Self-affinity of discs under glass-cut dissections

Abstract

A topological disc is called $n$-self-affine if it has a dissection into $n$ affine images of itself. It is called $n$-gc-self-affine if the dissection is obtained by successive glass-cuts, which are cuts along segments splitting one disc into two. For every $n \ge 2$, we characterize all $n$-gc-self-affine discs. All such discs turn out to be either triangles or convex quadrangles. All triangles and trapezoids are $n$-gc-self-affine for every $n$. Non-trapezoidal quadrangles are not $n$-gc-self-affine for even $n$. They are $n$-gc-self-affine for every odd $n \ge 7$, and they are $n$-gc-self-affine for $n=5$ if they aren't affine kites. Only four one-parameter families of quadrangles turn out to be $3$-gc-self-affine. In addition, we show that every convex quadrangle is $n$-self-affine for all $n \ge 5$.