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NEEDLE PROBLEM

Written by G. T. Hushion. Posted in Exploring Random

"You can't calculate possibilities with just algebra. The geometry must be taken into account."

Comte George Buffon, Essay on Moral Arithmetic.

 

Various terms and words used by mathematicians to describe the Needle have included: "unique", "charming", "quaint", "arresting" and "of the greatest notoriety."

The Needle was the first means to statistically determine pi (Snell, J. Laurie, Introduction to Probability. p.57 Dartmouth College. Random House/Birkhauser Mathematic Series, 1988).

From his Needle, Buffon is credited with an independent discovery of the binomial theorem (Gillespie, Charles Coulston, ed. dictionary of Scientific Biography. Vol ii, p.577, Scribner, 1970).

Buffon and his Needle are also credited with the first practical use of the "Monte Carlo Method" of probability analysis (Buslenko, N.P., D.I. Golenko, Yu. A. Shreider, I.M. Sobol, and V.G. Sragovich. The Monte Carlo Method. The Method of Statistical Trials. p.4,  Ed. Yu. A. Shreiver. Trans. G.J. Tee. Pergamon Press, 1966).

As well. from the composition of his Needle, Buffomd is credited with the first use of calculus to determine probabilities (Levin, S., Mng. ed. Lecture Notes in Biomathematics: Geometrical Probability and Biological Structures: Buffon's 200th Anniversary, Proceedings. Ed. by R.E. Miles and J. Serra. Proceedings. Paris, 1977).

Buffon's Needle Problem is considered history's first mathematical statement of geometric probability (see: Levin, above, p.29).

The Needle was the introduction of local geometric probability (Cajori, Florian. A History of Mathematics. Chelsea. 3rd ed., p.244, 1980).

The original Buffon Needle Problem cannot generally be found in modern scientific literature other than in this web site and the book cited above: "Geometrical Probability and Biological Structures: Buffon's 200th Anniversary, Springer-Verlag."

Even in Springer-Verlag, other than a quick look at the original Needle, the numerous studies of the Needle that are presented there do not reflect the original Needle, but rather Simon Laplace's warp of the Needle.

The numerous variations of the original Needle that are widely available and promoted on the web as the "Buffon Needle Problem" are the antithesis of the original Needle. They are based on the mathematical fraud perpetrated by Simon Laplace in 1812. In his intentionally misnamed book on "probabilities," Laplace disrespected the original Needle's random length. He disingenuously and maliciously substituted in a needle with the geometrically meaningless length of: any arbitrary given length.

The original Needle's length of relative 1/4 pi, relative to the diameter of the field being randomly measured, is gravity's own statement of geometric probability relative to the randomness of gravity. It is an immutable universal average measurement.

On the other hand, Laplace's variation is without random geometry. It is simply "random" algebra. It is simply an algebraic statement describing any arbitrary length of needle. It is pure algebra for its own sake. It is meaninglessly relative to life's perceptions and is without regard to the geometry of the universal random average.

Laplace is most famous for the "Laplace Transform." It is simply the reverse of his warp of the original Needle.

Laplace kept the original Needle out of public education. That effectively buried the original Needle. Why did Laplace effectively conceal the original Needle and replace it with a fraud?

The answer is simple but deeply buried in the circumstances leading to the French Revolution's Terror ...of which Laplace now appears to have been the architect. The reason Laplace needed the original Needle out of sight was because the history of its truth would have exposed him as a fraud. To avoid that embarrassment, he used Robespierre and Joseph Fouche to murder four specific men. The coverups were the mass killings that constituted the "Terror." There is a more detailed discussion in the History Sections of this site.

Georges Buffon (1707 – 1788) initially presented his Needle Problem to the Paris Academy of Sciences in 1733. In 1734, it was first published in a minor journal intended for worthy work of non Academy members. It was published again in a minor journal in 1776. Its first major popular publication was in 1777, in supplement (4) to Buffon's Histoire Naturelle.

The original Needle introduced geometric probability as it delivered the first random proof of pi. The geometric nature came from proving relative 1/4 pi, relative to the field's diameter, as both 1/4 of a circle (that is: 1/4 C) and, simultaneously, the universal random average: relative 1/4 pi.

The original Needle proved 1/4 pi to be a percentage of the diameter: .78539.... . Therefore, a randomly measured diameter has a random gravitational value: "1."

In delivering the universal random average, the original Needle introduced geometric probability. It also introduced the methodology of serial random measurements. Since WWII, this methodology has generally been known as: "Monte Carlo Methodology."

Buffon originally used a baguette (French loaf) on a plank floor. He suggested a needle on a checkerboard would be more convenient. The name understandably stuck.

The original Needle asks: on a series of equidistant parallel lines, if two players want an even chance a randomly dropped needle will cross a line, how long must the needle be?

“Je suppose que dans une chambre, dont le parquet est simplement divise par des joints paralleles, on jette en l’air une baguette, & que l’un des joucurs parie que la baguette ne croisera aucune des paralleles du parquet, & que l’autre au contraire parie que la baguette croisera quelques-unes de ces paralleles; on demande le sort de ces deux joucurs.”

Buffon concluded that the needle must be approximately 3/4 of the diameter distance (shortest possible distance) between the lines. More precisely, that percentage: .78539.... . That is: relative 1/4 pi, relative to the diameter.

“...que la longueur de la baguette doit saire a peu-pres les trios quarts de la distance des joints du parquet.”

Buffon also identified that percentage as one fourth of the circle described between and just touching two adjacent lines. That is: 1/4 C.

“...quart de la circonference du cercle don’t la longue de la baguette est le diameter...”

A unique and fascinating feature of the original Needle is that it readily and deductively demonstrates that, in the first instance, every series of random measurements --of anything-- automatically turns the circle or field or game or object (of any shape) into a dimensional game of relative pi in rotation. More specifically, into a series of random measurements tending to average 1/4 pi each (see: A Proof of String Theory, Pi and Mechanics).

The original Needle proves a field or circle or game’s diameter (or "pi-angle") has a random mathematical value: “1.”. Therefore, deductively, the radius of the diameter has a random mathematical value: .50 .

It is here that the original Needle makes its random point. It deductively and inferentially proves the relative cross diameter (the dimension of relative “width”) has a random mathematical value of relative pi in rotation. An examination of the relativity is found by making 1/4 pi relative to the Center of Rotation. This is effectuated by dividing 1/4 pi into the cross radius of 1/2 pi. This may also be found by dividing the metric length of a quadrant (the original Needle) by the metric length of the cross radius. The makes the universal random average relative to the COR of the game or field being measured.

By dividing the radius into the quadrant, the outcome is the mathematical value of the relative cross radius for every series of random measurements: relative 1/2 pi. This appears to hold true for any randomly measured field.

The original Needle coincidentally (since it apparently appeared a year before Boskovic's first apparent use of "action at a distance") supports Boskovic’s methodology of a geometric finesse for predicting the orbits of comets. Since the original Needle’s deductions and inferences invite the finesse, it is quite possible the original Needle inspired Boskovic. As noted in the history section, in 1733, Boskovic was teaching mathematics in the educational hierarchy of the Colegio Romano. He may have been asked to evaluate the Needle for the Jesuit's consideration. Both the original Needle and "action at a distance" are traceable to Newton. In 1733, "actio in distans" was already suppressed by the church.

The heart and soul of Laplace’s work was based on the random quadrature he usurped from the original Needle. It appears he may have been quietly handed the Needle by Buffon and Condorcet in 1770. It appears Laplace just as quietly used the Needle’s point (without mention or reference of the Needle or Buffon) in 1772, when he announced he had “discovered” that the “second degree of every equation lies in quadrature.” That, of course, is the point of the original Needle. This history indicates Laplace's announcement concerning quadrature was almost surely part of the set up to use him. That is, quadrature is fairly easy to understand. Having him make the announcement would help establish his credibility. With credibility established as the "greatest mathematician in France," Laplace could introduce "action at a distance" and make it credible. His professed status as an atheist would leave him immune from religious discipline. Since all close examinations of the original Needle naturally lead to "action at a distance," Buffon could then reintroduce his Needle Problem and it could be exposed to open examination.

It is a point of this history that Laplace never discovered anything. Virtually everything he published, with the possible exception of a minor (and disastrous) study of tides, appears to have arguably been stolen from others, mainly from the scientists he had murdered in the Terror.

“Amongst the minor discoveries of Laplace in pure mathematics I may mention his discussion (simultaneously with Vandermonde) of the general theory of determinants in 1772; his proof that every equation of an even degree must have at least one real quadratic factor….” (Ball, Rouse. A Short History of Mathematics. Macmillan, 1908. p.419).

When Buffon unexpectedly published the Needle, in both 1776 and 1777, in the middle of the Laplace/Boskovic “debate” over “action at a distance,” the geometric truth surely started its inexorable atomic fizz. It is here worth noting that Boskovic is the father of atom theory. Buffon, who was the driving force behind Laplace, was the first to conceptualize "quantum."

There for the taking was a .08333 flat bet advantage (or .16666 depending on the "game") over the very traditional random theory of quadrature that Laplace was advocating …and that would appear two centuries later as the same flat bet advantage in and of the Quantum sciences.

The flat bet advantage is the randomly measured geometric difference between a circle and diameter. In the Boskovic/Laplace debate, relative to the mean inclination of comets, relative to the totality of the field (360 possible degrees) it is also the difference between 45 degrees and 60 degrees, factored by two possible directions. That is: 60 – 45 = 15. Next: 15 /360 = .08333…. . Next: 2 (.08333) = .16666 .

It is the difference between our perception of randomly finding one of four algebraic poles on a circle or orbit, each with a .25 algebraic possibility …and gravity's eternal random delivery of one of three geometric poles on the circle or orbit’s diameter, each with a geometric probability of .33333, factored by two directions.

That is: 2 (.33333 - .25) = .16666 .

The same random advantage is found on a semi circle as discussed within.

As is the subject of this book, an arc of 60 degrees is not only .16666 of a circle, it is the geometric probability, over three random measurements, of a randomly measured field’s relative pi angle pole. Therein is the flat bet advantage: .16666.... . Again, it is only found with the methodology of "action at a distance."

In short, the original Needle structures the algebra of a randomly measured circle as the average of four Cardinal poles of 90 degrees each. To define the circle, just multiply a quadrant (length of the original Needle or Cardinal pole) by 4. This is the fundamental algebra on which the odds of every random table game are based.

Shorter yet, with a few fresh words to help out, when the original Needle is extended with “action at a distance,” the algebra of a randomly measured circle of four poles is geometrically changed to six poles of 60 degrees each. Each pole is .16666 of circle or "game." To mathematically define a circle (which is just a random statement of algebraic possibilities) after using "action at a distance," just algebraically multiply the randomly found arc of a geometric pi angle pole by 6.

That too, is just an appearance of the circle relative to the pi-angle pole.

Shortest of all, with some additional necessary explanation: ...when the same series of random events is made relative to the complete pi-angle pole (or: "diameter") with "action at a distance," relative random values change again. Deductively, the first random event in the series is geometrically structured in a pole of 1/12 of the wheel. On a 38 pocket wheel, that would be 3.16666.... pockets. Let a randomly released ball land anywhere. Let it be pocket 23 for example.

Relative to the game of and on a circle, the ball just landed in pocket 23.

As this entire website and discussion indicates, pocket 23 was an event on one of three pole/pockets of geometric probability on the circle or game's diameter. In the first instance of randomness a diameter has three poles: one end/Center of Rotation/other end. That gives pocket 23 an apparent initial random value of geometric probability: .33333.... .

Since it is the first in a series and the relativity cannot yet be established in the first of a series, pocket 23 also retains its random value on a circle: .25 as a quadratic pole. As well, since the circle has yet to be mathematically eliminated by the geometric finesse of "action at a distance," the circle statistically retains its algebraic influence.

Therefore, relative to perception relative to the randomness of gravity, the first random event in a series has a random geometric value: 1/12 C. That is: (.33333) .25 = .08333. Or, just as accurately, 1/12 pi.

Relative to the geometric probabilities delivered by "action at a distance," a random roulette ball lands in a "pole/pocket" that is 3.16666 units wide. This is the first ball in a series of three. It is not predictable since, mathematically, it is not yet geometrically identified by relativity ...and it is only through relativity (the relativity only found with "action at a distance") that prediction of randomness is possible. However, just because it was not mathematically (through relativity) identified as an event on the diameter, doesn't mean it wasn't a geometric event happening on the diameter. Therefore, without the geometric finesse of "action at a distance" (yet to be applied in events 2 and 3 of the series) the random event of a ball landing in pocket 23 simply appears as part of the circle. By the proof of the original Needle, pocket 23 is part of the circle as a continuing series of events of relative 1/4 C each. Therefore, by proof of the original Needle, pocket 23 has a dual random average value: .25 on the circle and .33333 on the diameter. That is: .08333 of the circle. Since a circle is pi relative to the randomness of gravity, that is: 1/12 pi.

The second event in the series has an inferential value: .25 C or .75 C. (that is: 1/4 pi or 3/4 pi) on a circle. That is: E or W relative to a first random event in, for example, S. Geometrically, it matches the middle pole of a three pole pi-angle (or "diameter"). This value may only be inferred and may not be directly statistically proven ....since it is the very Center of Rotation (that is: pi) that is eliminated from mathematical consideration through the geometric finesse in "action at a distance."

It is the third event that holds the flat bet advantage. It is predictable --with action at a distance-- as the geometric probability of the third pole on a diameter of three poles. Relative to both the "game" and our perceptions (they are one and the same since the "game" only exists in and of our "perceptions") the geometric probability is factored by the possibility of two directions on the circle of the "game." That is: .33333 / 2 = .16666 .

None of this game analysis is possible without the original Needle as the universal random unit of measure.

The underlying nature of these matters was apparently known both in the Vatican and the Paris Academy of Sciences in the 18th century. All was lost in the French Revolution. This occurred under the control of Laplace who was protected during the iron-fist 15 year tyranny of Napoleon ...who was in turn, mentored by Laplace.

As discussed within, geometrically, the use of "action at a distance" only makes mathematical sense when it is an extension of the original Needle. This is due to the requirement of "action at a distance" that Monte Carlo methodology be used to form the underlying mathematical matrix upon which "action at a distance geometrically feasts.

The original Needle also introduced the methodology of serial random measurements (it only acquired the name "Monte Carlo methodology" circa WWII). The original Needle introduced geometric probability. Since the original Needle proved random geometric probability to be a statement of relative 1/4 pi ...and since the original Needle was the first random proof of pi ..."action at a distance" is best understood in terms of pi (using relative 1/4 pi as the unit of measure and 1/2 pi or 1/6 pi as the predictive goal.

The flat bet advantage appears to shatter traditional random theory.

Did Laplace see it?

Of course!

Every indication is that it was given to him by Buffon between 1770 and 1773. As well, Laplace obtaining Buffon's papers held in estate after Laplace had Buffon's son guillotined (see History). Today, the Needle often bears Laplace's name and is sometimes referenced as the Buffon/Laplace Needle Problem.

In 1812, Laplace published the Needle under his own name without crediting or mentioning Buffon.

“Imaginons un plan divise par des lignes paralleles, equidistantes de la quantitie a; concevons de plus un cylindre tres-etroit don’t 2r soit la longueur, supposee eqale ou moindre que a. On demande la probabilite qu’en le projetant, il rencontrera une divisions du plan.” (Theorie Analytique des Probabilities, Simon Laplace, p. 569).

["Let us imagine a plane divided by equally spaced parallel lines of the distance a; and a cylinder of the length 2r and suppose the cylinder to be equal or less than the distance a. Give the probability of a toss touching a line of the plane."]

Buffon’s original take on the Needle left a clear inference that the value of a randomly measured field’s radius is: .50 .

When Laplace changed the Needle’s ("cylinder’s") length from .78539 (that is: the universal random average) of the field’s diameter to the complete length of the field’s diameter (or any other length) he eliminated the universal random average!

That ended geometric probability relative to the randomness of gravity. It left intact the appearance of geometric probability relative to life's perceptions ...but that was just unpredictable algebra. Laplace made the point stick by  controlled the roots of modern science and education That ended any effective study of relativity and gravity for the next two hundred years. Laplace's perfidy continues to this day as science continues to be without the original Needle and the geometric truth of "action at a distance." (The correct use of "action at a distance" requires a unit of measure that is: the  relative 1/4 pi of the original needle).

In his analysis, Laplace fundamentally changed the original Needle while discrediting its original length. He also, without giving reason or discussion, arbitrarily and casually mentioned that his calculated result (without mention of “pi angles” or the Needle or Boskovic or “action at a distance” or any other reason) must be multiplied by 16 ...?!  

The only reason to multiply the result of Laplace’s calculations of the Needle by “16” is if the randomly found arc on the Needle’s circle of 360 degrees is a relative geometric probability of 60 degrees (that is: 1/6 of a circle) without going through three random measurements and the finesse of “action at a distance.” This seemingly impossible mathematical phenomenon is the result of Laplace’s disingenuous convolutions (see History).

If this history of pi and gaming is correct, then it was precisely for this reason Laplace was brought into the Paris Academy of Sciences. Buffon needed someone to be a front for his Needle's geometric probability. Buffon's reason was that in its original form, his Needle inevitably leads to "action at a distance." The scenarios aren't hard to imagine.

Question: "If the two average random measurements of Buffon's Needle give the universal random average ...what do three give?"

Answer: "By the proof of the original Needle, two random measurements tend to give a universal random average that is only a mathematical perception. Three random measurements of "action at a distance" gives a flat bet advantage on the diameter of gravity's pull. This advantage is only found with when the middle measurement of perception is eliminated from consideration by the finesse. The elimination is geometrically legitimate since it is just a mathematical perception in the first instance of randomness.

When measuring the randomness of gravity, eliminating the middle measurement is a legitimate action. By the proof of the original Needle, the middle measurement is just a perception --just a mathematical average.

The original Needle --and only the original Needle-- naturally sets up a mathematical matrix for the geometric probability of "action at a distance." The basis of the matrix is the straight line of gravity with a value of "1." and the circle of pi that is subscribed by the rotating (randomly measured) diameter. The quadrants of he circle are just mathematical perceptions.

Sixty degrees of arc is 1/6 of a circle. However, when found randomly with “action at a distance,” its relativity to a diameter comes with a unique directional factor. Like a circle, the random measurements of a diameter are also subject to the possibility of two directions ...but with a difference.

Two possible algebraic directions on a diameter come with a geometric certainty of one direction.

Two possible algebraic directions on a circle come without geometry or geometric certainty. They are just two algebraic possibilities.

With “action at a distance,” all four directional factors come into play, but not equally. The randomness of gaming relative to a diameter is different from the randomness of gaming relative to a circle. This difference iis what this study is all about in the first place.

When two possibilities are factored into the Needle’s quadrant arc of 90 degrees (or into the average gaming quadrant) they also contain the two possible directions transferred from the diameter, through 1/2 pi (comes automatically with "action at a distance") with a unit of measure of relative 1/4 pi. This reduces a Quadrant to 45 degrees of inclination relative to a diameter base or, relative to the COR ...to 22.5 degrees of inclination.

When 22.5 degrees is made relative to a circle of 360 degrees (the “game”) the result is .08333 . This result (Laplace’s result ((and also the advantage of the Quantum Sciences))) must be multiplied by 16 to order to define the circle (or game or field or orbit or particle or randomly measured circumference of anything). That is: 16 (22.5) = 360. This completes the field (or "game") of 360 degrees, to which the result of a random measurement is relative. However, it is only relative as an algebraic statement relative to the circle being measured. Since a circle (or "game") is already only a statement of algebra ...this incestuously renders it meaninglessly relative to the quadrature that Laplace was promoting as well as to the geometry he was ostensibly looking for.  

That is what Laplace necessarily did to obtain the 60 degrees he needed to be able to multiply 22.5 degrees by 16 to obtain the results in degrees (or 60 degrees by 6 if the results are expressed as percentages). He simply changed the Needle’s length.  

The problem for Laplace and his take on the Needle is that he couldn’t use “action at distance” over three random measurements to get the result of 60 degrees that is delivered by the randomness of “action at a distance” and the original Needle. That would have exposed the sham of both his apparent acceptance/usurpation of the original Needle in 1770, and of his fraudulent "discovery" of quadrature (the Needle’s quadrature) in 1772, and his use of it to attack Boskovic’s finesse methodology of three random measurements in 1776.

To protect his plagiarism of the Needle, Laplace had to start with 60 degrees instead of the original Needle’s 45 degrees (straight line connecting the ends of a 90 degree arc which is the original Needle’s length) because otherwise, in 1812, he would have been right back where he was in 1776 …where he was ultimately proved wrong in the Boskovic debate although no one could admit it (to do so would invite the wrath of the Vatican). Laplace’s solution was to arbitrarily lengthen the Needle. This intentionally lost the unique geometry of the original Needle’s random length. It made the original Needle’s length geometrically meaningless and no more than algebraically equal with any or all other lengths.  

Without the original Needle and its insistence on relative 1/4 pi, Laplace could use the original Needle’s paradox of quadrature and twist it into an appearance of randomness without going through “action at a distance” and pi....  

...All Laplace had to do was make his "cylindre" longer than the original Needle’s random length.

Laplace made his "cylindre" the entire length of the diameter distance between the lines. Laplace claimed there were fewer “errors” when using his length. By that he meant his longer length crossed a line more often than a shorter length (such as --by implication-- Buffon’s length of the original Needle). Therefore, according to Laplace, pi could be calculated faster.

Instead of recognizing the original Needle’s relative valuation being relative to a field’s complete diameter (which is random “1.” and to which everything random is gravitationally relative) Laplace left the muddled inference that the value of a randomly measured field’s radius is “1.” ...from which it may be algebraically deduced that the complete diameter therefore has a random value of: "2." ...and therefore a circle is deductively valued: "2pi."

History has since interpreted Laplace's disingenuous comment to mean that Buffon made an “error” that Laplace somehow “corrected.”

Nothing could be further from the truth!

Ever since Laplace's malicious warp, the original Needle has been considered as no more that a quaint way of calculating pi. It is now sometimes called the "Buffon/Laplace Needle Problem" or the "Laplace/Buffon Needle Problem" or even the "Laplace Needle Problem."

In point of fact, there is no error in the original Needle and no one has ever found or identified one. The "error" was entirely Laplace’s.

Laplace did not make an error of algebraic calculation. Rather, it appears a malicious error in which Laplace was concealing the original Needle's random geometric truth of pi and his twisted usurpation of it.

Laplace's alteration of the original Needle is perhaps the greatest disaster in the history of commerce and science and education. It even derailed Albert Einstein's quest for the grail!

Laplace's change made the random geometry of relative 1/4 pi algebraically equal to any other length of Needle or "cylindre." In doing so, he dismissed and disdained the unique geometric values of the original Needle. That lost the unique fundamental geometric values of “1.” and pi that are only found with the random circumstances of the original Needle.

History, including Einstein, followed Laplace's lead. "One complete revolution corresponds to the angle 2pi in the absolute angular measure customary in physics...." (Einstein, Albert, tr. Lawson. Relativity. Crown, 1961. p.125). Like Laplace, Einstein also valued a radius as: "1.".

The original random values of "1." and pi ...are restored and explored here. As previously noted, even after it was altered, the power of the Needle is such that when physicists built the first atomic reactor they had to use the needle by randomly tossing nails on a grid floor to determine the probability of random particle collision so as to know how thick to build the walls.

The formula for the gravity bet is the formula for the original Needle, extended with the geometric finesse ("action at a distance") factored by two directions. This delivers the random flat bet advantage: .16666 !

In the world of science and education, the original Needle has never fully recovered since its loss through the Paris Academy of Sciences and Laplace. In short, the original Needle never had a chance to develop beyond its original introduction.

Laplace’s conduct and cover up was surely the most colossal fraud in history. Disastrously, it has been the basis of science ever since, including the stock market and the insurance and gaming industries!

It may be fairly said that any deep understanding of randomness and geometric probability and "action at a distance," must start with the original Needle.