Polyomino Reptiles

Hexominoes

While all rectifiable polyominoes are also reptiles, not a great deal is known about polyomino reptiles in general. One example of an hexomino which is rep-n for all n is shown below.

The hexomino replications above can also be formed from a affine transformation of the L-tromino replication. The diagram below shows how two of the L-tromino replications can be stretched horizontally to give replications for the above and for an enneomino.

Several other hexominos are also reptiles and below is some information (mostly from Mike Reid). The rectangles used in some of the constructions can be found at Michael Reid's Rectifiable polyomino page.

 rep-n for all n > 6

Since a 2x6 rectangle is possible so is a 6x6 square which allows us to construct any rep-(6n). The diagram below shows constructions of rep-n for n = 7, 8, 9, 10, 11.

A rep-(6n+7) can be extended to rep-(6n+13) as shown below. Similar constructions extend the rep-(6n+k) for k = 8, 9, 10 and 11.

 rep-n for n = 9 and all n >= 12

The diagram below shows rep-9 , rep-12, rep-13 and rep-14 .

Providing that the rectangles shown exist any rep-n can be extended to a rep-(n+12) as shown below.

 rep-n for n = 24k (k>2), 24k+1 (k>5)

The construction below is based on the fact that 24 x m rectangles exist for m = 23+6n and all m > 98. (see possible rectangles)

 rep-n for n = 12k+8 , 12k+9 , 12k , 12k+13 , 12k+15 , 12k+16 , 12k+17 , 12k+19 and 12k+23

Replications with this hexomino are generally made using combinations of one of three basic shapes. Note that for the odd replications there is a single separate piece.

 rep-n for n = 6k, 12k ± 1

 rep-n for n = 12k, 12k+1, 12k+11

The diagram below shows rep-11 and rep-13.

 rep-n for n = 6k

The known replications for this are made exclusively from rectangles as in the example below.