Abstract: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 $n=2$.
| Comments: | 19 pages, 11 figures. New version includes corrections of some typos, an extended proof of Lemma 13 and ancillary files (of computations for Section 4) |
| Subjects: | Combinatorics (math.CO); Metric Geometry (math.MG) |
| MSC classes: | 52C20 (Primary) 51N10 (Secondary) |
| Cite as: | arXiv:2502.15521 [math.CO] |
| (or arXiv:2502.15521v2 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2502.15521 arXiv-issued DOI via DataCite |
From: Christian Richter [view email]
[v1]
Fri, 21 Feb 2025 15:22:50 UTC (27 KB)
[v2]
Fri, 22 May 2026 10:40:01 UTC (17,049 KB)
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