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Alhazen’s problem

The International Year of Light aims to raise global awareness about how light-based technologies can promote sustainable development and provide solutions to global challenges in energy, education, agriculture, and health. To celebrate this special year over the course of March, three of our authors will write about light-based technologies and their importance to us. This week, Stephen R. Wilk discusses the contributions of Alhazen to the field of optics.

One of the reasons that 2015 has been declared the International Year of Light is that it marks the 1000th year since the publication of Kitāb al-Manāẓir, The Treasury of Optics, by the mathematician and physicist Abu Ali al-Hasan ibn al-Hasan ibn al-Haitham, better known in Western cultural history as Alhazen. Born in Basra in present-day Iraq, he is acknowledged as the most important figure in optics between the time of Ptolemy and of Kepler, yet he is not known to most physicists and engineers.

A quick survey of my own optics books reveals that almost none of them even mention him. My first introduction to him came in the form of a Ripley’s Believe it or Not! cartoon that claimed his interest in optics was sparked by his contemplation of the iridescent colors of a soap bubble. I have not, however, been able to find any reference to the story outside of Ripley.

Much of the material in the first three books of the Treasury is concerned with color theory and visual perception. But you would think that the material in the fourth and fifth books, being devoted to the problems of reflection and refraction, would be of greater relevance. Aside from his ideas about vision, about the only element that gets touched on in most mentions of Alhazen is his provision of the earliest description of the camera obscura.

Ibn_al-Haytham
“Ibn al-Haytham [Alhazen] as shown on the obverse of the 1982 Iraqi 10 dinar note.” Scan by Jacob L. Bourjaily; Banknote produced by Iraqi government. CC by SA 3.0 via Wikimedia Commons.

He did much more, of course. He came very close to discovering the law of refraction. He solved multiple problems in reflection, building upon the notion that the angle of incidence equals the angle of reflection, along with the tenets of geometry. That most people are unaware of this might be the real Alhazen’s Problem.

But there is one specific detail he examined that captured the attention of Christiaan Huygens and others who sought to find a simpler algebraic solution than the rather complex geometrical one found by Alhazen himself; Huygens described it as longa admodum ac tediosa–“too long and tedious”. The problem asks that, if light emanates from one point and reflects from a spherical mirror to a second given point, at which point does the ray strike the mirror? This has given rise to a whimsical name for the situation–Alhazen’s Billiard Problem–since it is equivalent to assuming that one strikes a ball on a circular billiard table with the aim of sending it to a designated second point. At what point on the circular cushion should one aim the ball?

It’s conceptually a simple problem and can be solved on the basis of the assumptions of circular geometry and the law of reflection. But the problem resists an easy solution using compass and straightedge.

Alhazen’s solution was achieved by use of six subsidiary proofs or lemmas (Arabic muqaddamāt). Alhazen was able to show, by purely geometrical arguments, that the location of the point on the circular mirror at which the reflection must occur lies at the intersection of that circle with a hyperbola.

Modern treatments solve the problem by a combination of analytical geometry, algebra, and trigonometry. At this point, the impatient engineer, who simply wants a solution, asks in exasperation, “Why don’t you simply divide up the circumference into equally spaced points, calculate the angles of incidence and reflection, and take the position where these are the closest? If you’re going to use a computer anyway, why not simply use the Brute Force approach?”

In fact, we may ask why it is that this problem is not more widely known among optical scientists and engineers. Reflection from a concave or convex spherical surface is one of the most basic problems in geometrical optics. It’s taught in every introductory class on geometrical optics, but Alhazen’s name is virtually never associated with it. Why not?

The problem, as stated above, is not one of great utility in optics and optical design. One rarely asks, “Where must a ray coming from point A and going to point B strike by spherical mirror in order to get there?” One asks, “Where do these rays, leaving point A, go after they have struck the spherical mirror?” In the usual paraxial optical formulation, the student uses the approximation that virtually every point from a point A that is further than a focal point away from a concave mirror will pass through the same point B in the corresponding image. Teaching them about Alhazen’s Problem will only confuse them, without the compensating virtue of teaching them a general principle.

So why is Alhazen’s problem important to anyone but circular billiards players? It’s because Alhazen himself did not see this as the end of his calculations, but as another step that was necessary for the solution of his ultimate problem–the way that light reflected from curved surfaces.

To cite historian A.J. Sabra:

“The problem of finding the reflection-point occurs in [The Treasury of Optics] as part of a long series of investigations of specular images which occupy the whole of Book V, and these investigations in turn presuppose a theory of optical reflection which is expounded in Book IV. Much of the character of Ibn Al-Haytham’s treatment of reflection-points can only be appreciated if understood with reference to this wider context… Ibn Al-Haytham was therefore aiming to solve a wider and more complex set of problems than “Alhazen’s problem” in Huygens’ limited sense.”

In other words, Alhazen’s “complicated and tedious” solution was simply one piece in his carefully argued development for calculating how light was reflected from non-planar mirrors. Concentrating only on this one piece of the puzzle might provide diversion for mathematicians, but that was not Alhazen’s goal. Modern beginning optical students do learn Alhazen’s results, although they no more use his methods than modern students of algebra use those of medieval scholars.

Image Credit: “Optics Final” by Bryan. CC by NC 2.0 via Flickr.

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