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You have just acquired a charming, yet “atypical” apartment, as the estate agent described it. Moving in, you realise that the rooms are quite uneven and that the layout of the walls creates numerous nooks and crannies. Where should you place a lamp that will light up the whole room? After looking for a while, you decide not to light the room directly. Following the advice of an interior designer, you cover the walls with mirrors to eliminate the dark corners. Yet you still have the impression that you cannot light the whole room. This is a mathematical problem called illumination.
The mathematical theory dates from the 1950s, when the American mathematician Ernst Straus asked whether a room covered in mirrors could always be fully lit from any point or whether any room could be lit from at least one point. The answer to this question of geometry is far from obvious. We need to understand that light radiates from the source to the walls, reflects once in the mirrors, reaches the walls again and reflects again in the mirrors... In this way, it can reach a “dark” corner after numerous reflections on the walls (similar to a billiard problem).
The inital problem is three-dimensional, but it could be considered two-dimensional by supposing that the walls of the room are a curve drawn on a plane. A first answer came from Roger Penrose, a British PhD student at Cambridge and now famous for his work in cosmology and on tiling. He proposed a counter-example by building a room that could not be fully lit up from any one point; its shape is an ellipse to which two corners are added.
The problem occurs in another form with the additional hypothesis that the room is a polygon. Several mathematicians have showsn examples of rooms which could not be lit from all of their points. The most famous is a room proposed by the Canadian George Tokarsky in 1995; it has 26 walls and there is one position for the light source that lights the whole room, except for one point (1). The case was simlified by the student David Castro, who obtained a similar room that “only” had 24 walls. Remarkable ingenious results, but not exactly satisfying for the original problem. Only mathematicians could conceive of a room missing one point.
While Straus’s initial question is resolved, you still don’t know whether you can light your room reasonably. You could deal with a single unlit point, but nothing confirms that your apartment is not a counter-example even more freakish than the ones previously shown. In a 2016 article, Samuel Lelièvre, Thierry Monteil and Barak Weiss provided an answer that should reassure you. In a special class of rooms known as translation surfaces, they showed that wherever you place the lamp, there will only be a finite number of unlit points (2).
This result is based on the work of mathematicians Alex Eskin, Maryam Mirzakhani (the only female Fields medal winner, recently deceased, see La Recherche no. 527, p. 20) and Amir Mohammadi. They offered in-depth understanding of the action of a group of transformations, called special linear group - SL2(Z) for short.
Your apartment is not really a translation surface though; to obtain such a surface, you have to “stick” together parallel walls from different rooms, which modifies the layout slightly. However, the result can be deduced (under reasonable hypotheses) using a technique of successive unfolding of the surface. So, don’t worry, your quirky apartment with its mirrored walls can be lit (almost) entirely.
(1) G. W. Tokarsky, Am. Math. Mon., 102, 867, 1995.
(2) S. Lelièvre et al., Geom. Topol., 20, 1737, 2016.
Howard Masur, mathematician, talks about the illumination problem on the Numberphile channel.
Roger is a teacher at the Louis-le-Grand secondary school in Paris. Evenry month, he explains a mathematical issue in a public talk.