{
  "id": "2110.02558",
  "version": "v3",
  "published": "2021-10-06T08:03:26.000Z",
  "updated": "2022-04-09T10:47:58.000Z",
  "title": "Transformations of 2-port networks and tiling by rectangles",
  "authors": [
    "Svetlana Shirokovskikh"
  ],
  "comment": "In English and in Russian. English translation added and minor improvement of exposition performed",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we study 2-port networks and introduce new concepts of voltage drop and $\\Pi$-equivalence. The main result is that each planar network is $\\Pi$-equivalent to a network with no more than 5 edges. This implies that if an octagon in the shape of the letter $\\Pi$ can be tiled by squares then it can be tiled by no more than 5 rectangles with rational aspect ratios. Kenyon's theorem from 1998 proves this only for 6 rectangles.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-04-09T10:47:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rectangles",
      "transformations",
      "rational aspect ratios",
      "main result",
      "planar network"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "ru",
      "license": "arXiv",
      "status": "editable"
    }
  }
}