Polyiamonds

A polyiamond is a plane figure composed of a number of equilateral triangles joined by common edges. Some polyiamonds are rep-tiles.

Moniamond

The moniamond is the equilateral triangle. This is a rep-tile (a special case of the rep-k² triangle). More details will be given in the discussion of triangles as IFS attractors.

Diamond

The diamond is a 60° rhombus. This is a rep-tile (a special case of the rep-k²parallelogram). More details will be given in the discussion of parallelograms as IFS attractors.

Triamond

The triamond is a isosceles trapezium (trapezoid in American usage), and also a demihexagon. This is also a rep-tile. Like the moniamond and diamond it is rep-k² tile. I first encountered this dissection on Gary Teachout's web site, where he presented it as an L-system. I have since encountered it at Eric Weisstein's World of Mathematics.

If a m copies of a n-iamondcan be fitted together to form an equilateral triangle, then this construction gives us a rep-mn tile corresponding to that n-iamond. The only polyiamond which I have created such a construction is the triamond.

Tetriamonds

There are 3 tetriamonds. 2 of these are rep-tiles.

In general any bar polyiamond consisting of an even number of equilateral triangles is a rep-tile, with the same construction as is shown for the diamond, bar tetriamond, bar hexiamond (below) and bar octiamond (below).

Pentiamonds

There are 4 pentiamonds. No investigation has been performed into which, if any, of these, are rep-tiles.

Hexiamonds

There are 12 hexiamonds. At least 4 (rhomboid, sphinx, lobster and bat hexiamonds) are rep-tiles.

Heptiamonds

There are 24 heptiamonds. No investigation has been performed into which, if any, of these, are rep-tiles.

Octiamonds

There are 66 octiamonds. At least 7 are rep-tiles.

1. diamond octiamond
2. bar octiamond
3. trapezoidal octiamond (3:2:1 isosceles trapezium)
4. "T" octiamond
5. "L+" octiamond
6. "L-" octiamond
7. sphinx octiamond

Polyiamond	IFS	Name	Min # in unit cell Rep-number
		bar octiamond	1 4
		diamond (octiamond)	1 4
		T octiamond	4 16
		L+ octiamond	2 4
		L+ octiamond	2 16
		L- octiamond	2 4
		L- octiamond	2 16
		trapezoidal octiamond (3:2:1 isosceles trapezium)	2 9
See A.L. Clarke's PolyPages	See A.L. Clarke's PolyPages	sphinx octiamond	2 49

Sources: The sphinx hexiamond is taken from Eric Weisstein's World of Mathematics. The trapezoidal octiamond (3:2:1 isosceles trapezium) is taken from A.L. Clarke's PolyPages

References:

© 2001, 2002 Stewart R. Hinsley