Polyforms

The number and names of the various orders of polyiamonds are as follows:

The numbers of polyiamonds can also be found in the following sequences from The On-Line Encyclopedia of Integer Sequences: A000577 (free) and A006534 (one-sided).

Examples of the polyiamonds from order 1 (moniamond) to order 7 (heptiamonds) are given in the tables below.

The polyiamonds are named with a letter-number 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 non-existant!), 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 "One-Sided" 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 "one-sided" 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.

Moniamond

There is only one moniamond (order-1 polyiamond):

Name	Image	Aspects	One-Sided	Disparity
T1 (from "Triangle")		2		1

Diamond

There is only one diamond (order-2 polyiamond):

Name	Image	Aspects	One-Sided	Disparity
D2 (from "Diamond")		3

Triamond

There is only one triamond (order-3 polyiamond):

Name	Image	Aspects	One-Sided	Disparity
I3		6		1

Tetriamonds

There are 3 free tetriamonds (order-4 polyiamonds) and 4 one-sided tetriamonds:

Name	Image	Aspects	One-Sided	Disparity
C4		6
I4		6	yes
T4 (from "Triangle")		2		2

Pentiamonds

There are 4 free pentiamonds (order-5 polyiamonds) and 6 one-sided pentiamonds:

Name	Image	Aspects	One-Sided	Disparity
C5		6		1
I5		6		1
L5		12	yes	1
P5		12	yes	1

Hexiamonds

There are 12 free hexiamonds (order-6 polyiamonds) and 19 one-sided hexiamonds:

Name	Image	Aspects	One-Sided	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

Heptiamonds

There are 24 free heptiamonds (order-7 polyiamonds) and 43 one-sided heptiamonds:

Name	Image	Aspects	One-Sided	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

Coordinate System

Polyiamonds use a pseudo-3-dimensional 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), (x-1, y, 1), (x, y-1, 1)}

The neighbors of the triangle at coordinates (x, y, 1) are:

{(x, y, 0), (x+1, y, 0), (x, y+1, 0)}