Six vertical mirrors are arranged to form a regular hexagon with only a small gap (of negligible size) at each vertex. A beam of light is shone through one of these gaps toward the one directly opposite, through which it exits.
However, the beam can be aimed slightly off this direct route to the other side, so that it is reflected off each of the mirrors exactly once before finally exiting the arrangement through that same opening opposite its entrance.
What is the smallest angle off of straight across which will make this possible?
Let ABCDEF be the hexagon with A and D
the source and target vertices respectively.
Reflect the hexagon six times about edges
DE, B_1C_1, A_2F_2, C_3D_3, E_4F_4,
Where the vertices of the hexagon are
labeled thus for each reflection
X --> X_1 --> X_2 --> X_3 --> X_4
--> X_5 --> X_6.
Draw the line AD_6.
The angle we desire is
Angle BAD_6 - Angle BAD.
These angles are easily determined from
the reflection diagram.
arctan(3*sqrt(3)/2) - 60
Posted by Bractals
on 2009-05-08 15:03:37