]]
*[[Feynman point]]
*[[Pi Day]]
*[[Lucy Tuning]]
*[[Cadaeic Cadenza]]
==References==
* Isaac Bonewits "The Second Epistle of Isaac" in "the Druid Chronicles (Evolved)" Berkeley Drunemeton Press, 1974.
*[[Petr Beckmann]], ''A History of Pi''
*Isaac Bonewits "Real Magic" Creative Arts Book Co., 1979
*Charlotte Caron "To Make and Make Again: Feminist Ritual Thealogy" NY Crossroad 1993
== External links ==
*Carol Christ "Rebirth of the Goddess:Finding meaning in feminist spirituality" Routledge 1997
*Cynthia Eller "Living in the Lap of the Goddess: The Feminist Spirituality Movement in America" Crossroad 1993
===Digit resources===
*Naomi Goldenberg "The Changing of the Gods" 1979
*[http://www.gutenberg.net/etext/50 Project Gutenberg E-Text containing a million digits of Pi]
*Ursula King "Women and Spirituality" Macmillan 1989
*[http://3.141592653589793238462643383279502884197169399375105820974944592.com/ Pi to a million places]
*Melissa Raphael "Thealogy & Embodiment" 1997 Sheffield Academic Press
*[http://www.solidz.com/pi/ Archives of Pi calculated to 1,000,000 or 10,000,000 places.]
*Melissa Raphael "Introducing Thealogy: Discourse on the Goddess" 1999 Sheffield Academic Press
*[http://www.pisearch.de.vu Search π] – search and print π's digits (up to 3.2 billion places)
*Letty M. Russell & J Shannon Clarkson "Dictionary of Feminist Theologies" Mowbray 1996.
*[http://www.super-computing.org/pi-decimal_current.html Statistics about the first 1.2 trillion digits of Pi]
*Starr* Saffa "Tahirih Thealogy: Female Christ Spirit of the Age" OzForUs Publishing 2004; Zeus-publications 2005.
*[http://3.14.maxg.org/ A banner of approximately 220 million digits of pi]
*Valerie Saiving "Androcentrism in Religious Studies" in Journal of Religion 56:1976:177-97
===Calculation===
*[http://projectpi.sourceforge.net/ Calculating Pi: The open source project for calculating Pi.]
*[http://backpi.sourceforge.net Background Pi: An open source project for calculating Pi over many computers. (Inspired by "Calulating Pi", Above)]
*[http://numbers.computation.free.fr/Constants/PiProgram/pifast.html PiFast: a fast program for calculating Pi with a large number of digits]
*[http://www.cecm.sfu.ca/projects/pihex/index.html PiHex Project]
*[http://files.extremeoverclocking.com/file.php?f=36 Super Pi: Another program to calculate Pi to the 33.55 millionth digit. Also used a benchmark]
*[http://www.pislice.com/ PiSlice: A distributed computing project to calculate Pi]
*[[wikisource:Calculating the digits of pi|Calculating the digits of π using generalised continued fractions]] - open source [[Python programming language|Python]] code
===General===
*[http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Pi_through_the_ages.html J J O'Connor and E F Robertson: ''A history of Pi''. Mac Tutor project]
*[http://machination.mysite.freeserve.com/ A collection of Machin-type formulas for Pi]
*[http://www.lrz-muenchen.de/~hr/numb/pi-irr.html A proof that Pi Is Irrational]
*[http://www.joyofpi.com/pifacts.html PiFacts-Record Broken]
*[http://www.joyofpi.com/thebook.html The Joy of Pi-About the Book]
*[http://mathworld.wolfram.com/PiFormulas.html From the Wolfram Mathematics site lots of formulae for π]
*[http://www.pisymphony.com/gpage.html Pi Symphony : An orchestral work by Lars Erickson based on the digits of pi and 'e'.]
*[http://planetmath.org/encyclopedia/Pi.html PlanetMath: Pi]
*[http://groups.yahoo.com/group/pi-hacks The pi-hacks Yahoo! Group]
*[http://mathforum.org/isaac/problems/pi1.html Finding the value of Pi]
*[http://cf.geocities.com/ilanpi/pi-exists.html Proof that Pi exists]
*[http://pi314.at/ Friends of Pi Club ''(German and English)'']
*[http://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml Determination of Pi]
*[http://www.lucytune.co.uk LucyTuning - musical tuning derived from Pi]
*[http://dse.webonastick.com/pi/ The Pi Is Rational Page]
===Mnemonics===
*[http://users.aol.com/s6sj7gt/mikerav.htm One of the more popular mnemonic devices for remembering pi]
*[http://www.cilea.it/~bottoni/www-cilea/F90/piph.htm Andreas P. Hatzipolakis: ''PiPhilology''. A site with hundreds of examples of π mnemonics]
*[http://www.startfromhere.freeserve.co.uk/nudesci/abc/pi.htm Pi memorised as poetry]
*[http://bangalore.sancharnet.in/viveknayak/riddles.htm Phrase to easily remember upto 8 decimal places of the value of Pi (See Item #3 on page)]
*[http://brianbondy.com/other/pi.aspx Free software to help memorise Pi]
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