Illuminating a Convex Solid

[gavinksong]

I'm not sure if this has been posted yet, but it is a nice puzzle.

 

Can three lights always be placed outside of any convex solid such that the solid is completely illuminated?

 

You can assume that:

• We are working in an otherwise empty three-dimensional space
• The lights are point sources.
• The solid is illuminated at every point on its surface where a straight line can be drawn uninterrupted between it and a light source.

[bonanova]

Consider illuminating a sphere. Two antipodal lights at infinity (providing collimated light) will do the job. The circular edges of the two illuminated areas will lie on a common great circle, which has zero area. Since a convex solid will not cast local shadows, the same result holds.

If the lights must be within finite distance, two lights will not suffice for the sphere. If the lights are antipodal, a ring-shaped shadow will remain. If they are sufficiently off-axis, the circular edges of their illuminated areas will intersect, leaving an "orange slice" - shaped sector of dark area having vertices  greater than 180 degrees apart. Even a third light placed at infinity would not suffice to illuminate this area. So three lights (at finite distance) will not suffice to illuminate a sphere. Four are required. That result applies to convex solids in general.

Edited January 22, 2015 by bonanova
Reworded my answer by describing illuminated areas, rather than shadows

[gavinksong]

Consider illuminating a sphere. Two antipodal lights at infinity (providing collimated light) will do the job. The circular edges of the two illuminated areas will be coinciding great circles. Since a convex solid will not cast local shadows, the same result holds.

If the lights must be within finite distance, two lights will not suffice for the sphere. If the lights are antipodal, a ring-shaped shadow will remain. If they are sufficiently off-axis, the circular edges of their illuminated areas will intersect, leaving an "orange slice" - shaped sector of dark area having vertices  greater than 180 degrees apart. Even a third light placed at infinity would not suffice to illuminate this area. So three lights (at finite distance) will not suffice to illuminate a sphere. Four are required. That result applies to convex solids in general.

 

Let the solid be a sphere. Consider two lights in arbitrary positions. Together with the center of the sphere, they define a plane. The two points on the sphere farthest from the plane are antipodal, and are not illuminated by the two lights. The third light cannot illuminate both antipodal points.