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<p>A topological disc is called <i>n</i>-self-affine if it has a dissection into <i>n</i> affine images of itself. It is called <i>n</i>-gc-self-affine if the dissection is obtained by successive glass-cuts, which are cuts along segments splitting one disc into two. For every <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we characterize all <i>n</i>-gc-self-affine discs. All such discs turn out to be either triangles or convex quadrangles. All triangles and trapezoids are <i>n</i>-gc-self-affine for every <i>n</i>. Non-trapezoidal quadrangles are not <i>n</i>-gc-self-affine for even <i>n</i>. They are <i>n</i>-gc-self-affine for every odd <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n \ge 7\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>, and they are <i>n</i>-gc-self-affine for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n=5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <i>n</i>-self-affine for all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n \ge 5.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Self-Affinity of Discs Under Glass-Cut Dissections

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  • 发表日期：

    2025年10月15日

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  • Christian Richter

摘要

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 .