This is a question about tiling puzzle jargon. What are the types of moving a tiling piece? I mean the single word or a phrase for rearrangement move of a piece. I would be grateful for proper mathematical names as well as contemporary English names. Suppose we are talking about 2D puzzle that lay on the table.
- Sliding the item along the plane on the table without picking it up off the table, without rotating it. Translate.
- Rotate the piece clockwise or counterclockwise without picking it up from the table Rotate or Twist.
- Flip the piece to the other side, just like a pancake. It is necessary to take the piece off the table to do that. Here I found some inconsistency. It is also called Twist as in twist-hinged dissection. Flip sounds good but I have not found it in use. Instead, I found Reflect - reflecting the item in the plane as if being viewed in a mirror.
- Scale up or down. This transformation is called dilation although I am not sure if both ways up and down.
I especially ask for the third type of move.
Reference:
http://www.mathpuzzle.com/Tessel.htm
Move a polygon to a specified position using only allowed rotations, reflections, and dilations
If you wish for the pieces to stay the same size and shape (which precludes the dilation), then the motion taking one piece to another is an isometry of the plane. These can be classified into four types:
Translation, as you mentioned.
Rotation about a point, again pretty self-explanatory.
Reflection about a line, which roughly corresponds to the "flip" you describe.
Glide reflection, which consists of a reflection followed by a translation along the line of reflection, and is shown in this diagram from Wikipedia:
A similar classification can be performed in higher dimensions, though the number of possible motions increases; a list for three-dimensional Euclidean space is given here.