The best known packings of equilateral triangles into an equilateral triangle are illustrated above for the first few cases (Friedman).

The best known packings of equilateral triangles into a circle are illustrated above for the first few cases (Friedman).

The best known packings of equilateral triangles into a square are illustrated above for the first few cases (Friedman).

Stewart (1998, 1999) considered the problem of finding the largest convex area that can be nontrivially tiled with equilateral triangles whose sides are integers for a given number of triangles and which have no overall common divisor. There is no upper limit if an arbitrary number of triangles are used. The following table gives the best known packings for small numbers of triangles.

max. area | reference | max. area | reference | ||

1 | 1 | Stewart 1997 | 11 | 495 | Stewart 1997 |

2 | 2 | Stewart 1997 | 12 | 860 | Stewart 1998 |

3 | 3 | Stewart 1997 | 13 | 1559 | Stewart 1998 |

4 | 7 | Stewart 1997 | 14 | 2831 | Stewart 1998 |

5 | 11 | Stewart 1997 | 15 | 4782 | Stewart 1999 |

6 | 20 | Stewart 1997 | 16 | 8559 | Stewart 1998 |

7 | 36 | Stewart 1997 | 17 | 14279 | Stewart 1998 |

8 | 71 | Stewart 1997 | |||

9 | 146 | Stewart 1997 | |||

10 | 260 | Stewart 1997 |