Author:  David Goodger <goodger@python.org> 

Date:  20150224 
Revision:  600 
Web site:  http://puzzler.sourceforge.net/ 
Copyright:  © 19982015 by David J. Goodger 
License:  GPL 2 
Polyominoes are polyforms constructed from unit squares joined edgetoedge on a regular twodimensional Cartesian plane. The name comes from dominoes; one domino is composed of two ("d", short for "di") unit squares ("omino").
Here is a puzzle containing all the polyominoes of order 1 through 5:
See Polyominoes: Puzzles & Solutions, Pentominoes: Puzzles & Solutions, and Hexominoes: Puzzles & Solutions for many more puzzles.
Polyominoes were named and studied by Solomon Golomb and popularized by Martin Gardner in Scientific American magazine in the 1950s. Pentominoes may have been first featured in a puzzle in 1907 (Dudeney's The Canterbury Puzzles), although the history of polyforms may go back much further. The tetrominoes are well known from the video game "Tetris".
Kadon Enterprises, vendors of quality polyform puzzles in wood and acrylic, have an excellent introduction to polyominoes on their site.
When the polyominoes are formed from unit cubes intead of flat squares, they are called "solid polyominoes" or "planar polycubes". See An Introduction to Polycubes for background, and Polyominoes: Puzzles & Solutions for many puzzles.
The number and names of the various orders of polyominoes are as follows:
Order  Polyform
Name

Free
Polyominoes

OneSided
Polyominoes


1  monomino  1  1 
2  domino  1  1 
3  trominoes  2  2 
4  tetrominoes  5  7 
5  pentominoes  12  18 
6  hexominoes  35  60 
The numbers of polyominoes can also be found in the following sequences from The OnLine Encyclopedia of Integer Sequences: A000105 (free) and A000988 (onesided).
Examples of the polyominoes from order 1 (monomino) to order 6 (hexominoes) are given in the tables below.
The polyominoes are named with letters (like the pentomino "X") or a letternumber scheme (like the "I3" tromino or the "X06" hexomino). The traditional names for the pentominoes are used by Polyform Puzzler, without a number suffix. The names of the hexominoes below roughly correspond to the Kadon naming system for hexominoes. The rest of the polyomino names correspond loosely to Kadon's names for their Poly4 Supplement set (exceptions noted in the tables below).
The initial letter of each name is the letter of the alphabet that the polyomino most closely resembles, or an initial. In some cases, that resemblance is weak, and the letters are arbitrary. The final digit of the number represents the polyform order (how many unit squares are in the polyomino). For the hexominoes, some letters are used for multiple pieces (there are more pieces than letters in the alphabet), so a middle index digit is used to differentiate (e.g. "X06" and "X16").
In the tables below, "Aspects" refers to the number of unique orientations that a polyform may take (different rotations, flipped or not). This varies with the symmetry of the polyform.
The "OneSided" column identifies polyforms that are asymmetrical in reflection. Treating the flipped and unflipped versions of asymmetrical polyominoes as distinct polyforms (and disallowing further reflection or "flipping"), results in "onesided" polyominoes and puzzles.
Alternate names and name origins are noted in the "Name" column.
There is only one monomino (order1 polyomino):
Name  Image  Aspects  OneSided 

O1
("M", from "Monomino")

1 
There is only one domino (order2 polyomino):
Name  Image  Aspects  OneSided 

I2
("D", from "Domino")

2 
Sometimes called "triominoes", there are 2 trominoes (order3 polyominoes):
Name  Image  Aspects  OneSided 

I3  2  
V3  4 
There are 5 free tetrominoes (order4 polyominoes) and 7 onesided tetrominoes:
Name  Image  Aspects  OneSided 

I4  2  
L4  8  yes  
O4  1  
T4  4  
Z4
("S4")

4  yes 
There are 12 free pentominoes (order5 polyominoes) and 18 onesided pentominoes:
Name  Image  Aspects  OneSided 

F  8  yes  
I  2  
L  8  yes  
N  8  yes  
P  8  yes  
T  4  
U  4  
V  4  
W  4  
X  1  
Y  8  yes  
Z  4  yes 
There are 35 free hexominoes (order6 polyominoes) and 60 onesided hexominoes:
Name  Image  Aspects  OneSided 

A06
(Kadon's "A")

4  
C06  4  
D06  4  
E06  4  
F06
("hi F")

8  yes  
F16
("low F")

8  yes  
F26
("hi 4")

8  yes  
F36
("low 4")

8  yes  
G06  8  yes  
H06  8  yes  
I06  2  
J06  8  yes  
K06  4  
L06  8  yes  
M06  8  yes  
N06 [*]
("short N")

4  yes  
N16
("long N")

8  yes  
O06  2  
P06  8  yes  
Q06  8  yes  
R06  8  yes  
S06
("long S")

4  yes  
T06
("long T")

4  
T16
("short T")

8  yes  
U06  8  yes  
V06  8  yes  
W06
("Wa")

8  yes  
W16
("Wb")

4  yes  
W26
("Wc")

8  yes  
X06  4  
X16
("italic X")

4  yes  
Y06
("hi Y")

8  yes  
Y16
("low Y")

4  
Z06
("long Z")

4  yes  
Z16
("short Z")

8  yes 
[*]  Kadon's Sextillions set includes a second copy of N06 called "short S", also adopted by Polyform Puzzler's Hexominoes Plus as "S16". 
Polyominoes use a standard 2dimensional (x,y) Cartesian coordinate system.
Each unit square has 4 immediate neighbors. The neighbors of the square at coordinates (x, y) are:
{(x+1, y), (x1, y), (x, y+1), (x, y1)}