N-D polyominoes

[PWrong]

I found a 3D version of the game tetris recently, and I've been trying to find patterns in the number of possible shapes.

There are 7 tetris blocks in 2D, counting reflections but not counting rotations. i.e. there are 2 different L shapes and 2 different Z shapes.

In 3D, analogues of all the 2D shapes are possible, but they can now be rotated around another axis. So there is now only 1 L block and 1 Z block. However, there are three completely new shapes, two of which are reflections. So we lose two shapes and gain three.
In all, there are 8 3D tetronimoes, as shown here:
http://mathworld.wolfram.com/Polycube.html

Now in 4D, no new shape can be formed, because with only four hypercubes, you can only occupy one cell at a time. What I mean is, you can't push out a block into all four dimensions, because
But two shapes are reflections of each other in 3D, so one can now be rotated to become the other. Hence there are only 7 tetris blocks in 4D.

There are also 7 shapes in 5D and beyond, because there can be no new shapes, and there are no new mirror images.

So for shapes of 4 blocks, we have
2D-7
3D-8
>4D-7

Before I go onto pentominoes, I'll show a table for 3 blocks, just for completeness.
2D- 2 shapes
3D- 2
4D- 2
...
Pretty simple.

Now pentominoes are far too numorous. Mathworld tells me that there are 18 in 2D and 29 in 3D, twelve of which come in mirror pairs.
In 4D, these pairs are the same, so there are at least 23, and probably much more. In 5D, no new shapes will be available, so the number will decrease. In 6D onwards, the number will remain the same as for 5D.

blocks 2D 3D 4D 5D
2 1 1 1 1
3 2 2 2 1
4 7 8 7 7
5 18 29 >23

Apparently there's no known formula for the number of n-ominoes even in 2 dimensions, so I guess my first question is, "why not?".

More importantly, is there another way to represent polyominoes in any dimension, using numbers rather than drawing them? Matrices are the only option I can think of. For a 3D pentomino, for instance, you would have a 3 by 5 matrix, with a set of coordinates for each block.

But you have to impose the conditions that each point has to be separated from another point by 1 cell orthogonally, and no two points can be the same. Finally, you have to make sure that rotations and translations don't count, but reflections do. So you have to apply all possible transformation matrices to sort it out.

Frankly, I don't think matrices are particularly well suited for this. So does anyone know a simpler representation?

[pat]

I'm not sure if this will extend and it's not a very easy notation to gleen stuff from and I'm not sure if you'll be able to spot reflections.... but.....

In thinking how to efficiently encode the shapes of the 2-D Tetris pieces, I had come up with this:

• 1 = move right
• 2 = move diagonally up and left
• 3 = move up

Now, the Tetris pieces are: 111, 121, 131, 313, 113, 331, and 112.

There is a requirement that you not begin with 2. And, there is a requirement that you not have adjacent 2's. And, the reflection of piece is just swapping all its 1's and 3's and making two be diagonally down and left.

But, this doesn't nearly cover the possibilities for pentominos. And, it seems like extending it to more dimensions will require a great number of restrictions.

[pat]

Another way of thinking of them which may aid counting is: Connected lattice graphs with n-vertices.

[pat]

Actually, I suppose the graph view isn't as useful as I had hoped. I was hoping that some of the graph-counting up-to-isomorphism sorts of things would help. But, there are isomorphic, connected grid graphs that represent different polyominos. And, there are dimorphic grid graphs that represent the same polyomino. Hmmm...

[wendy]

Last time i looked, there were only 12 planar pentimos, not 18.

Of the 29 three-dimensional ones, they are counting the chiral ones only once, not twice, so there would be 41 including the enamatiphors.

In four dimensions, there are further ones, including some chiral ones too, but all of these fall in a 2-unit tesseract.