(Image source: SAEDNews, 2020)

Maps showing the extent of the Islamic empire from the 8th C - approximately the 14th C CE. The medieval Islamic world stretched from the Indian Ocean to the Atlantic, from Persia, Afghanistan and Uzbekistan in the east to Algeria, Morocco and Spain in the west. Learning, research and the arts were highly valued and supported in this multicultural empire, and the House of Wisdom in Baghdad was a centre for learning in mathematics, astronomy and medicine as well as poetry and architecture and other arts and sciences.

Here is a nice geometric introduction to some aspects of beautiful Islamic design, from architecture and other fields: https://www.youtube.com/watch?v=pg1NpMmPv48

Here is a link from Len Bergen's book, Episodes in the Mathematics of Medieval Islam, detailing
a very different set of arithmetic algorithms from this mathematical culture. We will try out some of the 'dustboard arithmetic' algorithms today.

Some of the many notable Islamic mathematicians/ scientists/ poets from this era:

al-Khwārizmī (780-850)

Muḥammad ibn Mūsā al-Khwārizmī

al-Khwārizmī was born near Baghdad and spent much of his life at the Bayt al-Ḥikmah. He wrote, among many things, Hisab Al-jabr wa’l-muqabala (Algebra).

Al-jabr: completion (+ & x)

Al-muqabala: balancing (-)

In Saiya's presentation we saw al-Khwārizmī's geometric/ algebraic method for solving three cases of quadratics by completing the square: For example, she showed us how to solve to find the positive x value of the following equation using al-Khwārizmī’s method.

You can read more about this in THE BOOK AL-JABR WA AL-MUQĀBALAH (in the original Arabic language version, and here in a 19th C English translation)

Ibn Sīnā (c. 980-1037)

Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn al-Ḥasan ibn ʿAlī ibn Sīnā

Ibn Sīnā, known in Europe as Avicenna, was born in the Northeast of the Islamic Empire near Bukhara in current day Uzbekistan. His studies focused mainly in medicine but extended to mathematics, astronomy, and philosophy (of education!).

The picture to the left is a page of Ibn Sīnā’s medical textbook al-Qanun (The Canon) from the National Library of Medicine. Here is an open-source al-Qanun English translation with explanation.

al-Khayyāmī (1048-1123)

Abū’l-Fath Ghiyāth al-Dīn ‘Umar ibn Ibrāhīm al-Khayyāmī

Known more popularly as Omar al-Khayyām, he was born in North-East Iran in a city called Neyshabur. al-Khayyāmī was known as a poet before his works of mathematics and astronomy made it to Europe. He was influenced by Ibn Sīnā and is most famous for his contributions to solving cubic equations. Image source to the right from Math Tutor

Activities:

1) Exploring one of al-Khayyāmī’s solutions:

Consider the following: For any cubic equation, where a and b are constants, a solution to the equation will be the x value of the intersection point of the parabola& the circle whose diameter isand whose centre lies on the x-axis.

• Go to https://www.desmos.com/calculator

• Graph a cubic equation of your choosing that aligns with the above form. Add graphs for the circle and parabola that correspond to al-Khayyām’s solution.

• What do you notice? What questions do you have ?

• Try on paper with a compass!

2) Exploring numerical patterns from Ibn Sīnā: Sums of odd numbers in square arrays

On a piece of paper, sketch out two more square arrays of sequential odd numbers nxn, which are bigger than 3x3. Make one of them an even number "length" and another an odd number "length".

Play around with sums (or other operations?)

Reflect: What patterns do you notice? What might you conjecture about these patterns?

3) Choose a construction from Medieval Islam Mathematics (Centre for Asian and Middle Eastern Studies from University of Illinois) and go through the process of constructing it, on Desmos and/or using compass and straightedge on papr.