Omar Khayyâm’s doodle

Omar Khayyam was born in Nishapur on May 18, 1048. He went to school and two of his best friends there were Nizam al-Mulk and Hassan al-Sabbah. The three boys agreed that the first of them who could get a rich position will help his friends in a future. Al-Mulk became vizier in Alp Arslam’s palace and al-Sabbah and Khayyam also benefited of his nrew position although Khayyam got a job which let him to study Astronomy, Literature and Mathematics. Khayyam’s life wasn’t easy but his astronomical studies and his participation in the reform of the calendar were so decisive that he always had a city or a place where being able to life and work. According to al-Arudî al-Samarcandî, Khayyâm died on December 4, 1131.

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Khayyâm studied Euclid’s Elements and Data, Apollonius’ Conics and al-Khwârizmî’s Algebra and wrote his major work on Algebra around 1074 whre was able to solve geometrically the cubic equation. The treatise begins with three introductory lemmas:

  1. To find two segments x and y which a/x = x/y = y/b (Khayyâm finds a point (x,y) which is the crossing point of the parabolas x2 = ay and y2 = bx).
  2. To determine the height of a parallelepiped with known squared base b if we know that its volume must be the same as another parallelepiped with squared base a and height (Khayyâm determines a line k such that a/b =b/m and the searched height K is k/a = h/K).
  3. To determine the side of the base of the second parallelepiped.

Khayyâm solves fourteen canonic cubic equations (he didn’t know the negative numbers!) from these three geometric lemmas. For example:

Lemma 1. From lemma 1, we can find x and y such that 1/x = x/y = y/c and this point (x,y) satisfies x3 = c.

The other thirteen cases are solved crossing parabolas, circles and hiperbolas.

In 1857 E.Fitzgerald discovered Khayyam’s Rubâyyât in the British Museum and trabslated some verses from this “new” manuscript. The translation to English was so popular since 1861 and the Khayyâm’s name was very famous in the literary circles. The Rubâyyât contains over 400 quatrains written in Persian and were translated to English again in 1970s by Robert Graves:

Wake! for Morning in the Bowl of Night

Has flung the Stone that puts the Stars to Flight:

And Lo! the Hunter of the East has caught

The Sultán’s Turret in a Noose of Light

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Dreaming when Dawn’s Left Hand was in the Sky

I heard a Voice within the Tavern cry,

Awake, my Little ones, and fill the Cup

Before Life’s Liquor in its Cup be dry.

The mathematical doodle was published by Google in the Arabic countries two years ago to commemorate Khayyâm’s birthday.


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