Opaque Squares and Cubes

An old puzzle asks: A man has a square back yard. What is the least length fence he can put in it to keep anybody from looking through his yard from any angle? Equivalently, what is the least length set of lines that will intersect every straight line passing through a unit square? A square with such a fence is called an "opaque square". Two opaque squares are shown here, but these are not the best possible. If you know the principles of soap films or Steiner networks, you can recognize that the first is not optimal since edges in soapfilms or Steiner networks always meet in threes. The second satisfies these conditions, and is the minimal soap film or Steiner network spanning the corners of a square, but it is not the optimal opaque fence. Try figuring out the better one before looking at the answer.

opaque X opaque H

The Opaque Cube Problem poses the analogous challenge in three dimensions: What is the least area surface that intersects all straight lines passing through a cube?

opaque cube opaque cube

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