There's a simple mechanical device that provies a simplified representation of wave propigation from a given normal to point on path from which to start/generate search for '4 square points'. aka radar 180 wave propigation with two points returning from same time interval would generate canidate points to check for corresponding normal triangle arc points on the other 180 degrees from normal. aka physics wave tank & wave generate from physical sheet movement (in this case just up/down oscillation of path point normal). -- way more simple to capture via math than describe, but some pretty heavy math theory to understand/interpret equations. problem with math is generating/capturing a time interval that links the 'pond ripples" from path point normal in order to derive the 4 equi-distant points at 90 degree arc intervals from a canidate 'ping' point.
aka moving the normal to path point up/down in 'wave tank' and using ripples out from normal path point to simulated changes in normal to path point length to find a point that lies along the closed path to see if 3 other derivable square points exist on the closed path. max propigation to look at is the minimal 45 degree angle square that can contain the full closed path.
It's a variation on the question of can two or more base terms be used in give context. (classical vs. quantum instance)
a & b are two back to back radars. parallel line of 180 radar a, with parallel line of 180 b. at time index a, if results of a - b cancel out, then there's a square along closed path.
time t of 4 points has to correspond to normal of point on path. Maximum propigation distance is the size of the square box that can completely contain the closed path.
On a ligher note: It's the practical application of enclosing oneself inside of a box, with a board & saw. In order to get out, you cut the board in half and then put the two halves back together to get a whole/hole. The whole/hole in this instance being the normal/rope which provides a singularity link to both the square box with 4 points on the closed path and the associated normal to point on closed path line. Which kinda makes it sound like all one really needs is a naked singularity within proximity to schrodinger's cat to figure things out.
So, if one used water transport device upon which one moves up and down to stay afloat & move along the 'close path" of interst; should be able to find the 'squares' of the path by tracing back circular water ripple intersections on path points via video analysis. 4 ripple radii == 1 point on path == 1 square.
ANN should be able to work out the likely hood of the line path proximity density to ripple density relative to bounding box enclosing the closed path in order to get the likely hood of number of square boxes that exist with 4 points on the closed path.
Somewhat related: A video by 3Blue1Brown about the same problem https://www.youtube.com/watch?v=IQqtsm-bBRU