Minimum opaque barrier for a square

A diagonal across a square misses lines running parallel to it. How little total barrier length can meet every complete straight line that touches a unit square? The pieces may be disconnected and may extend outside the square.[1] [2]

How short can a collection of curves be while meeting every complete line through the unit square? For $[0,1]^2$, the diagonal $y=x$ misses the parallel line $y=x+1/2$, even though that line crosses the square. The four boundary edges do work and have total length four, without being claimed optimal.

The permitted class uses at most countably many rectifiable curves: each has finite curve length. The barrier may be outside the square or disconnected. The condition concerns entire lines meeting the square, including its boundary, rather than only selected directions or finite rays.[1] [2]

No exact answer is known, even for a unit square.[2]

The cited 2026 theorem raises the lower bound slightly above two and still leaves a gap to known constructions. The infimum is the lowest length that can be approached; proving that one permitted barrier attains it is a separate unresolved issue. Restricting to inside-square or connected barriers would change the problem.[2]

It reveals how many viewing directions can share the same blocking material.[1]

Looking from each direction gives a projection of the square and of the barrier. The barrier’s projection must cover the square’s projection in every direction. Overlapping projected pieces can spend length on the same coverage twice, so efficiency links total curve length with widths and orientations across all viewing angles.[1] [2]