# From Prisoner to Polymath: The Remarkable Story of Ibn Al-Haytham

Abu Ali Al-Hasan Ibn Al Hasan Ibn Al-Haytham (Latinized ** Alhazen **or simply known as

**) was an Arab polymath that specialized in astronomy, physics, mathematics, medicine and theology. He is widely recognized as a hero of the Islamic Golden Age and is sometimes called “the father of modern optics.”**

*Ibn Al-Haytham***Ibn Al-Haytham the Prisoner**

Ibn Al-Haytham was born is Basra and eventually became a *vizier* or Minister given his intellectual prowess, specifically in applied mathematics. He even claimed the ability to use regulate the flooding of the notorious Nile river (a problem which had plagued Egyptians since the time of the Pharaohs).

Around the year 1000, Ibn Al-Haytham was invited to put his scientific theory to practical use by the Fatimid Calipha al-Hakim via a huge hydraulic project at Aswan. Thankfully for us all, Ibn Al-Haytham failed was forced to concede the impracticability of his project.

This failure resulted in one of three legends: 1) Ibn Al-Haytham was imprisoned by the vengeful Caliph or 2) Ibn Al-Haytham was forced into hiding until the Calipha’s death in 1021 or 3) Ibn Al-Haytham feigned madness and was kept under house arrest.

In either case, Ibn Al-Haytham invested his time wisely and quickly bounced back from his failure. It took him almost a decade to write his influential opus; *Kitab al-Manazir* (*Book of Optics**) *which would go on to shatter the 1500 year prevailing theory of emission that was supported by such classic Greek thinkers such as Euclid and Ptolemy, who believed that sight worked like our modern flash lights (for the millennials, a flash light is like a phone without a screen or calling capabilities). His seminal work, would go on to inspire countless Western scientists for the next 700 years including the fallen scientific angel from Italy *Galileo Galilei, Johannes Kepler* of Germany and *Sir Isaac Newton*.

Indeed, Ibn Al-Haytham’s all important moment of insight, is believed to have come to him during those dark days in captivity, when a small hole-in-the-wall reflected an upside down image of the world outside. A great analogy to the another-door-always-opens mentality towards failure.

Ibn Al-Haytham’s military discipline and dedication to the scientific process is unparalleled and distinguished by his memorable quote:

“The duty of the man who investigates the writings of scientists, if learning the truth is his goal, is to make himself an enemy of all that he reads, and … attack it from every side. He should also suspect himself as he performs his critical examination of it, so that he may avoid falling into either prejudice or leniency.”

— Alhazen

**Post Script Problem**

An interesting part of Ibn Al-Haytham’s lasting legacy is a theorem he published that would go on to be unsolved for almost a thousand years.

Ibn Al-Haytham’s work on catoptrics, in Book V of the Book of Optics, contains a discussion of what is now known as Alhazen’s problem, first formulated by Ptolemy in 150 AD.

It comprises drawing lines from two points in the plane of a circle meeting at a point on the circumference and making equal angles with the normal at that point. This is equivalent to finding the point on the edge of a circular billiard table at which a player must aim a cue ball at a given point to make it bounce off the table edge and hit another ball at a second given point. Thus, its main application in optics is to solve the problem, “Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.” This leads to an equation of the fourth degree or quadratic equation.

This eventually led Ibn Al-Haytham to derive a formula for the sum of fourth powers, where previously only the formulas for the sums of squares and cubes had been stated. His method can be readily generalized to find the formula for the sum of any integral powers, although he did not himself do this (perhaps because he only needed the fourth power to calculate the volume of the paraboloid he was interested in).

Instead, he used his result on sums of integral powers to perform what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. Ibn Al-Haytham eventually solved the problem using conic sections and a geometric proof. His solution was extremely long and complicated and may not have been initially understood by mideval mathematicians reading him in Latin translation. Later mathematicians would be forced to use Descartes’ analytical methods to analyse the problem.

A new algebraic solution was eventually found in 1997 by the Oxford mathematician Peter M. Neumann. In 2011, Mitsubishi Electric Research Laboratories (MERL) researchers Amit Agrawal, Yuichi Taguchi and Srikumar Ramalingam solved the extension of Alhazen’s problem to general rotationally symmetric quadric mirrors including hyperbolic, parabolic and elliptical mirrors. They showed that the mirror reflection point can be computed by solving an eighth degree equation in the most general case. If the camera (eye) is placed on the axis of the mirror, the degree of the equation reduces to six. Alhazen’s problem can also be extended to multiple refractions from a spherical ball. Given a light source and a spherical ball of certain refractive index, the closest point on the spherical ball where the light is refracted to the eye of the observer can be obtained by solving a tenth degree equation.

One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten, Eleven, Twelve, 965