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 $$n \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , 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 $$n \ge 7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> , and they are n -gc-self-affine for $$n=5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> 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 $$n \ge 5.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>