Polyiamonds are an interesting field when it comes to convex shapes. Unlike polyhexes and polyominoes, they offer some variety -- the only convex polyhex shape is the monohex itself and the only convex polyomino shapes are rectangles. With polyiamonds we can have anything from triangles to hexagons, at least in theory.
Generally speaking, a convex polyiamond shape is necessarily a hexagon whose edges come in three parallel pairs. That's why such a shape can be expanded into a regular isoscele triangle, some of the angles of which are cut.
The area of a side-N triangle is N2 unit triangles. This means that all convex polyiamond shapes of Pn cells have areas that can be expressed in the form:
However, since there are six sides in a hexagon, there are two ways to construct such triangular expansions of each of the convex shapes. These are related by a simple equality:
It is convenient to always choose the smaller enclosing triangle, as shown in the example above, where a second possibility was a side 13 triangle since 13 = 2 × 9 − (1 + 2 + 2). This bigger triangle can be imagined by mentally continuing the three sections line until they intersect into a downwards pointing shape.
For each shape below, I attemp to provide a link to the oldest known appearance. However, this is still very inaccurate. Whenever I haven't seen a shape anywhere, I indicate myself as the author.
For a more detailed analysis of the convex shapes that can be tiled with a complete set of hexiamonds, see this page.
This set hasn't been completely checked: a shape resulting from the equality 301 = 52² − 49² - 2 × 1² remains to be checked for tilability.
Theoretically there exist 34 convex shapes of 528 triangles, 25 of which have some kind of symmetry. However, the full set of octiamonds can tile only shapes with parity a multiple of 4 (4k). Parity is calculated as the difference between upwards and downwards pointing triangular cells.
The remaining non symmetric shapes can be classified as follows:
