Orthodiagonal Maps, Tilings of Rectangles, and their Convergence to Conformal Maps
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| Format: | Preprint |
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2024
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| _version_ | 1866908373145878528 |
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| author | Binder, Ilia Pechersky, David |
| author_facet | Binder, Ilia Pechersky, David |
| contents | A classic result of Brooks, Smith, Stone and Tutte associates to any finite planar network with distinguished source and sink vertices, a tiling of a rectangle by smaller subrectangles whose aspect ratios are given by the conductances of corresponding edges in the network. This tiling can be viewed as a discrete analogue of the uniformizing conformal map that maps a simply connected domain with four distinguished prime ends to a rectangle, so that the four prime ends are mapped to the four corners of the rectangle. \\ \\ We make this intuition precise by showing that if $Ω$ is a simply connected domain with four distinguished prime ends $A,B,C,D$ in counterclockwise order and $(Ω_{n})_{n\geq{1}}$ is a sequence of orthodiagonal maps with distinguished boundary vertices $A_{n}, B_{n}, C_{n}, D_{n}$ in counterclockwise order, that are finer and finer approximations of $Ω$ with its distinguished boundary points $A,B,C,D$, then the corresponding ``rectangle tiling maps" converge uniformly on compacts to the aforementioned conformal map on $Ω$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_20851 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orthodiagonal Maps, Tilings of Rectangles, and their Convergence to Conformal Maps Binder, Ilia Pechersky, David Complex Variables Mathematical Physics Probability A classic result of Brooks, Smith, Stone and Tutte associates to any finite planar network with distinguished source and sink vertices, a tiling of a rectangle by smaller subrectangles whose aspect ratios are given by the conductances of corresponding edges in the network. This tiling can be viewed as a discrete analogue of the uniformizing conformal map that maps a simply connected domain with four distinguished prime ends to a rectangle, so that the four prime ends are mapped to the four corners of the rectangle. \\ \\ We make this intuition precise by showing that if $Ω$ is a simply connected domain with four distinguished prime ends $A,B,C,D$ in counterclockwise order and $(Ω_{n})_{n\geq{1}}$ is a sequence of orthodiagonal maps with distinguished boundary vertices $A_{n}, B_{n}, C_{n}, D_{n}$ in counterclockwise order, that are finer and finer approximations of $Ω$ with its distinguished boundary points $A,B,C,D$, then the corresponding ``rectangle tiling maps" converge uniformly on compacts to the aforementioned conformal map on $Ω$. |
| title | Orthodiagonal Maps, Tilings of Rectangles, and their Convergence to Conformal Maps |
| topic | Complex Variables Mathematical Physics Probability |
| url | https://arxiv.org/abs/2407.20851 |