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Comment by mg | Hacker News Reader

Important to note that the solution uses reflection. It uses the tile and a mirrored version of it. In my book, this means it is rather using two tiles than one.

So the journey is not over. The question if a nonperiodic tiling of the plane is possible with one (non-mirrored) tile is still open.

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Seems a bit stingy to say you don't get to flip the tiles over, though I guess it makes sense if you are talking tiles with only one "nice" side.

Also if you’re talking about the platonic ideal of a plane. There is no “flip” transformation purely within the 2D plane.

What do you mean? Flipping is same as reflection, one of the rigid transformations

> The rigid transformations include rotations, translations, reflections, or any sequence of these. [...] All rigid transformations are examples of affine transformations.

https://en.wikipedia.org/wiki/Rigid_transformation

I mean, cut out paper shapes and work on a table as a 2D plane. There is no way to get a reflected shape without lifting the shapes off the plane.

And now I’m curious: what additional transformations could we do with 3D shapes if we could move them in a 4D space?

I think there's none. Flipping in 3d can reduce the shape to a line or a point or even make it disappear entirely from initial 2d space. Transformation in 4d would be capable of the same, but it could also remove the shape from 3d space that contains initial 2d space.

But as you transform 2d shape it remains embedded in it's own local 2d space. So you are not really transforming the shape but it's associated 2d space. And since the only thing can do to 2d space to transform it into another 2d space without deforming it is translation, rotation and flipping going into higher dimensions doesn't give you anything extra.

Apologies, I was thinking about 4D transformations of 3D shapes. Like, if I can transform a 3D cube in 4D space, can it produce a shape beyond what 3D transformations can, analogous to the way 3D space lets you flip a 2D shape.

I think using 4d transformation you could turn 3d object into mirror image of itself swapping chirality. For example changing left-handed molecule into right-handed one and vice versa.

It depends on if you're a mathematician interested in rigid transformations or if you manufacture ceramics.

> 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.

Chirality and handedness maybe?

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