# Polyiamond Bireptiles

## Introduction

In combinatorial geometry a *reptile* is a geometric figure,
equal copies of which can be joined to form an enlarged form of the figure.
For example, four copies of the P-hexiamond can form a P-hexiamond at
double scale, or four times as large:

Reptiles are known for polyominoes, polyiamonds, polyaboloes,
and other polyforms.

Few polyforms of any kind form reptiles.
A *bireptile* is a figure of which copies can be joined to
form two joined, equally enlarged copies of the original figure.

Any figure with a reptiling trivially has a bireptiling, but not every
figure with a bireptiling has a reptiling.
That is, bireptiles are more common than reptiles.

Below I show minimal known bireptilings for various polyiamonds.

Number of Cells | Number of Reptiles | Number
of Bireptiles |

1 | 1 | 1 |

2 | 1 | 1 |

3 | 1 | 1 |

4 | 2 | 2 |

5 | 1 | 3 |

6 | 4 | 5 |

7 | 0 | 1 |

8 | 7 | 8 |

## Tetriamonds

## Pentiamonds

Yoshiaki Araki has found that the J pentiamond can form
a bireptile at any scale that is a multiple of 5.
Click here to see his first few cases.

Click here for a bireptile at scale 4.

Click here for ten bireptiles at scale 5.

The J pentiamond also has bireptilings at scales 6, 7, and 8.

## Hexiamonds

## Heptiamonds

## Octiamonds

*Last revised 2022-12-07.*

Back to Polyform Tiling
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Polyform Curiosities

Col. George Sicherman
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