### Abstract

We consider a tiling of a square by finitely many tiles each of which is a rectangle. We do not assume that the tiles are mutually congruent. Such a tiling is called irreducible if for any two tiles the union of them is not a rectangle. A tiling is called generic if no four tiles meet in a point. A tilling is trivial if it has only one tile. A tile r in a generic tiling of a square is called a spiral if it is contained in the interior of the square and for each edge e of r there is a tile s adjacent to r such that the straight line containing e intersects the interior of s. We show that a nontrivial generic irreducible tiling of a square has a spiral.

Original language | English |
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Pages (from-to) | 175-184 |

Number of pages | 10 |

Journal | Journal of Geometry |

Volume | 90 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2008 Dec |

### Fingerprint

### Keywords

- Spiral
- Tiling

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Geometry*,

*90*(1-2), 175-184. https://doi.org/10.1007/s00022-008-1923-5

**An irreducible rectangle tiling contains a spiral.** / Motohashi, Tomoe; Taniyama, Kouki.

Research output: Contribution to journal › Article

*Journal of Geometry*, vol. 90, no. 1-2, pp. 175-184. https://doi.org/10.1007/s00022-008-1923-5

}

TY - JOUR

T1 - An irreducible rectangle tiling contains a spiral

AU - Motohashi, Tomoe

AU - Taniyama, Kouki

PY - 2008/12

Y1 - 2008/12

N2 - We consider a tiling of a square by finitely many tiles each of which is a rectangle. We do not assume that the tiles are mutually congruent. Such a tiling is called irreducible if for any two tiles the union of them is not a rectangle. A tiling is called generic if no four tiles meet in a point. A tilling is trivial if it has only one tile. A tile r in a generic tiling of a square is called a spiral if it is contained in the interior of the square and for each edge e of r there is a tile s adjacent to r such that the straight line containing e intersects the interior of s. We show that a nontrivial generic irreducible tiling of a square has a spiral.

AB - We consider a tiling of a square by finitely many tiles each of which is a rectangle. We do not assume that the tiles are mutually congruent. Such a tiling is called irreducible if for any two tiles the union of them is not a rectangle. A tiling is called generic if no four tiles meet in a point. A tilling is trivial if it has only one tile. A tile r in a generic tiling of a square is called a spiral if it is contained in the interior of the square and for each edge e of r there is a tile s adjacent to r such that the straight line containing e intersects the interior of s. We show that a nontrivial generic irreducible tiling of a square has a spiral.

KW - Spiral

KW - Tiling

UR - http://www.scopus.com/inward/record.url?scp=58149201140&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58149201140&partnerID=8YFLogxK

U2 - 10.1007/s00022-008-1923-5

DO - 10.1007/s00022-008-1923-5

M3 - Article

VL - 90

SP - 175

EP - 184

JO - Journal of Geometry

JF - Journal of Geometry

SN - 0047-2468

IS - 1-2

ER -