A minimal succinct cover of the rectangular pentominoes:
A minimal succinct cover of the rhombic pentominoes:
An interesting set of types of polyominoes can be made by using the symmetries of the square grid. For example, the one-sided polyominoes are those for which reflected polyominoes are considered different.
Notice that a minimal succinct cover of the one-sided pentominoes is actually smaller than one of the set where relections are equivalent. Despite the greater number of pentominoes, the narrower definition of each type of pentomino makes it easier to avoid making duplicates.
Rectangular and rhombic polyominoes are those with cells having the same symmetry as rectangles and rhombi respectively. Sets of rhombic pentominoes made out of plastic are sold by Kadon under the name Rhombiominoes. The pieces of the minimal succinct cover of the rhombic pentominoes shown to the right are both symmetrical, which I found surprising and elegant.
Next: Polyomino Anatomy