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 
Contents
Polyiamonds are polyforms constructed from unit equilateral triangles joined edgetoedge on a triangular grid. The name comes from diamonds the same way "polyominoes" comes from "dominoes"; one diamond is composed of two ("di") unit triangles ("iamond").
Here is a puzzle containing all the polyiamonds of order 1 through 6:
See Polyiamonds: Puzzles & Solutions, Hexiamonds: Puzzles & Solutions, and Heptiamonds: Puzzles & Solutions for many more puzzles.
The number and names of the various orders of polyiamonds are as follows:
Order  Polyform
Name

Free
Polyiamonds

OneSided
Polyiamonds


1  moniamond  1  1 
2  diamond  1  1 
3  triamonds  1  1 
4  tetriamonds  3  4 
5  pentiamonds  4  6 
6  hexiamonds  12  19 
7  heptiamonds  24  43 
8  octiamonds  66  120 
The numbers of polyiamonds can also be found in the following sequences from The OnLine Encyclopedia of Integer Sequences: A000577 (free) and A006534 (onesided).
Examples of the polyiamonds from order 1 (moniamond) to order 7 (heptiamonds) are given in the tables below.
The polyiamonds are named with a letternumber scheme (like the "P6" hexiamond). The initial letter of each name is the letter of the alphabet that the polyiamond most closely resembles, or an initial. In some cases, that resemblance is weak (or nonexistant!), and the letters are arbitrary. The final digit of the number represents the polyform order (how many unit triangles are in the polyiamond).
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 polyiamonds as distinct polyforms (and disallowing further reflection or "flipping"), results in "onesided" polyiamonds and puzzles.
The "Disparity" column shows the discrepancy in parity of the polyiamond's unit triangles, i.e. the difference between the number of z=0 and z=1 units in the polyiamond. Left blank when there is no difference.
Alternate names and name origins are noted in the "Name" column.
There is only one moniamond (order1 polyiamond):
Name  Image  Aspects  OneSided  Disparity 

T1
(from "Triangle")

2  1 
There is only one diamond (order2 polyiamond):
Name  Image  Aspects  OneSided  Disparity 

D2
(from "Diamond")

3 
There is only one triamond (order3 polyiamond):
Name  Image  Aspects  OneSided  Disparity 

I3  6  1 
There are 3 free tetriamonds (order4 polyiamonds) and 4 onesided tetriamonds:
Name  Image  Aspects  OneSided  Disparity 

C4  6  
I4  6  yes  
T4
(from "Triangle")

2  2 
There are 4 free pentiamonds (order5 polyiamonds) and 6 onesided pentiamonds:
Name  Image  Aspects  OneSided  Disparity 

C5  6  1  
I5  6  1  
L5  12  yes  1  
P5  12  yes  1 
There are 12 free hexiamonds (order6 polyiamonds) and 19 onesided hexiamonds:
Name  Image  Aspects  OneSided  Disparity 

C6
(a.k.a. "Chevron" or "Bat")

6  
E6
(a.k.a. "Crown")

6  
F6
(a.k.a. "Yacht")

12  yes  2  
G6
(a.k.a. "Shoe" or "Hook")

12  yes  
H6
(a.k.a. "Pistol" or "Signpost")

12  yes  
I6
(a.k.a. "Bar" or "Rhomboid")

6  yes  
J6
(a.k.a. "Club" or "Crook")

12  yes  
O6
(a.k.a. "Hexagon")

1  
P6
(a.k.a. "Sphinx")

12  yes  2  
S6
(a.k.a. "Snake")

6  yes  
V6
(a.k.a. "Lobster")

6  
X6
(a.k.a. "Butterfly")

3 
There are 24 free heptiamonds (order7 polyiamonds) and 43 onesided heptiamonds:
Name  Image  Aspects  OneSided  Disparity 

A7  12  yes  1  
B7  12  yes  1  
C7  6  1  
D7  6  1  
E7  12  yes  1  
F7  12  yes  1  
G7  12  yes  1  
H7  12  yes  1  
I7  6  1  
J7  12  yes  1  
L7  12  yes  1  
M7  6  3  
N7  12  yes  1  
P7  12  yes  1  
Q7  12  yes  1  
R7  12  yes  1  
S7  12  yes  1  
T7  12  yes  1  
U7  12  yes  1  
V7  6  1  
W7  4  yes  1  
X7  12  yes  1  
Y7  12  yes  1  
Z7  12  yes  1 
Polyiamonds use a pseudo3dimensional skewed (x,y,z) Cartesian coordinate system. The X and Y axes are 60° apart instead of 90°, and the Z dimension is used to represent the orientation of triangles:
____ /\ \ / z=0 /__\, z=1 \/
Example (with coordinates):
x=0 1 2 3 4 5 6 7 8 ____________________________________ / / / \ / / 3 / /___________/ ___\ / / / / \ / / \ /\ / 2 /_______/ \____/ /______\/ \/ / /\ / /\ \ / 1 / ____/ \____/ / \_______\ / / / \ \ / / y=0 /___/__________\_______\________/___/ x=0 1 2 3 4 5 6 7 8
Each coordinate triple locates a triangle. The (x,y) pair locates an equilateral parallelogram (not a point), and the z coordinate identifies the triangle (half of the parallelogram).
Another example:
. /\ .. . .G /F \H . .I . ______/____\_ _ _.. /\ /\ /\ . y=0 /A \B /C \D /E \ . /____\/____\/____\ x=0
The coordinates in the figure above are as follows:
The coordinate (0,1,1) would be at G, (1,1,1) would be at H, and (2,1,1) would be at I. The (x,y) coordinates (0,0) identify the parallelogram formed from triangles A & B, (1,0) is C & D together, etc.
Each unit triangle has 3 immediate neighbors. The neighbors of the triangle at coordinates (x, y, 0) are:
{(x, y, 1), (x1, y, 1), (x, y1, 1)}
The neighbors of the triangle at coordinates (x, y, 1) are:
{(x, y, 0), (x+1, y, 0), (x, y+1, 0)}