# Supporting flexible square plate of uniform density in equilibrium with only 4 rolling contacts placed along the square's diagonals

Consider a flexible square of uniform density with vertices (0,0), (0,1), (1,0), (1,1). I need the 4 coordinates for rolling contacts along the diagonals that will stably support it. Someone could just test it. Open to all thoughts and ideas.

This seems to be similar to the Bessel Points problem, but now in 2D. Guessing the answer is around (.22, .22), (.22, .78), (.78, .22), (.78, .78). How can I find this answer?

• What do you mean by "stably support it?" Obviously, the closer the points are to the center the less stable it will be in a practical sense, but there isn't a position where it flips from stable to unstable. – alephzero Dec 28 '18 at 22:55
• @alephzero Good point - sorry for using "stable" colloquially. Technically the solution is unstable. I just meant that these exact coordinates for the rolling contacts will not be too close to the center (causing the contacts to roll toward the center) and not too far (causing the contacts to roll toward the vertices). – Larry Panozzo Dec 28 '18 at 23:44
• I am not sure if we can expect an infinitesimal section of the plate directly above the contacts to be parallel to the ground, or if this position is not the equilibrium where the net torques around the contacts is 0. There may not even be a dynamic equilibrium solution. Practically, I'm most interested in making the plate parallel above the contacts. – Larry Panozzo Dec 28 '18 at 23:45