4 ms·

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.

brookst 4y ago
Also if you’re talking about the platonic ideal of a plane. There is no “flip” transformation purely within the 2D plane.
- zokier 4y ago
  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 https://en.wikipedia.org/wiki/Rigid_transformation
  - brookst 4y ago
    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?
    - scotty79 4y ago
      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.
      - brookst 4y ago
        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.
        scotty79 4y ago
        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.
evanb 4y ago
It depends on if you're a mathematician interested in rigid transformations or if you manufacture ceramics.

brookst 4y ago

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

zokier 4y ago
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 https://en.wikipedia.org/wiki/Rigid_transformation
- brookst 4y ago
  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?
  - scotty79 4y ago
    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.
    - brookst 4y ago
      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.
      - scotty79 4y ago
        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.

evanb 4y ago

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