Polyforms derived from the removal of half of one of the basic tiling shapes.
If take the set of tetrominoes and remove half a square (cut across the square) so that no original join is divided then we get the six truncated tetrominoes below.
This set is quite small and does not produce any interesting constructions. Removing a half square from the pentominoes gives us this set of 19 pieces. The only possible rectangles with the truncated pentominoes would be 4½ x 19 or 8½ x 9. Brendan Owen has shown that no solution exists for the 4½ x 19 and Patrick Hamlyn has found two solutions so far for the 8½ x 9.
An easier problem is to make the rectangle below where one of the pieces is used twice.
The 35 one-sided pieces shown below can also form rectangles. At least some of 2½ x 63, 3½ x 45, 4½ x 35, 5 x 31½, 7 x 22½, 7½ x 21, 9 x 17½ and 10½ x 15 are possible (the first two are not possible). The first two solutions are by Peter Esser and the three colour version of the 5 x 31½ is by Patrick Hamlyn. Peter Esser has a page on his website devoted to the one-sided truncated pentominoes.
There are 67 truncated hexominoes shown here in an 11 x 33½ rectangle by Patrick Hamlyn.The only other rectangle with the correct area is a 5½ x 67 which is almost certainly not possible. The second rectangle is an 11x64 with the 128 one-sided pieces.
Another solution for the 11 x 33½ by Roel Huisman shows the symmetric pieces.
The one-sided set consists of 128 pieces which probably form a number of rectangles - 4 x 176, 5½ x 128, 8 x 88, 11 x 64, 22 x 32, 16 x 44 - with the first two at least probably not possible.
Livio Zucca has looked at sets where the half square is added to a polyomino (extended polyominoes). There are seven pieces from the trominoes (12 with the one sided set) and 25 from the tetrominoes (47 from the one-sided set).
Livio's constructions are shown below.
One-sided extended trominoes
Extended tetrominoes.
Peter Esser fouind a number of pentuplications of pieces and Livio Zucca completed the solutions and produced the diagram below.
Extended pentominoes (96 pieces).
This set has a piece which has a gap which cannot be filled by other pieces and so no rectangle is possible. The solutions shown here are by Patrick Hamlyn
and the 36 sliced tetracubes -
The number of pieces in the set depends on whether to just allow single cuts which produce the pieces at the left or to allow for halving cubes diagonally which will produce extra pieces as at the right (these are only some of the possible extra pieces). Peter Esser has done some work on these and similar pieces - see his Clipped and Extended Polycubes.