Self-affinity of discs under glass-cut dissections
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909172843413504 |
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| author | Richter, Christian |
| author_facet | Richter, Christian |
| contents | 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 $n \ge 5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11460 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-affinity of discs under glass-cut dissections Richter, Christian Combinatorics Metric Geometry 52C20, 05B45, 51N10, 52B45 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 $n \ge 5$. |
| title | Self-affinity of discs under glass-cut dissections |
| topic | Combinatorics Metric Geometry 52C20, 05B45, 51N10, 52B45 |
| url | https://arxiv.org/abs/2404.11460 |