If we consider reflections (in the X-axis, Y-axis, diagonal(Z) or antidiagonal(N)) and
rotations (in 90 degree steps), chains can come in 8 variations if there are no symmetries:

http://i.imgur.com/O9bObip.jpg

For chains that have 1 symmetry (meaning they come in 4 variations), we can have different
ways for them to be symmetric:

http://i.imgur.com/mKgAKd3.jpg

Apart from that, chains can also have 2 symmetries (meaning they come in 2 variations) or
3 symmetries (meaning they are unique).

If a chain has only 1 symmetry, are there only three types of such chains, matching up with
the three cases in this image?

http://i.imgur.com/mKgAKd3.jpg

Umm... maybe you already knew this, maybe not, but the term Variation
also has an existing meaning in Go that's very well established,
and which has no connection to the way you're using it. (A bit confusing, to me.)

Could you also define what you mean by "symmetry" ?
I'm a bit slow and don't quite follow what you're asking,
but would like to learn. Thanks.

If you're interested in learning about symmetries in general, perhaps you'd like to pick up an undergraduate textbook on group theory. In that language, you are attempting (with decent success) to list the subgroups of the group of symmetries of the square.

Symmetry in the sense that an operation (like reflecting a chain in the X-axis) leaves a chain unchanged.
Some operations will always map a chain to itself (like rotating a chain 360 degrees), while
a chain with a rotational symmetry will also map to itself under a rotation of 180 or 90 degrees
(meaning that if you rotate such a chain 180 degrees, you end up with exactly the same chain).

If a chain is most symmetric (having 3 symmetries), you end up with the same chain, for each
of the possible operations (reflecting it horizontally, vertically or diagonally or rotating it
by 90 or 180 degrees).
If a chain lacks symmetries, you end up with a different chain, for each of those possible operations.

I've never followed a course on group theory, but I'm aware it's the math that deals with invariants like symmetries.
I reckon that it can also be assessed on an intuitive level as it applies to chains on the goban without considering
the topic in general at a more abstract level, the way it's covered in group theory.

Yes, I'm sure it can. There are a few advantages to learning basic group theory, though: (1) it gives you a precise language to explain your thoughts in, (2) finding answers to questions like this (when you have formulated them precisely) will become very easy, (3) your intuition will sharpen.

To answer your question: if my understanding is right, you are trying to find the subgroups of D₈ of size 4 modulo conjugation ("chains with 4 variations" in your language), which you seem to have done correctly - there are three of them. The diagrams you have drawn are (rather ornate) Cayley diagrams. Note that the two diagrams on the left are not quite the same - because the reflections are different - but they're "essentially" the same, i.e. isomorphic.

You've counted the subgroups of size 2 differently. Compare these two shapes:

Click Here To Show Diagram Code

[go]$$
$$ . . . . . . . . . . . . . .
$$ . . . . . . . . . . X . . .
$$ . . X X X . . . X X X X . .
$$ . X X X X X . . X X X . . .
$$ . . X X X . . X X X X . . .
$$ . . . . . . . . X . . . . .
$$ . . . . . . . . . . . . . .[/go]

These are also "essentially" the same: they have a 90-degree rotation 'variant' and nothing else. But that 90-degree rotation is generated by different reflections: a reflection in either diagonal line in the first case, and in the horizontal/vertical line in the second case. I'm sure you've already found a shape that has complete rotational symmetry but reflected 'variants'.

I don't know if this belongs in General Go Chat - it seems only barely relevant to the game, if it's relevant at all. Maybe Off-Topic?

I think it's very impressive that you came up with Cayley graphs on your own.
Still, all this stuff is extremely well-researched and you are basically reinventing the wheel.

See for example:
http://en.wikipedia.org/wiki/List_of_pl ... try_groups

Ok, so this provides a more exhaustive overview of all possible variations of chains regarding their symmetries:

http://i.imgur.com/fs7AR6Y.jpg

http://i.imgur.com/N46bdkO.jpg


I've also found an interesting math course at youtube that deals with groups:

http://www.youtube.com/watch?v=yuzoHGEYWaU

http://imgur.com/a/ZhUNJ