He made such a name for himself that the Seljuq sultan Malik-Shāh invited him to Eṣfahān to undertake the astronomical observations necessary for the reform of the calendar. (See The Western calendar and calendar reforms.) To accomplish this an observatory was built there, and a new calendar was produced, known as the Jalālī calendar. Based on making 8 of every 33 years leap years, it was more accurate than the present Gregorian calendar, and it was adopted in 1075 by Malik-Shāh. In Eṣfahān he also produced fundamental critiques of Euclid’s theory of parallels as well as his theory of proportion. In connection with the former his ideas eventually made their way to Europe, where they influenced the English mathematician John Wallis (1616–1703); in connection with the latter he argued for the important idea of enlarging the notion of number to include ratios of magnitudes (and hence such irrational numbers as Square root of√2 and π).
His years in Eṣfahān were very productive ones, but after the death of his patron in 1092 the sultan’s widow turned against him, and soon thereafter Omar went on a pilgrimage to Mecca. He then returned to Neyshābūr where he taught and served the court as an astrologer. Philosophy, jurisprudence, history, mathematics, medicine, and astronomy are among the subjects mastered by this brilliant man.
Omar’s fame in the West rests upon the collection of robāʿīyāt, or “quatrains,” attributed to him. (A quatrain is a piece of verse complete in four lines, usually rhyming aaaa or aaba; it is close in style and spirit to the epigram.) Omar’s poems had attracted comparatively little attention until they inspired FitzGerald to write his celebrated The Rubáiyát of Omar Khayyám, containing such now-famous phrases as “A Jug of Wine, a Loaf of Bread—and Thou,” “Take the Cash, and let the Credit go,” and “The Flower that once has blown forever dies.” These quatrains have been translated into almost every major language and are largely responsible for colouring European ideas about Persian poetry. Some scholars have doubted that Omar wrote poetry. His contemporaries took no notice of his verse, and not until two centuries after his death did a few quatrains appear under his name. Even then, the verses were mostly used as quotations against particular views ostensibly held by Omar, leading some scholars to suspect that they may have been invented and attributed to Omar because of his scholarly reputation.