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How Not to Square the Circle
Nicholas of Cusa was attacking a problem dating back to the ancient Greeks. The solution would have made him famous forever...
Tony Phillips
Stony Brook University
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Introduction
In 1965 my late friend and colleague John Stallings wrote "How not to prove the Poincaré conjecture." This work appeared in the Proceedings of the Wisconsin Topology Seminar and is still available on John's Berkeley website. It begins with the declaration: "I have committed the sin of falsely proving Poincaré's Conjecture. But that was in another country; and besides, until now no one has known about it." It continues with the exposition of the main ideas relating the conjecture to statements in algebra, and is certainly what Stephen Miller had in mind when he wrote, in the AMS Notices, after John's death, "His 1960s papers on the 3-dimensional Poincaré Conjecture are both brilliant and hilarious at the same time."
In 1445 Nicholas of Cusa wrote De Geometricis Transmutationibus (On Geometric Transmutations); my account is based on a recent translation into French of all of Nicholas' mathematical works: Nicolas de Cues, Les Écrits mathématiques by Jean-Marie Nicolle (Honoré Champion, Paris, 2007). This was the first of Cusa's writings in which he addressed the problem of squaring the circle. Literally, squaring the circle means devising the straightedge-and-compass construction of a square whose area equals that of a given circle. This means a construction relating a segment of length 1 (the radius of the circle) to a segment of length π√ (the side of the square). Nicholas's plan was start from an equilateral triangle and construct an isoperimetric circle; this is the content of the First Premise in De Geometricis Transmutationibus. If the triangle had perimeter 1, the circle would have diameter 1/π. Then the composition of two more standard straightedge and compass constructions could start from that diameter and generate first a segment of length 1/π−−−√, and from that one a segment of the reciprocal length π√.
Examples of straight-edge and compass arithmetic:
Left: square root. A segment AB of length x is extended by a segment BC of length 1. Choose one of the points D where the (green) circle with diameter AC intersects the perpendicular through B. Then by plane geometry DB2=AB⋅BC=AB, so DB has length x√.
Right: reciprocal. The construction starts with a segment EF of length 2 extended by FG of length 12. A circle (green) is constructed with EG as diameter. For any x between 1/2 and 2, for example 1/π−−−√, a circle (blue) of radius x is drawn with center F. Choose one of the intersection points X of the two circles and draw the line through X and F. It will intersect the green circle at a second point Y; the length y of the segment FY will be the reciprocal of x, since by standard plane geometry XF⋅FY=EF⋅FG=1.
Other circles, lines and points used in the constructions are shown in black.
Nicholas of Cusa
Nicholas of Cusa (1401-1464) was one of the leading intellectual figures in early 15th-century Europe. He is often described as a transitional figure between the Middle Ages and the Renaissance, and in fact he was personally involved in one of the great events that mark that transition: Pope Eugene IV sent him to Constantinople in 1437 as part of a delegation to negotiate the participation of the Eastern Orthodox hierarchy in the Council of Florence. They came, with an entourage of distinguished Greek scholars who stayed, and lectured, in Florence; contributing to the surge of interest in humanistic learning which led to the new age.
Nicholas' principal occupations were ecclesiastical politics and administration (he was named Cardinal in 1449) and, relatedly, theology/philosophy. Those were tumultuous times for the Church; Nicholas was at the center both of bitter jurisdictional controversies and of intense disputation about the exact wording of dogmatic texts, where the placement of a comma could assume cosmic importance. In those days philosophy, theology and natural science were closely linked: the physical structure of the universe had deep theological implications. Nicholas' energetic and erudite mind, in a priori meditation, led him to scientific insights that turned out to be prophetic. For example, he understood that the earth, the sun and the moon were objects moving through space; and he rejected the idea that all orbits had to be circular or even that the universe had a center (De Docta Ignorantia, Book II). Here he was a predecessor of Kepler (who referred to him as "divine") and of Giordano Bruno.
Nicholas' interest in mathematics seems to have been its status as an impregnable logical system. He believed that by testing his philosophical theories in mathematics he could produce convincing evidence of their validity. He outlines the parallelism between geometry and theology in De Circuli Quadratura, dated July 1450. "Transport yourself by assimilation from these mathematics to theology. ... Just as the circle is perfection in a figure, since any perfection of figures is worked into it, its surface contains all the surface of all figures and has nothing in common with all the other figures, but is absolutely one and simple in itself; likewise absolute eternity is the form of all forms ... having nothing in common with any other form. And whatever the figure of the circle therein may be, since it has neither beginning nor end, it has resemblance with eternity ... . ... Likewise, if a triangle wanted to triangulate the circle, or a square to square it and so forth for the other polygons, thus also intellectual nature wants to understand [God]."
The First Premise and its "proof"
Nicholas of Cusa's First Premise: a is the center of the equilateral triangle bcd. "You divide the side bc into four equal parts which you mark e,f,g: I assert that, if one extends the line drawn from a to e by its fourth, which gives ah, this will be the radius of the circle whose circumference is equal to the three sides of the triangle."
One of the thought schemata Nicholas devised for use in theology was the "concidence of opposites." Here is how he applied the principle to the proof of his First Premise. The construction involves a parameter, namely the position of the point e on the line cb. Nicholas observes that when e is at the midpoint f the length of the segment ah is smaller than the desired radius, and that when e is at b the length is larger. He applies the principle: ubi est dare magis et minis, quod ibi sit dare aequale (where one can give a greater and a lesser, one can also give an equal; essentially the Intermediate Value Theorem) and concludes correctly that for some intermediate position x the length ah must be exactly equal to that radius, "and that is the point e equidistant between b and f." The last statement made with no justification.
The construction is in fact plausible: suppose the sides of the triangle have length 1. Then ef=14; similarity of triangle abf with a half-equilateral triangle, and the Pythagorean theorem, yield af=123√; so ae=748−−√, and ah=(5/4)748−−√; the First Premise states that 2π⋅ah=3, which implies π=65487−−√=3.1423... . This value, which Nicholas could have calculated but never mentions explicitly, was within the bounds [22371,227]=[3.14084...,3.14285...] established by Archimedes. Therefore, until better approximations to π were available, there was no way to prove Nicholas's construction wrong, even though there were obvious gaps in his proof.
Later developments: Things get worse
Nicholas circulated copies of his work among his friends, who included Paolo Toscanelli (1397-1482), a Florentine astronomer and physician. He had been Nicholas' classmate, and they remained good friends for life. Toscanelli wrote back with objections. To us, now, it is clear that there was no way the argument could be repaired. Nicholas' solution was to devise a different, and considerably more complicated, construction.
The diagram for Nicholas of Cusa's second quadrature construction, from Quadratura Circuli, 1450. The construction starts from a triangle cde, superimposes an isoperimetric square ilkm and yields rq as the radius of the isoperimetric circle.
Nicholas would have done better to stay with his first construction. The new one was reprinted and minutely analyzed by Regiomontanus (Johannes Müller, 1436-1476) who showed that the implied value for π was outside the Archimedean bounds (Nicolle calculates it as 3.154); this is part of a 60-odd page appendix to his De triangulis omnimodis, dated 1464, published in 1533. There Regiomontanus takes up all of Nicholas' constructions one by one and "does the math" (Nunc ad numeros descendendum), using his knowledge of trigonometry to show "that Nicholas' approximations to π were --except one-- not even within the limits established by Archimedes," according to Menso Folkerts, who characterizes Regiomontanus as "a gifted student of Archimedes," and Nicholas of Cusa as "an amateur in mathematics." The one exception is presumably the First Premise above.
The moral of the story
Nicholas of Cusa was attacking a problem dating back to the ancient Greeks. The solution would have made him famous forever, and might even have helped bolster his side in theological disputations. No one knew at the time that squaring the circle is impossible: the proof requires calculus, which was 200 years away; and even then it was not discovered until 1882.
John Stallings was also attacking a famous problem: 50 years old, a very long time in modern mathematics. In this case the problem was not impossible, but the methods that led to its solution lay far in the future. Richard Hamilton's introduction of the Ricci flow, which led to Gregory Perelman's ultimate victory, came out in 1982, some 17 years after Stallings wrote "How not to prove the Poincaré Conjecture." But Stallings discovered his error by himself, before publishing, whereas Nicholas seems to have believed until the end that he had squared the circle, but perhaps had not been able to find the right argument to substantiate his claim.
Here is how Stallings ends his story: "... I was unable to find flaws in my 'proof' for quite a while, even though the error is very obvious. It was a psychological problem, a blindness, an excitement, an inhibition of reasoning by an underlying fear of being wrong. Techniques leading to the abandonment of such inhibitions should be cultivated by every honest mathematician."
Why circle-squaring is impossible
We will see that any length occurring in a compass and straightedge construction starting from length one must be an algebraic number, i.e. it must be a root of a polynomial with integer coefficients. Considerably more intricate is the proof that π, and therefore π√, is transcendental, i.e. not algebraic. Some references are given here.
A random compass-straightedge construction: all the coordinates of the vertices produced by the construction are of a special form: they are obtained from 1 by composing a finite number of operations, which can be arithmetic (sum, product, quotient, etc.) or the extraction of a square root. For future use, let's call the set of these numbers S. In this example, the construction starts with the vertices O=(0,0) and A=(1,0); the line they span is the x-axis. The circle of center O and of radius OA intersects the circle of center A and of radius OA at B=(12,3√2), the x-axis at C=(−1,0) and the y-axis (the perpendicular bisector of AC, constructed as usual by two circles and a line) at E=(0,1). The circle of center E and radius EA intersects the line through O and B at D=(3√ 7√4,3 21√4). The circle of center A and radius AD intersects the x-axis at F=(1 AD,0)=(1 127 21−−√−3√−7√−−−−−−−−−−−−−−−−√,0). As the construction continues, the number of embeddings of radicals into radicals tends to rise, but the numbers always have this general form. They are clearly algebraic, since the radicals can be peeled off by continued squaring and rearranging. In fact these constructible numbers form a special class of algebraic numbers: those that can be reached from the rational numbers by a finite number of quadratic extensions, i.e. by arithmetic operations and taking square roots a finite number of times. To show squaring the circle is impossible, "algebraic" is sufficient; but other impossibilities (duplicating the cube, trisecting the angle) require this additional information.
To see why this works in general, note first that if points P and P′ have their coordinates in S, then by the Pythagorean theorem their distance PP′ = r must also belong to S. So the circle of radius PP′ about P, say, has the equation (x−p1)2 (y−p2)2=r2. Another circle constructed from two points with coordinates in S will have a similar equation, say (x−q1)2 (y−q2)2=s2. All these coefficients lie in S. The coordinates of the intersection points of the two circles (if they intersect) will be the pairs (x,y) satisfying both equations. From the first equation we can write y=±r2−(x−p1)2−−−−−−−−−−−√ p2. Substituting this value in the second equation yields a polynomial equation in x; it looks like it might have degree 4, but the higher powers cancel and it is a quadratic equation with coefficients in S. The quadratic formula involves arithmetic and a square root, so the solutions it produces will again belong to S. For intersections of a circle and a line no cancellation is needed; the equation is quadratic; and for the intersection of two lines it is linear.
Squaring the circle
From Wikipedia, the free encyclopedia
For other uses, see Square the Circle.
Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.
Impossibility[edit]
The solution of the problem of squaring the circle by compass and straightedge demands construction of the number , and the impossibility of this undertaking follows from the fact that pi is a transcendental (non-algebraic and therefore non-constructible) number. If the problem of the quadrature of the circle is solved using only compass and straightedge, then an algebraic value of pi would be found, which is impossible. Johann Heinrich Lambert conjectured that pi was transcendental in 1768 in the same paper in which he proved its irrationality, even before the existence of transcendental numbers was proven. It was not until 1882 that Ferdinand von Lindemann proved its transcendence.
The transcendence of pi implies the impossibility of exactly "circling" the square, as well as of squaring the circle.
It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of pi, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations.
Bending the rules by allowing an infinite number of compass-and-straightedge operations or by performing the operations on certain non-Euclidean spaces also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can be in Gauss–Bolyai–Lobachevsky space. Indeed, even the preceding phrase is overoptimistic.[7][8] There are no squares as such in the hyperbolic plane, although there are regular quadrilaterals, meaning quadrilaterals with all sides congruent and all angles congruent (but these angles are strictly smaller than right angles). There exist, in the hyperbolic plane, (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area. However, there is no method for starting with a regular quadrilateral and constructing the circle of equal area, and there is no method for starting with a circle and constructing a regular quadrilateral of equal area (even when the circle has small enough radius such that a regular quadrilateral of equal area exists).
Some apparent partial solutions gave false hope for a long time. In this figure, the shaded figure is the Lune of Hippocrates. Its area is equal to the area of the triangle ABC (found by Hippocrates of Chios).
Archimedes Liu Hui Zu Chongzhi Madhava of Sangamagrama William Jones John Machin John Wrench Ludolph van Ceulen Aryabhata
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Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. It had been known for some decades before then that the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.
The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.[1]
The term quadrature of the circle is sometimes used synonymously or may refer to approximate or numerical methods for finding the area of a circle.
Liu Hui
Mathematical work[edit]
Along with Zu Chongzhi (429–500), Liu Hui was known as one of the greatest mathematicians of ancient China.[1] Liu Hui expressed all of his mathematical results in the form of decimal fractions (using metrological units), yet the later Yang Hui (c. 1238-1298 AD) expressed his mathematical results in full decimal expressions.[2][3]
Liu provided commentary on a mathematical proof identical to the Pythagorean theorem.[4] Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known".[5]
In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry. For example, he found that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge.[6] He also found that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. In his commentaries on the Nine Chapters, he presented:
An algorithm for calculation of pi (π) in the comments to chapter 1.[7] He calculated pi to with a 192 (= 64 × 3) sided polygon. Archimedes used a circumscribed 96-polygon to obtain the inequality , and then used an inscribed 96-gon to obtain the inequality . Liu Hui used only one inscribed 96-gon to obtain his π inequalily, and his results were a bit more accurate than Archimedes'.[8] But he commented that 3.142074 was too large, and picked the first three digits of π = 3.141024 ~3.14 and put it in fraction form . He later invented a quick method and obtained , which he checked with a 3072-gon(3072 = 512 × 6). Nine Chapters had used the value 3 for π, but Zhang Heng (78-139 AD) had previously estimated pi to the square root of 10.
Gaussian elimination.
Cavalieri's principle to find the volume of a cylinder,[9] although this work was only finished by Zu Gengzhi. Liu's commentaries often include explanations why some methods work and why others do not. Although his commentary was a great contribution, some answers had slight errors which was later corrected by the Tang mathematician and Taoist believer Li Chunfeng.
Survey of sea island
Liu Hui also presented, in a separate appendix of 263 AD called Haidao suanjing or The Sea Island Mathematical Manual, several problems related to surveying. This book contained many practical problems of geometry, including the measurement of the heights of Chinese pagoda towers.[10] This smaller work outlined instructions on how to measure distances and heights with "tall surveyor's poles and horizontal bars fixed at right angles to them".[11] With this, the following cases are considered in his work:
The measurement of the height of an island opposed to its sea level and viewed from the sea
The height of a tree on a hill
The size of a city wall viewed at a long distance
The depth of a ravine (using hence-forward cross-bars)
The height of a tower on a plain seen from a hill
The breadth of a river-mouth seen from a distance on land
The depth of a transparent pool
The width of a river as seen from a hill
The size of a city seen from a mountain.
Liu Hui's information about surveying was known to his contemporaries as well. The cartographer and state minister Pei Xiu (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a rectangular grid and graduated scale for accurate measurement of distances on representative terrain maps.[12] Liu Hui provided commentary on the Nine Chapter's problems involving building canal and river dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.[13]
Although translated into English long beforehand, Liu's work was translated into French by Guo Shuchun, a professor from the Chinese Academy of Sciences, who began in 1985 and took twenty years to complete his translation.
Zu Chongzhi
The majority of Zu's great mathematical works are recorded in his lost text the Zhui Shu. Most scholars argue about his complexity since traditionally the Chinese had developed mathematics as algebraic and equational. Logically, scholars assume that the Zhui Shu yields methods of cubic equations. His works on the accurate value of pi describe the lengthy calculations involved. Zu used the Liu Hui's π algorithm described earlier by Liu Hui to inscribe a 12,288-gon. Zu's value of pi is precise to six decimal places and for a thousand years thereafter no subsequent mathematician computed a value this precise. Zu also worked on deducing the formula for the volume of a sphere.
Brothers Brick posted a review on the upcoming 70922 The Joker Manor set (comes out on Black Friday). I kinda scrolled through the page and looked at the pictures assessing whether or not I want to spend the time reading. I passed because there’s just… too much. And the thing is,...
www.fbtb.net/lego/2017/11/22/sometimes-reviews-can-be-too...
© J. Antonio Herrera F. - All Rights Reserved
"La Muerte es como una ecuación algebráica, una serie de factores "x" y valores "y", sumados y multiplicados, divididos y restados que al final dan una solución simple y definitiva = 0"
"Death is like an algebraic equation, a number of "x" and "y" factors, added and multiplied, divided and subtracted that in the end result into a simple and final = Ø"
We just had to check some claim about certain Lie algebras the other day... no rocket science but a fun computation.
You may recognize what this is, but it's presented in a "non-representational" way, in that the colors and the shapes and the underlying pattern are the focus of the picture and not the symbols themselves. I believe this would be classified as Abstract.
1. Abstract. From wikipedia: Abstract art is now generally understood to mean art that does not depict objects in the natural world, but instead uses color and form in a non-representational way. You are free to create your abstraction entirely in-camera, in post-processing, or as a combination of the two.
Take a Class With Dave and Dave
NOTE: This is a direct photo of the cover of my abstract algebra book - the only photoshop I used was to crop it a bit :-) Cool to see a book cover so close, huh? Should be looked at large to get the cool pixel effect!!
Explore #351
More ceiling detail from the Getty Museum with false color added by Picnik. The purple seemed to make the most visually striking overlay.
Hello, may I help you?
Yeah, I'm the painter. I'm here to paint the building.
Painter? We called no painter.
I wrote down the address. It's correct, right?
Well, yes.
Ok then. You want one coat or two?
No no no. I want no paint. The building is covered in mosaic tile, you see. Tile doesn't get painted.
And nice tile it is too. But that don't solve my problem, which is I got a truck full of paint you are telling me you don't need.
That is a problem, yes.
So how you gonna fix it?
What color is the paint?
Well, you got your reds, your yellows, your blues.
I suppose a little color here and there might be. . . acceptable.
Fantastic. We'll have this place looking like Disneyland in no time.
Actually, I'm not sure that would go over too well.
No? How come?
Just a feeling.
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i wish they had constructed a university at his burrial place - a place oflearning for algebra -sciences and astronomy
Khayyám wrote a book entitled Explanations of the difficulties in the postulates in Euclid's Elements. The book consists of several sections on the parallel postulate (Book I), on the Euclidean definition of ratios and the Anthyphairetic ratio (modern continued fractions) (Book II), and on the multiplication of ratios (Book III).
The first section is a treatise containing some propositions and lemmas concerning the parallel postulate. It has reached the Western world from a reproduction in a manuscript written in 1387-88 AD by the Persian mathematician Tusi. Tusi mentions explicitly that he re-writes the treatise "in Khayyám's own words" and quotes Khayyám, saying that "they are worth adding to Euclid's Elements (first book) after Proposition 28."[13] This proposition[14] states a condition enough for having two lines in plane parallel to one another. After this proposition follows another, numbered 29, which is converse to the previous one.[15] The proof of Euclid uses the so-called parallel postulate (numbered 5). Objection to the use of parallel postulate and alternative view of proposition 29 have been a major problem in foundation of what is now called non-Euclidean geometry.
The treatise of Khayyám can be considered as the first treatment of parallels axiom which is not based on petitio principii but on a more intuitive postulate. Khayyám refutes the previous attempts by other Greek and Persian mathematicians to prove the proposition. And he, as Aristotle, refuses the use of motion in geometry and therefore dismisses the different attempt by Ibn Haytham too.[16] In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate,[17]
Geometric algebra[edit]
Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved by propositions five and six of Book two of Elements.
Omar Khayyam[18]
Omar Khayyám's geometric solution to the cubic equation x3 + 200x = 20x2 + 2000.
This philosophical view of mathematics (see below) has had a significant impact on Khayyám's celebrated approach and method in geometric algebra and in particular in solving cubic equations. In that his solution is not a direct path to a numerical solution and in fact his solutions are not numbers but rather line segments. In this regard Khayyám's work can be considered the first systematic study and the first exact method of solving cubic equations.[19]
In an untitled writing on cubic equations by Khayyám discovered in the 20th century,[18] where the above quote appears, Khayyám works on problems of geometric algebra. First is the problem of "finding a point on a quadrant of a circle such that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal." Again in solving this problem, he reduces it to another geometric problem: "find a right triangle having the property that the hypotenuse equals the sum of one leg (i.e. side) plus the altitude on the hypotenuse.[20] To solve this geometric problem, he specializes a parameter and reaches the cubic equation x3 + 200x = 20x2 + 2000.[18] Indeed, he finds a positive root for this equation by intersecting a hyperbola with a circle.
This particular geometric solution of cubic equations has been further investigated and extended to degree four equations.[21]
Regarding more general equations he states that the solution of cubic equations requires the use of conic sections and that it cannot be solved by ruler and compass methods.[18] A proof of this impossibility was only plausible 750 years after Khayyám died. In this paper Khayyám mentions his will to prepare a paper giving full solution to cubic equations: "If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared."[18]
This refers to the book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, part of the body of Persian Mathematics that was eventually transmitted to Europe.[19] In particular, he derived general methods for solving cubic equations and even some higher orders.
Binomial theorem and extraction of roots[edit]
See also: History of binomial theorem
From the Indians one has methods for obtaining square and cube roots, methods which are based on knowledge of individual cases, namely the knowledge of the squares of the nine digits 12, 22, 32 (etc.) and their respective products, i.e. 2 × 3 etc. We have written a treatise on the proof of the validity of those methods and that they satisfy the conditions. In addition we have increased their types, namely in the form of the determination of the fourth, fifth, sixth roots up to any desired degree. No one preceded us in this and those proofs are purely arithmetic, founded on the arithmetic of The Elements.
Omar Khayyam Treatise on Demonstration of Problems of Algebra[22]
This particular remark of Khayyám and certain propositions found in his Algebra book has made some historians of mathematics believe that Khayyám had indeed a binomial theorem up to any power. The case of power 2 is explicitly stated in Euclid's elements and the case of at most power 3 had been established by Indian mathematicians. Khayyám was the mathematician who noticed the importance of a general binomial theorem. The argument supporting the claim that Khayyám had a general binomial theorem is based on his ability to extract roots.[23]
Khayyám-Saccheri quadrilateral[edit]
Main article: Saccheri quadrilateral
The Saccheri quadrilateral was first considered by Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.[24] Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyám was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle):
Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.[25]
Khayyám then considered the three cases (right, obtuse, and acute) that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.
It wasn't until 600 years later that Giordano Vitale made an advance on Khayyám in his book Euclide restituo (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.
Astronomer[edit]
The Jalali calendar was introduced by Omar Khayyám alongside other Mathematicians and Astronomers in Nishapur, today it is one of the oldest calendars in the world as well as the most accurate solar calendar in use today. Since the calendar uses astronomical calculation for determining the vernal equinox, it has no intrinsic error, but this makes it an observation based calendar.[26][27][28][29]
Like most Persian mathematicians of the period, Khayyám was also an astronomer and achieved fame in that role. In 1073, the Seljuq Sultan Jalal al-Din Malik-Shah Saljuqi (Malik-Shah I, 1072–92), invited Khayyám to build an observatory, along with various other distinguished scientists. According to some accounts, the version of the medieval Iranian calendar in which 2,820 solar years together contain 1,029,983 days (or 683 leap years, for an average year length of 365.24219858156 days) was based on the measurements of Khayyám and his colleagues.[30] Another proposal is that Khayyám's calendar simply contained eight leap years every thirty-three years (for a year length of 365.2424 days).[31] In either case, his calendar was more accurate to the mean tropical year than the Gregorian calendar of 500 years later. The modern Iranian calendar is based on his calculations.
Heliocentric Theory[edit]
It is sometimes claimed that Khayyam demonstrated that the earth rotates on its axis[32] by presenting a model of the stars to his contemporary al-Ghazali in a planetarium.[33]
The other source for the claim that Khayyam believed in heliocentrism are Edward Fitzgerald's popular but anachronistic renderings[34] of Khayyam's poetry, in which the first lines are mistranslated with a heliocentric image of the Sun flinging "the Stone that puts the Stars to Flight".[35]
Calendar Reform[edit]
Khayyám is claimed to be a member of a panel that introduced several reforms to the Iranian calendar.[citation needed] On March 15, 1079, Seljuk Sultan Malik Shah I accepted this corrected calendar as the official Persian calendar.[36]
This calendar was known as the Jalali calendar after the Sultan, and was in force across Greater Iran from the 11th to the 20th centuries. It is the basis of the Iranian calendar which is followed today in Iran and Afghanistan. While the Jalali calendar is more accurate than the Gregorian, it is based on actual solar transit, similar to Hindu calendars, and requires an ephemeris for calculating dates. The lengths of the months can vary between 29 and 31 days depending on the moment when the sun crosses into a new zodiacal area (an attribute common to most Hindu calendars). This meant that seasonal errors were lower than in the Gregorian calendar.
The modern-day Iranian calendar standardizes the month lengths based on a reform from 1925, thus minimizing the effect of solar transits. Seasonal errors are somewhat higher than in the Jalali version, but leap years are calculated as before.
Poet[edit]
Main article: Rubáiyát of Omar Khayyám
Omar Khayyám was a notable poet during the reign of the Seljuk ruler Malik-Shah I and his contributions to the developments of mathematics, astronomy and philosophy inspired later generations.
He is believed to have written about a thousand four-line verses or rubaiyat (quatrains). In the English-speaking world, he was introduced through the Rubáiyát of Omar Khayyám which are rather free-wheeling English translations by Edward FitzGerald (1809–1883). Other English translations of parts of the rubáiyát (rubáiyát meaning "quatrains") exist, but FitzGerald's are the most well known.
A well decorated plaque containing poems from the Rubáiyát of Omar Khayyám.
Ironically, FitzGerald's translations reintroduced Khayyám to Iranians "who had long ignored the Neishapouri poet." A 1934 book by one of Iran's most prominent writers, Sadeq Hedayat, Songs of Khayyam, (Taranehha-ye Khayyam) is said to have "shaped the way a generation of Iranians viewed" the poet.[37]
Omar Khayyám's poems have been translated to many languages.[38] Many translations were made directly from Persian and more literal than translation by Edward Fitzgerald.[38]
The Moving Finger writes; and, having writ,
Moves on: nor all thy Piety nor Wit,
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.
But helpless pieces in the game He plays,
Upon this chequer-board of Nights and Days,
He hither and thither moves, and checks… and slays,
Then one by one, back in the Closet lays.
And, as the Cock crew, those who stood before
The Tavern shouted— “Open then the Door!
You know how little time we have to stay,
And once departed, may return no more.”
A Book of Verses underneath the Bough,
A Jug of Wine, a Loaf of Bread—and Thou,
Beside me singing in the Wilderness,
And oh, Wilderness is Paradise enow.
Myself when young did eagerly frequent
Doctor and Saint, and heard great Argument
About it and about: but evermore
Came out of the same Door as in I went.
With them the Seed of Wisdom did I sow,
And with my own hand labour’d it to grow:
And this was all the Harvest that I reap’d—
“I came like Water, and like Wind I go.”
Into this Universe, and why not knowing,
Nor whence, like Water willy-nilly flowing:
And out of it, as Wind along the Waste,
I know not whither, willy-nilly blowing.
And that inverted Bowl we call The Sky,
Whereunder crawling coop’t we live and die,
Lift not thy hands to It for help—for It
Rolls impotently on as Thou or I.
Views on religion[edit]
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There have been widely divergent views on Khayyám. According to Seyyed Hossein Nasr no other Iranian writer/scholar is viewed in such extremely differing ways. At one end of the spectrum there are nightclubs named after Khayyám, and he is seen as an agnostic hedonist.[39] On the other end of the spectrum, he is seen as a mystical Sufi poet influenced by platonic traditions.
An Ottoman Era inscription of a poem written by Omar Khayyám at Morića Han in Sarajevo, Bosnia and Herzegovina.
Seyyed Hossein Nasr, after examining the philosophical works of Khayyám, maintains that it is really reductive to just look at the poems (which are sometimes doubtful) to establish his personal views about God or religion; in fact, he even wrote a treatise entitled "al-Khutbat al-gharrå˘" (The Splendid Sermon) on the praise of God, where he holds orthodox views, agreeing with Avicenna on Divine Unity.[5] In fact, this treatise is not an exception, and S.H. Nasr gives an example where he identified himself as a Sufi, after criticizing different methods of knowing God, preferring the intuition over the rational (opting for the so-called "kashf", or unveiling, method):[5]
"... Fourth, the Sufis, who do not seek knowledge by ratiocination or discursive thinking, but by purgation of their inner being and the purifying of their dispositions. They cleanse the rational soul of the impurities of nature and bodily form, until it becomes pure substance. When it then comes face to face with the spiritual world, the forms of that world become truly reflected in it, without any doubt or ambiguity. This is the best of all ways, because it is known to the servant of God that there is no reflection better than the Divine Presence and in that state there are no obstacles or veils in between. Whatever man lacks is due to the impurity of his nature. If the veil be lifted and the screen and obstacle removed, the truth of things as they are will become manifest and known. And the Master of creatures [the Prophet Muhammad]—upon whom be peace—indicated this when he said: “Truly, during the days of your existence, inspirations come from God. Do you not want to follow them?” Tell unto reasoners that, for the lovers of God, intuition is guide, not discursive thought."
—Omar Khayyám[40]
The same author goes on by giving other philosophical writings which are totally compatible with the religion of Islam, as the al-Risālah fil-wujūd (الرسالة في الوجود, "Treatise on Being"), written in Arabic, which begin with Quranic verses and asserting that all things come from God, and there is an order in these things. In another work, Risālah jawāban li-thalāth masāʾil (رسالة جوابان لثلاث مسائل, "Treatise of Response to Three Questions"), he gives a response to question on, for instance, the becoming of the soul post-mortem. S.H. Nasr even gives some poetry where he is perfectly in favor of Islamic orthodoxy, but expressing mystical views (God's goodness, the ephemerical state of this life, ...):[5]
Thou hast said that Thou wilt torment me,But I shall fear not such a warning.For where Thou art, there can be no torment,And where Thou art not, how can such a place exist?The rotating wheel of heaven within which we wonder,Is an imaginal lamp of which we have knowledge by similitude.The sun is the candle and the world the lamp,We are like forms revolving within it.A drop of water falls in an ocean wide,A grain of dust becomes with earth allied;What doth thy coming, going here denote?A fly appeared a while, then invisible he became.
Considering possible misunderstandings about Khayyám in the West and elsewhere, Hossein Nasr concludes by saying that if a correct study of the authentic rubaiyat is done, but along with the philosophical works, or even the spiritual biography entitled Sayr wa sulak (Spiritual Wayfaring), we can no longer view the man as a simple hedonistic wine-lover, or even an early skeptic, but a profound mystical thinker and scientist whose works are more important than some verses.[5] C.H.A. Bjerregaard earlier summarised the situation:
"The writings of Omar Khayyam are good specimens of Sufism but are not valued in the West as they ought to be, and the mass of english-speaking people know him only through the poems of Edward Fitzgerald which is unfortunate. It is unfortunate because Fitzgerald is not faithful to his master and model, and at times he lays words upon the tongue of the Sufi which are blasphemous. Such outrageous language is that of the eighty-first quatrain for instance. Fitzgerald is doubly guilty because he was more of a Sufi than he was willing to admit."[41]
A French orientalist named Franz Toussaint was so dissatisfied with Fitzgerald's translation (and with some works just translating Fitzgerald from English to French) that he wrote his own directly from the Persian texts, trying to express the spirit of the verses rather than to versify.[42] His translation was published from 1924 to 1979 uninterrupted by Editions d'Art Henri Piazza before that editor disappeared. That translation was itself translated in other languages on Internet sites.
Abdullah Dougan, a modern Naqshbandi Sufi, provides commentary[43] on the role and contribution of Omar Khayyam to Sufi thought. Dougan says that while Omar is a minor Sufi teacher compared to the giants – Rumi, Attar and Sana’i – one aspect that makes Omar’s work so relevant and accessible is its very human scale as we can feel for him and understand his approach. The argument over the quality of Fitzgerald’s translation of the Rubaiyat has, according to Dougan, diverted attention from a fuller understanding of the deeply esoteric message contained in Omar’s actual material – "Every line of the Rubaiyat has more meaning than almost anything you could read in Sufi literature".
Philosopher[edit]
Tomb of Omar Khayyám Neishapuri in Nishapur, Iran
Khayyám himself rejects to be associated with the title falsafī "philosopher" in the sense of Aristotelianism and stressed he wishes "to know who I am". In the context of philosophers he was labeled by some of his contemporaries as "detached from divine blessings".[44]
It is now established that Khayyám taught for decades the philosophy of Avicena, especially the Book of Healing, in his home town Nishapur, till his death.[5] In an incident he had been requested to comment on a disagreement between Avicena and a philosopher called Abu'l-Barakāt al-Baghdādī who had criticized Avicena strongly. Khayyám is said to have answered "[he] does not even understand the sense of the words of Avicenna, how can he oppose what he does not know?"[44]
Khayyám the philosopher could be understood from two rather distinct sources. One is through his Rubaiyat and the other through his own works in light of the intellectual and social conditions of his time.[45] The latter could be informed by the evaluations of Khayyám's works by scholars and philosophers such as Abul-Fazl Bayhaqi, Nizami Aruzi, and al-Zamakhshari and Sufi poets and writers Attar of Nishapur and Najm-al-Din Razi.
Mathematical philosophy[edit]
As a mathematician, Khayyám has made fundamental contributions to the philosophy of mathematics especially in the context of Persian Mathematics and Persian philosophy with which most of the other Persian scientists and philosophers such as Avicenna, Abū Rayḥān al-Bīrūnī and Tusi are associated. There are at least three basic mathematical ideas of strong philosophical dimensions that can be associated with Khayyám.
1.Mathematical order: From where does this order issue, and why does it correspond to the world of nature? His answer is in one of his philosophical "treatises on being". Khayyám's answer is that "the Divine Origin of all existence not only emanates wujud "being", by virtue of which all things gain reality, but It is the source of order that is inseparable from the very act of existence."[45]
2.The significance of axioms in geometry and the necessity for the mathematician to rely upon philosophy and hence the importance of the relation of any particular science to prime philosophy. This is the philosophical background to Khayyám's total rejection of any attempt to "prove" the parallel postulate, and in turn his refusal to bring motion into the attempt to prove this postulate, as had Ibn al-Haytham, because Khayyám associated motion with the world of matter, and wanted to keep it away from the purely intelligible and immaterial world of geometry.[45]
3.Clear distinction made by Khayyám, on the basis of the work of earlier Persian philosophers such as Avicenna, between natural bodies and mathematical bodies. The first is defined as a body that is in the category of substance and that stands by itself, and hence a subject of natural sciences, while the second, called "volume", is of the category of accidents (attributes) that do not subsist by themselves in the external world and hence is the concern of mathematics. Khayyám was very careful to respect the boundaries of each discipline, and criticized ibn al-Haytham in his proof of the parallel postulate precisely because he had broken this rule and had brought a subject belonging to natural philosophy, that is, motion, which belongs to natural bodies, into the domain of geometry, which deals with mathematical bodies.[45]
Ambridge
Camera scan, school annual, 1962, adapted. The original photographer was not credited and is unknown.
In code theory class today. It looks empty, but we are actually at least six people taking this course this semester. There are some more people to my left. (Actually, half an arm got in the frame, but I cropped it off).
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After a night's rest, some neurofen and a good cup of Earl Grey, Mr Fox was ready to make a second attempt at beginning the new school term!
Today was algebra; one of if his personal favourites! After rounding up the unruly rabble, whilst little Squirt waited pen at the ready in his front row desk, Mr Fox took the register.
Concerned their attention wouldn't stretch beyond a couple of minutes, in an entrepreneurial bid for their cooperation, he thought he'd give interactive learning another go.
Introducing himself as Pythagofox after a quick wardrobe switcheroo, he certainly got their attention.
But Mrs Polar Bear had heard that name before... and Mr Fox probably ought to have known better.
Pythagorus was a renowned hater of beans. It's even fabled that he regarded them as such an abomination that, faced with a choice of whether to stay and get murdered, or escape by running through a field of the crop, he actually chose the former. As a result he was promptly slain.
A figure of hate amongst bean lovers the world over, Mrs PB was left with no real choice. She was obliged to splat Mr Fox. Subconsiocusly he must have known this. He had time to taze her as he saw her fluffy white paw reach for her rolling pin. But he didn't. How could he? Pythagorus stood for everthing she hated in the world and he couldn't be a part of that.
Good job then that right on cue, Mr Completely Mad (Helmet Wearing) Raccoon stepped into the room wearing a toga and a Pythagorus namebadge! At least with Mr Raccoon, no matter how many times he gets hit on the head no one would ever be able to tell the difference!
An example jigsaw that has students using multiple types of representation, including models/symbolic/simplified models
I have decided four years after getting my Bachelor of Arts, that I will go back to school to obtain my Masters of Business Administration with a Project Management Certification. Yesterday I took an algebra assesment test. I was a little worried since I hadn't taken math since 2005, but I actually did really well! I will start classes July 10th!!
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I was sitting on Math class last thursday and I was bored out of my mind, couldn't focus and as usual I couldn't undersatand jack about what she was talking about. I turned to a friend of mine and made a finger gun, pointed it at my head and blew my brains out. It gave me this idea almost as soon as I pulled the trigger. Maybe at some point I'll take a shot with it painted at the wall. Thank you AyLeks (http://www.flickr.com/photos/ayleks) for helping me problem solve on this, on how to get the numbers on my wall. and yes, I am a huge Ravens fan.
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One of my colleague PhD students wrote a whole lot of magic on the blackboard in our room the other day. He says he understands it...
...and three is the magic number.
I love algebra. I teach algebra to young people. As a teacher I often push my students to check their work to make sure they don't make mistakes. I set up this shot by solving a simple two-step algebraic equation with a solution of 2. Then I realized I wanted to have x = 3, so I started from there and 'created' the equation by working backward... anyway, I ended up doing it wrong: 4x - 9 = 21... as I was editing the pictures I realized my mistake, but was too lazy to set up again and re-shoot. So I put a funky tight crop on it and voila.
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Das "Rohmaterial" wurde mit einem Lensbaby aufgezeichnet, anschließend folgte noch eine aufwendige Bearbeitung in Photoshop.
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Ota Darvoza edo Mendebaldeko Atearen aurrean dagoen monumentua. Muhammad Ibn Mūsā al-Khwārizmī (Al-Khwārizmī edo al-Juarismi) hemen jaiotakoa dela diote, eta hortik omen dator “algoritmo” (latinez Algoritmi), eta berak idatzitako lan baten izenburutik “aljebra” hitza. Pertsonaia hau aljebraren eta zenbaketa sistema hamartarraren sortzailetzat jotzen da.
Situado delante de la entrada de occidente (Ota Darvoza). Dicen que Muhammad Ibn Mūsā al-Khwārizmī (Al-Khwārizmī edo al-Juarismi) que nació en Khiva . La palabra algoritmo parece que viene de su nombre latinizado (Algoritmi) y la palabra “álgebra” del título de una de sus obras escritas. Se le conoce como el creador del sistema numérico basado en diez y del álgebra.