Self-affine quadrangles
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918516992507904 |
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| author | Richter, Christian Zimmermann, Felix |
| author_facet | Richter, Christian Zimmermann, Felix |
| contents | A quadrangle in the Euclidean plane is called $n$-self-affine if it has a dissection into $n$ affine images of itself. All convex quadrangles are known to be $n$-self-affine for every $n \ge 5$. The only $2$-self-affine convex quadrangles are trapezoids. Here we characterize all $3$-self-affine convex quadrangles, obtaining $5$ one-parameter families and $13$ singular examples of affine types. This way we reduce the quest for all $n$-self-affine convex quadrangles to the open case $n=4$.
In addition, we show that there are $n$-self-affine non-convex quadrangles for all $n \ge 3$, but not for $n=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Self-affine quadrangles Richter, Christian Zimmermann, Felix Combinatorics Metric Geometry 52C20 (Primary) 51N10 (Secondary) A quadrangle in the Euclidean plane is called $n$-self-affine if it has a dissection into $n$ affine images of itself. All convex quadrangles are known to be $n$-self-affine for every $n \ge 5$. The only $2$-self-affine convex quadrangles are trapezoids. Here we characterize all $3$-self-affine convex quadrangles, obtaining $5$ one-parameter families and $13$ singular examples of affine types. This way we reduce the quest for all $n$-self-affine convex quadrangles to the open case $n=4$. In addition, we show that there are $n$-self-affine non-convex quadrangles for all $n \ge 3$, but not for $n=2$. |
| title | Self-affine quadrangles |
| topic | Combinatorics Metric Geometry 52C20 (Primary) 51N10 (Secondary) |
| url | https://arxiv.org/abs/2502.15521 |