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Self-Affinity of Discs Under Glass-Cut Dissections.

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Title: Self-Affinity of Discs Under Glass-Cut Dissections.
Authors: Richter, Christian1 (AUTHOR) christian.richter@uni-jena.de
Source: Discrete & Computational Geometry. Jul2026, Vol. 76 Issue 1, p466-490. 25p.
Subjects: Quadrilaterals, Triangles, Affine transformations, Trapezoids, Topology
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 ≥ 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 ≥ 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 ≥ 5. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Jul2026, Vol. 76 Issue 1, p466-490. 25p.
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  Data: 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 ≥ 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 ≥ 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 ≥ 5. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00454-025-00777-8
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      – Code: eng
        Text: English
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      – SubjectFull: Quadrilaterals
        Type: general
      – SubjectFull: Triangles
        Type: general
      – SubjectFull: Affine transformations
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      – SubjectFull: Trapezoids
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      – SubjectFull: Topology
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      – TitleFull: Self-Affinity of Discs Under Glass-Cut Dissections.
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              Text: Jul2026
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