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