> In my book, this means it is rather using two tiles than one.
I expected to see more about this. Are a given 2D shape and its mirror image generally considered the same shape by... the people who study this stuff? That would surprise me. So much so that calling this an "aperiodic monotile" doesn't feel right.
Yes, "same shape" in this context means "isometric". Rotations and reflections are considered differences in the way the shape is placed into the plane, not differences in the shape itself.
Fascinating. Rotation feels like it should be "allowed" to me, because you can continuously rotate a 2D shape to any other orientation, in 2D, without it ever becoming not the same shape.
Reflections feel like a totally different thing, because there's no continuous path to go from a shape to its reflection in 2 dimensions: you either have to have it instantaneously jump to its reflection, or introduce an extra imaginary "third dimension" for it to move through.
I'm not arguing with the definitions, because that's pointless. I'm just trying to explain why I find it so surprising as a lay person that reflections would be admitted in this way.
I expected to see more about this. Are a given 2D shape and its mirror image generally considered the same shape by... the people who study this stuff? That would surprise me. So much so that calling this an "aperiodic monotile" doesn't feel right.