Cellominoes or fixed diameter polyominoes are defined as being those polyominoes which can fit into and span an m x n rectangle. For example the twelve 2-4 cellominoes are shown below.
The tables below give number of pieces with and without holes. As can be seen, these numbers get very large for even relatively small values of the sides of the rectangle.
A few examples of constructions are shown below. Roel Huismann has many more on his site and you can download a program by Peter Esser to search for your own.
One sided 2-4 cellominoes:
It is also possible to make a number of constructions with holes in a shape similar to the main figure.
3-3 cellominoes without holes:
If one piece is used twice then we can form a number of rectangles.
One sided 3-3 cellominoes:
One sided 3-3 cellominoes without holes:
Peter Esser's solver also allows for subsets of cellominoes. For example the 3-4 cellominoes with area seven can for a 7 x 39 rectangle.
There are 54 3-4 cellominoes with area 8 covering a total area of 432 squares. Sets of three congruent rectangles are possible with this set.
There are 77 one-sided 3-4 cellominoes with area 7 which can form two rectangles.