Four-self-affine convex quadrangles

Characterize all convex quadrangles that admit a dissection into four affine images of themselves, thereby resolving the remaining case in the classification of n-self-affine convex quadrangles.

Background

The paper completely characterizes 3-self-affine convex quadrangles and recalls that every convex quadrangle is n-self-affine for n ≥ 5, while the only 2-self-affine convex quadrangles are trapezoids. Consequently, the classification of n-self-affine convex quadrangles reduces to determining precisely which convex quadrangles admit a 4-self-affinity. The authors explicitly identify this as an open case and state that its investigation is ongoing.