Bullseye-shaped interference pattern in seminar-room chair

Question

During a break n a seminar today, I noticed that the chairs in front of me all had slightly transparent black mesh fabric. The backs of the chairs were in the shape of a hyperbolic paraboloid. The fabric covering it had two layers a couple of centimeters apart (it was just a fabric sheath over a frame), and the mesh was a hexagonal tiling with hole size approximately equal to thread size, both small.

I noticed that the interference between the two meshes made a bullseye pattern. The circles in the bullseye moved inwards or outwards as I moved my head, but the center of the bullseye seemed to be constantly at the critical point of the hyperbolic paraboloid (although I couldn't move my head much from side to side).

Why did this shape appear?

Answer 1

I have not seen the chairs in question, so below is just pure speculation. I suspect, but am not sure, that what you saw was a combination of two effects caused by the chair frame.

1. Because the chair frame is a hyperbolic paraboloid, from where you sit the pattern of "holes" in the fabric appears effectively aperiodic while roughly uniformly distributed.
2. Because the chair frame has a physical dimension, one layer of the chair fabric is further away from you then the other. From where you sit the layer that is further away from you appears to be "smaller".

Apparently if you start with an aperiodic screen pattern $A$ and produce a Moiré pattern based on $A$ and a rotation and/or rescaling $A'$, you tend to observe circular patterns. See figures 2.1c and 2.1e here. In general, I think you may be able to find an answer to your question in this book.

update: Links refer to The Theory of the Moiré Phenomenon by Isaac Amidror. Original published in paperback in 2000 (hardcover earlier) and then expanded to two volumes in 2007/2009: volume I and volume II.