In the early 1950s, mathematician Ernst Straus came up with an interesting problem. He wondered whether it would be possible to design a room completely lined with mirrors, but designed in such a way that it would not be 100% illuminated by a single match. For this thought experiment, he assumed that the light from the match could travel infinitely far, but only in straight lines and at mathematically precise angles, with no scattering.

It turns out that there are indeed theoretically unilluminable rooms. The first one was discovered in 1958, by Roger Penrose. It uses the magic of ellipses to guarantee that no matter where the light is ignited, there will be somewhere that remains in shadow. The large ellipses focus light on the O's (located on the lateral surfaces of the apex of the cornua), leaving dark spaces in at least two of the pockets. If the light-bearer stands in one of the pockets, more than half of the room is left in darkness; if she stands in the center of the room, all four pockets are dark. (further illustration). In the illustration below, the light source is marked by the X, and the shadows are marked by dots.


   MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
   MMMMMMMMMMMMMMMMMMMMMM                         MMMMMMMMMMMMMMMMMMMMMM
   MMMMMMMMMMMMMMMMM                                  MMMMMMMMMMMMMMMMMM
   MMMMMMMMMMMMMM                                         MMMMMMMMMMMMMM
   MMMMMMMMMMM                                              MMMMMMMMMMMM
   MMMMMMMMM                                                    MMMMMMMM
   MMMMMMM                                                       MMMMMMM
   MMMMM                                                           MMMMM
   MMMM                X                                            MMMM
   MMM                                                               MMM
   MM                                                                 MM
   MM                                                                 MM
   MM      OMMM                                             MMMO      MM
   MM      MMMMM                                           MMMMM      MM
   MM      MMMMMM                                         MMMMMM      MM
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   MMMMMMMMMMMMMM                                         MMMMMMMMMMMMMM
   MMMMMMMMMMMMMM                                         MMMMMMMMMMMMMM
   MMMMMMMMMMMMMM                                         MMMMMMMMMMMMMM
   MM .... MMMMMM                                         MMMMMM .... MM
   MM .... MMMMMM                                         MMMMMM .... MM
   MM .... MMMMM                                           MMMMM .... MM
   MM .... OMMM                                             MMMO .... MM
   MM                                                                 MM 
   MM                                                                 MM   
   MMM                                                               MMM   
   MMMMM                                                            MMMM   
   MMMMMM                                                          MMMMM   
   MMMMMMM                                                       MMMMMMM   
   MMMMMMMMM                                                   MMMMMMMMM   
   MMMMMMMMMMM                                               MMMMMMMMMMM   
   MMMMMMMMMMMMMM                                         MMMMMMMMMMMMMM   
   MMMMMMMMMMMMMMMMMM                                 MMMMMMMMMMMMMMMMMM   
   MMMMMMMMMMMMMMMMMMMMMMMM                      MMMMMMMMMMMMMMMMMMMMMMM   
   MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM       


In 1995 the first polygonal unilluminable room was designed by George Tokarsky, who proved that a light ray starting from one corner of a mirrored right isosceles triangle would never return to that original corner. Armed with this information and some topological trickery, he created a very pointy 26-sided room in which a single light source, placed in just the right spot, leaves a corner in darkness -- but unfortunately this did not hold true for every spot in the room; you could find many places to stand in which you could light to whole room.


          ZZZZZZZZZZ                                       +                    
          ZZZZZZZZZZ                                       ZZ+                  
          ZZZZZZZZZZ                                       ZZZ?                 
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          ZZZZZZZZZZ                                       ZZZZZZZ              
          ZZZZZZZZZZ                                       ZZZZZZZZN            
         ?ZZZZZZZZZZZZZZZZZZZZ                   ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
        ZZZZZZZZZZZZZZZZZZZZZZ                   ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
      +ZZZZZZZZZZZZZZZZZZZZZZZ                   ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     ZZZZZZZZZZZZZZZZZZZZZZZZZ                   ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
   NZZZZZZZZZZZZZZZZZZZZZZZZZZ                   ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
  ZZZZZZZZZZZZZZZZZZZZZZZZZZZZ                   ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
8ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZD 
          ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ           
          ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ           
          ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ           
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          ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ           
                    ZZZZZZZZM          ZZZZZZZZZ           ZZZZZZZZ+            
                    ZZZZZZZ            ZZZZZZZN            ZZZZZZZ              
                    ZZZZZN             ZZZZZZ              ZZZZZN               
                    ZZZZ               ZZZZM               ZZZZ                 
                    ZZ+                ZZZ                 ZZN                  
                    +                  Z?                  Z                 

These remain the most well-known of the unilluminable rooms, although in 1997 D. Castro managed to create a polygonal room with only 24 sides (illustration). It also is only 'unilluminable' given certain light source placements, but the resulting areas of shadow are much larger. To the best of my knowledge these rooms remain mathematical constructs, never having been built. Granted, due to the sad absence of perfectly flat mirrors and a scatter-free atmosphere, a real-world room would not have truly unilluminable spaces (given the hypothesized infinitely-strong light source, anyway). Perhaps even more disappointing, I can find no work being done on the smallest possible unilluminable room given two light sources; perhaps thesis material for an up-and-coming topologist?




Other References:
http://plus.maths.org/issue8/xfile/index.html
http://mathworld.wolfram.com/IlluminationProblem.html

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