### Abstract

In order to tile the unit square with rational triangles, at least four triangles are needed. There are four candidate configurations: one is conjectured not to exist; two others are dealt with elsewhere; the fourth is the "nuconfiguration, " corresponding to rational points on a quartic surface in affine 3-space. This surface is examined via a pencil of elliptic curves. One rank-3 curve is treated in detail, and rational points are given on 772 curves of the pencil. Within the range of the search there are roughly equal numbers of odd and even rank and those of rank 2 or more seem to be at least 0.45 times as numerous as those of rank 0. Symmetrical solutions correspond to rational points on a curve of rank 1, which exhibits an almost periodic behavior.

Original language | English (US) |
---|---|

Pages (from-to) | 195-202 |

Number of pages | 8 |

Journal | Mathematics of Computation |

Volume | 59 |

Issue number | 199 |

DOIs | |

State | Published - 1992 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*59*(199), 195-202. https://doi.org/10.1090/S0025-5718-1992-1134716-2

**Nu-configurations in tiling the square.** / Bremner, Andrew; Guy, Richard K.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 59, no. 199, pp. 195-202. https://doi.org/10.1090/S0025-5718-1992-1134716-2

}

TY - JOUR

T1 - Nu-configurations in tiling the square

AU - Bremner, Andrew

AU - Guy, Richard K.

PY - 1992

Y1 - 1992

N2 - In order to tile the unit square with rational triangles, at least four triangles are needed. There are four candidate configurations: one is conjectured not to exist; two others are dealt with elsewhere; the fourth is the "nuconfiguration, " corresponding to rational points on a quartic surface in affine 3-space. This surface is examined via a pencil of elliptic curves. One rank-3 curve is treated in detail, and rational points are given on 772 curves of the pencil. Within the range of the search there are roughly equal numbers of odd and even rank and those of rank 2 or more seem to be at least 0.45 times as numerous as those of rank 0. Symmetrical solutions correspond to rational points on a curve of rank 1, which exhibits an almost periodic behavior.

AB - In order to tile the unit square with rational triangles, at least four triangles are needed. There are four candidate configurations: one is conjectured not to exist; two others are dealt with elsewhere; the fourth is the "nuconfiguration, " corresponding to rational points on a quartic surface in affine 3-space. This surface is examined via a pencil of elliptic curves. One rank-3 curve is treated in detail, and rational points are given on 772 curves of the pencil. Within the range of the search there are roughly equal numbers of odd and even rank and those of rank 2 or more seem to be at least 0.45 times as numerous as those of rank 0. Symmetrical solutions correspond to rational points on a curve of rank 1, which exhibits an almost periodic behavior.

UR - http://www.scopus.com/inward/record.url?scp=84966200826&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966200826&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1992-1134716-2

DO - 10.1090/S0025-5718-1992-1134716-2

M3 - Article

AN - SCOPUS:84966200826

VL - 59

SP - 195

EP - 202

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 199

ER -