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 $$n \ge 2$$ n ≥ 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 $$n \ge 7$$ n ≥ 7 , and they are n-gc-self-affine for $$n=5$$ 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 $$n \ge 5.$$ n ≥ 5 .