A polyomino is a generalization of the domino to a collection of squares of equal size arranged with coincident
sides. Polyominos were originally called "super-dominoes" by Gardner (1957).
A polyomino with
squares is known as an -polyomino
or "-omino."
Free polyominoes can be picked up and flipped, so mirror image pieces are considered identical. One-sided polyominoes may not be flipped,
but may be rotated, so different rotational orientations are the same, but pieces
having different chiralities are considered distinct. Fixed
polyominoes (also called "lattice animals") are considered distinct if
they have different chirality or orientation.
When the type of polyomino being dealt with is not specified, it is usually assumed that they are free. There is a single unique 2-omino (the domino),
and two distinct 3-ominoes (the straight- and -triominoes). The 4-ominoes (tetrominoes) are known as the straight,
L, T, square,
and skew tetrominoes. The 5-ominoes (pentominoes)
are called ,
, , ,
, , ,
, , ,
, and (Golomb 1995). Another common naming scheme replaces , , ,
and with , ,
, and so that all letters from O to Z are used (Berlekamp et
al. 1982).
The first few polyominoes with holes are illustrated above (Myers).
Redelmeier (1981) computed the number of free and fixed polyominoes for ,
and Mertens (1990) gives a simple computer program. The following table gives the
number of free (Lunnon 1971, 1972; Read 1978; Redelmeier
1981; Ball and Coxeter 1987; Conway and Guttmann 1995; Goodman and O'Rourke 1997,
p. 229), fixed (Redelmeier 1981), and one-sided polyominoes
(Redelmeier 1981; Golomb 1995; Goodman and O'Rourke 1997, p. 229), as well as
the number containing holes (Parkin et al. 1967, Madachy 1969, Golomb 1994)
for the first few .
The best currently known bounds on the number of -polyominoes are
(Eden 1961, Klarner 1967, Klarner and Rivest 1973, Ball and Coxeter 1987).
Generalizations of polyominoes to other base shapes other that squares are known as polyforms, with the best-known examples being the
polyiamonds and polyhexes.
squares of equal size arranged with coincident
sides. Polyominos were originally called "super-dominoes" by Gardner (1957).
A polyomino with 
squares is known as an 
-polyomino
or "
-omino."
-triominoes). The 4-ominoes (tetrominoes) are known as the straight,
L, T, square,
and skew tetrominoes. The 5-ominoes (pentominoes)
are called 
,
, 
, 
,
, 
, 
,
, 
, 
,
, and 
(Golomb 1995). Another common naming scheme replaces 
, 
, 
,
and 
with 
, 
,
, and 
so that all letters from O to Z are used (Berlekamp et
al. 1982).
,
and Mertens (1990) gives a simple computer program. The following table gives the
number of free (Lunnon 1971, 1972; Read 1978; Redelmeier
1981; Ball and Coxeter 1987; Conway and Guttmann 1995; Goodman and O'Rourke 1997,
p. 229), fixed (Redelmeier 1981), and one-sided polyominoes
(Redelmeier 1981; Golomb 1995; Goodman and O'Rourke 1997, p. 229), as well as
the number containing holes (Parkin et al. 1967, Madachy 1969, Golomb 1994)
for the first few 
.
-polyominoes are
-ominoes." Can. J. Math. 25, 585-602, 1973.