The idea is to surround shapes with a channel that is only 1/2 of a triangle thick. Since we want to keep all pieces placed on the polyiamond grid (except for the ones in the island, obviously), only islands composed of same direction parallelograms are possible. This is a severe restriction.
Because of the restriction mentioned above, only three hexiamonds can be surrounded by a narrow channel and tiled with the whole set. The same symmetric envelopping shape is used below:
A convex 82-iamond envelop is also possible:
The following has been found by George Sicherman who also has proved that a convex envelop is impossible for the Bar-hexiamond:
When convexity is sacrificed, islands of two pieces are easy to find, although the envelopping shapes look terribly bad. A symmetric island is possible two:
I found one convex envelopping shape, corresponding to the 86-iamond shown below. George Sicherman informs me that no other such shape exists:
An island of three pieces surrounded by a regular narrow channel isn't difficult to produce. Once the island, composed of same direction parallelograms, has been composed, I surround it manually with the remaining pieces. Here are three examples obtained "by hand":
Here is another neat solution by Roel Huisman, done with a computer:
And another one by George Sicherman, also using a computer:
Is an island of four pieces possible? George Sicherman has proved that a 3x4 parallelogram, the most compact possible island, is impossible. This casts a serious doubt on the possibility for a more complex island.
Is it possible to surround two islands with a narrow channel? I tried a few times by hand with no success; George Sicherman was victorious of this problem, also working with physical pieces and obtaining the following solution: