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
111
211
311
422
513
645
701
878

Tetriamonds

Pentiamonds

Hexiamonds

Heptiamonds

Octiamonds

Last revised 2015-12-10.


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Col. George Sicherman { HOME | MAIL }