Characterization of 4-self-affine convex quadrangles

Characterize all 4-self-affine convex quadrangles, thereby completing the classification of n-self-affine convex quadrangles for the unresolved case n=4.

Background

The paper completely characterizes 3-self-affine convex quadrangles and recalls that every convex quadrangle is n-self-affine for every n greater than or equal to 5, while the only 2-self-affine convex quadrangles are trapezoids. Consequently, the classification of self-affine convex quadrangles reduces to determining precisely which convex quadrangles admit a dissection into four affine copies of themselves.

References

Here we characterize all $3$-self-affine convex quadrangles, obtaining $5$ one-parameter families and $13$ singular examples of affine types. This way we reduce the quest for all $n$-self-affine convex quadrangles to the open case $n=4$.

— Self-affine quadrangles  (2502.15521 - Richter et al., 21 Feb 2025) in Abstract; Section 1, Introduction