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Unit square overlap II (Posted on 2023-05-18) Difficulty: 4 of 5
Two unit squares overlap with an area 1/16. The shape of this overlap may be a triangle or a quadrilateral.

Find the minimum distance between their centers.

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further soln Comment 2 of 2 |
This is a great problem: easy to pose, hard to answer. 

I refined the solution. The best answer is: minimum separation distance
d = 0.933012705 +/- .000000003

It is plotted here.

I had to determine two things via simulations: 
(A table from simulations is here.) 

1) The minimal separation is made with an overlap shape that 
is 3-sided, not 4-sided. (Note that the overlap of two squares
can have between 3-8 sides.) 

Here are examples of small overlaps. 

Many 4 sided overlaps came close, but the triangle cases win.

2) In the minimal separation the squares centers are connected 
by a line perpendicular to the sides. I learned this via 
simulation, without a proof. It seems natural however, since 
one square is poking a corner onto the other while trying to 
stay as close as possible. 

The best rotation for the 2nd square around its axis is 
about 9.7307 degrees. 
Statement 2 does make the problem amenable to calculus. The overlap 
area is now determined by only two factors - theta (rotation of 2nd 
square on it's axis, and separation d. Since A = f(d , theta) and A is 
constant, one can make d a function of theta and differentiate for a 
minimum. The math is a bit messy - Euler rotation angles allow the 
solution of intersecting lines. Rather than invert and differentiate - 
I just ran the math forward and the simulations further. 

The parameters listed produce a 1/16 area overlap to an accuracy of 10^(-9).

Edited on July 6, 2023, 3:25 pm
  Posted by Steven Lord on 2023-07-06 13:52:46

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