Khayyám's full name was Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim Al-Nishapuri al-Khayyami () and was born in Nishapur, Iran, then a Seljuk capital in Khorasan (present Northeast Iran), rivaling Cairo or Baghdad.

He is thought to have been born into a family of tent makers (literally, al-khayyami in Arabic means "tent-maker"); later in life he would make this into a play on words:

He spent part of his childhood in the town of Balkh (present northern Afghanistan), studying under the well-known scholar Sheik Muhammad Mansuri. Subsequently, he studied under Imam Mowaffaq Nishapuri, who was considered one of the greatest teachers of the Khorassan region.

Khayyám was famous during his times as a mathematician. He wrote the influential Treatise on Demonstration 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. In particular, he derived general methods for solving cubic equations and even some higher orders.In the Treatise he wrote on the triangular array of binomial coefficients known as Pascal's triangle. In 1077, Khayyám wrote Sharh ma ashkala min musadarat kitab Uqlidis (Explanations of the Difficulties in the Postulates of Euclid) published in English as "On the Difficulties of Euclid's Definitions" . An important part of the book is concerned with Euclid's famous parallel postulate, which attracted the interest of Thabit ibn Qurra. Al-Haytham had previously attempted a demonstration of the postulate; Khayyám's attempt was a distinct advance, and his criticisms made their way to Europe, and may have contributed to the eventual development of non-Euclidean geometry.

Khayyám had notable works in geometry, specifically on the theory of proportions.

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 us 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." This proposition 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. 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 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. In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate,

Geometric algebra

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.

In an untitled writing on cubic equations by Khayyám discovered in 20th century, 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. To solve this geometric problem, he specializes a parameter and reaches the cubic equation x^3+200x=20x^2+2000. 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.

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. A proof of this impossibility was plausible only 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."

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. In particular, he derived general methods for solving cubic equations and even some higher orders.

Binomial theorem and extraction of roots

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.

Khayyám-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. 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.

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.

Like most Persian mathematicians of the period, Khayyám was famous as an astronomer.In 1073, the Seljuk Sultan Sultan Jalal al-Din Malekshah Saljuqi (Malik-Shah I, 1072-92), invited Khayyám to build an observatory, along with various other distinguished scientists. Eventually, Khayyám and his colleagues measured the length of the solar year as 365.24219858156 days. Khayyám's calendar was more accurate than the Gregorian calendar of 500 years later. The modern Iranian calendar is based on his calculations.

Calendar reform

Khayyám was part of a panel that introduced several reforms to the Persian calendar. On March 15, 1079, Sultan Malik Shah I accepted this corrected calendar as the official Persian calendar.

This calendar was known as 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 isfollowed 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 32 days depending on the moment when the sun crossed 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 areform 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.

Khayyám built a star map (now lost), which was famous in the Persian and Islamic world.

Khayyám's poetic work has eclipsed his fame as a mathematician and scientist.

He is believed to have written about a thousand four-line verses or quatrains (rubaai's). 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 translations of parts of the rubáiyát (rubáiyát meaning "quatrains") exist, but FitzGerald's are the most well known. Translations exist in languages other than English.

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.

Khayyám's personal beliefs are not known with certainty, but much is discernible from his poetic oeuvre.

Poetry

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."

Alike for those who for TO-DAY prepare,  And that after a TO-MORROW stare,A Muezzin from the Tower of Darkness cries  "Fools! your reward is neither Here nor There!"

Why, all the Saints and Sages who discuss'd  Of the Two Worlds so learnedly, are thrustLike foolish Prophets forth; their Words to Scorn  Are scatter'd, and their mouths are stopt with Dust.

Oh, come with old Khayyam, and leave the Wise  To talk; one thing is certain, that Life flies;One thing is certain, and the Rest is Lies;  The Flower that once has blown for ever dies.

Myself when young did eagerly frequent  Doctor and Saint, and heard great ArgumentAbout 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.

The Moving Finger writes; and, having writ,  Moves on: nor all thy Piety nor WitShall lure it back to cancel half a Line,  Nor all thy Tears wash out a Word of it.

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

In his own writings, Khayyám rejects strict religious structure and a literalist conception of the afterlife.

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 night clubs named after Khayyám and he is seen as an agnostic hedonist. On the other end of the spectrum, he is seen as a mystical Sufi poet influenced by platonic traditions.

Robertson (1914) believes that Khayyám was not devout and had no sympathy for popular religion, but the verse: "Enjoy wine and women and don't be afraid, God has compassion," suggests that he wasn't an atheist. He further believes that it is almost certain that Khayyám objected to the notion that every particular event and phenomenon was the result of divine intervention. Nor did he believe in an afterlife with a Judgment Day or rewards and punishments. Instead, he supported the view that laws of nature explained all phenomena of observed life. One hostile orthodox account of him shows him as "versed in all the wisdom of the Greeks" and as insistent that studying science on Greek lines is necessary.. Roberston (1914) further opines that Khayyám came into conflict with religious officials several times, and had to explain his views on Islam on multiple occasions; there is even one story about a treacherous pupil who tried to bring him into public odium. He then wrote that Khayyám "performed pilgrimages not from piety but from fear" of his contemporaries who divined his unbelief.

The following two quatrains are representative of numerous others that serve to reject many tenets of religious dogma:

[[Image:Khayam21.jpg|thumb|right|O cleric, we are more active than you,even so drunk, we are more attentive than you,You drink the blood of men, we drink the blood of grapes [wine],Be fair, which one of us is more bloodthirsty?]]

???? ??? ? ???? ???? ??? ???

?? ??? ??? ??? ????? ??? ???

??? ????? ??? ???? ????? ???

????? ?? ?????? ?? ???? ??? ???

which translates in FitzGerald's work as:

And if the Wine you drink, the Lip you press,

End in the Nothing all Things end in ... Yes ...

Then fancy while Thou art, Thou art but what

Thou shalt be ... Nothing ... Thou shalt not be less.

A more literal translation could read:Wake! For the Sun, who scatter'd into flight

The Stars before him from the Field of Night,

Drives Night along with them from Heav'n, and strikes

The Sultan's Turret with a Shaft of Light.

Before the phantom of False morning died,

Methought a Voice within the Tavern cried,

"When all the Temple is prepared within,

Why nods the drowsy Worshipper outside?"

And, as the Cock crew, those who stood before

The Tavern shouted--"Open then the Door!

You know how little while we have to stay,

And, once departed, may return no more."

Wake! For the Sun, who scatter'd into flightThe Stars before him from the Field of Night,Drives Night along with them from Heav'n, and strikesThe Sultan's Turret with a Shaft of Light.

Before the phantom of False morning died,Methought a Voice within the Tavern cried,"When all the Temple is prepared within,Why nods the drowsy Worshipper outside?"

And, as the Cock crew, those who stood beforeThe Tavern shouted--"Open then the Door!You know how little while we have to stay,And, once departed, may return no more."

If with wine you are drunk be happy,

If seated with a moon-faced (beautiful), be happy,

Since the end purpose of the universe is nothing-ness;

Hence picture your nothing-ness, then while you are, be happy!

?????? ? ??? ???????? ?? ????

????? ???? ???????? ?? ????

?? ???? ??? ? ????? ?? ?? ????

??? ??? ?????? ???????? ?? ????

which FitzGerald has boldy interpreted as:

Why, all the Saints and Sages who discuss’d

Of the Two Worlds so learnedly ... are thrust

Like foolish Prophets forth; their Words to Scorn

Are scatter’d, and their Mouths are stopt with Dust.

A literal translation, in an ironic echo of "all is vanity", could read:

Those who have gone forth, thou cup-bearer,

Have fallen upon the dust of pride, thou cup-bearer,

Drink wine and hear from me the truth:

(Hot) air is all that they have said, thou cup-bearer.

Now the New Year reviving old Desires,

The thoughtful Soul to Solitude retires,

Where the White Hand Of Moses on the Bough

Puts out, and Jesus from the Ground suspires

But some specialists, like Seyyed Hossein Nasr who looks at the available philosophical works of Khayyám, maintain 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. 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):

"... 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 andobstacle 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."

—‘Umar Khayyåm

The same author goes on by giving other philosophical writings which are totally compatible with the religion of Islam, as the "al-Risålah fil-wujud" (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, "Risålah jawåban li-thalåth maså?il" (Treatise of Responseto 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, ...):

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 misunderstandings about Khayyám in the West and elsewhere, S.H. 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. 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 the 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. "

Khayyám himself rejects to be associated with the title falsafi- (lit. philosopher) in the sense of Aristotelian one 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".

It is now established that Khayyám taught for decades the philosophy of Aviccena, especially "the Book of Healing", in his home town Nishapur, till his death. In an incident he had been requested to comment on a disagreement between Aviccena and a philosopher called Abu'l-Barakat (known also as Nathanel) who had criticized Aviccena 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?"

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. The latter could be informed by the evaluations of Khayyám's works by scholars and philosophers such as Bayhaqi, Nezami Aruzi, and Zamakhshari and Sufi poets and writers Attar Nishapuri and Najmeddin Razi.

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, Biruni, 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 wojud or 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."
2. The significance of postulates (i.e. axiom) 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.
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.

http://classics.mit.edu/Khayyam/rubaiyat.html