2022/11/05

A polyhexagon, or a polyhex for short, is a figure made from regular hexagons joined at their sides. If we take four hexagons we get seven different two-sided pieces, which can fill many areas of size 28=4*7. Let's add stripes to these polyhexes. Now 16 different two-sided pieces are possible, but we get some parity problems. Therefore I added the five trihexes in the picture above. All stripes are othogonal to two sides of the hexagons and all stripes of the pieces are aligned. You can also use stripes parallel to two sides of the hexagon or stretched hexagons. Furthermore you can print these pieces with the provided OBJ files.

It's always more difficult for the striped pieces to solve a given area because the pieces can only be rotated by 180° instead of multiples of 60° for the usual pieces. In some cases the required direction of the stripes for a given shape prevents or allows a solution. Here are some set of pieces. Click the numbers to see possible constructions.

Number of Hexagons for Each Piece | Two-sided Pieces | One-sided Pieces | ||||
---|---|---|---|---|---|---|

No Stripes | With Stripes | Total Area | No Stripes | With Stripes | Total Area | |

3 | 3 | 5 | 15 | 3 | 7 | 21 |

2 or 3 | 4 | 7 | 19 | 4 | 10 | 27 |

4 | 7 | 16 | 64 | 10 | 28 | 112 |

3 or 4 | 10 | 21 | 79 | 13 | 35 | 133 |

5 | 22 | 55 | 275 | 33 | 99 | 495 |

6 | 82 | 225 | 1350 | 147 | 433 | 2598 |

Another version of printed pieces with compressed hexagons instead of stripes is shown below.