> Shapes capable of tiling a 2D space need a minimum of two corners, but in 3D the rules are different
...for 3d they have bent the definition of "corner" so that a section of the 3d shape showing a corner is not considered a corner in the 3d space. It remains interesting, but.
I had the same reaction. Perhaps “vertex” vs “edge” should be used in place of the ambiguous “corner” (which is a vertex in 3D geometry).
The 2D projection of the top or side face of object the article shows repeated can be tiled in exactly the same way. It has two vertexes, but those become edges in 3D.
In that example the top and one side have 2D projections which are identical except one is rotated 180º from the other. The third face, when projected on a 2D plane, however, is a square.
So these shapes have no vertexes (like a cylinder), but they all have edges (like a cylinder).
I also think the 3D equivalent - what is the minimum number of edges required for a 3D tessellation - would be interesting.
I’m not sure what you mean. The shapes in the article do not have corners by the definition I would use… they do have edges and it left me wondering if you could tile 3d space with edge-free shapes.
You would speak of "vertexes". Tile a 3d space with cubes: you'll have faces, edges and vertexes. Curve the edges in smooth paths ("with continuous derivatives"): the vertexes disappear. But the equivalent of the 2d "corners" remain, as apparent in the 2d sections: no miracle happens.
The interesting part is the induction that by adding a dimension we may lose a "constraint".
I felt the same way. But after a second it was again sort of interesting to me wondering if it generalizes so that a 4D space can be tiled by a shape with only 3D portions meeting at an angle.
Source article has a lot more details and illustrations: https://www.scientificamerican.com/article/mathematicians-di...
> Shapes capable of tiling a 2D space need a minimum of two corners, but in 3D the rules are different
...for 3d they have bent the definition of "corner" so that a section of the 3d shape showing a corner is not considered a corner in the 3d space. It remains interesting, but.
I had the same reaction. Perhaps “vertex” vs “edge” should be used in place of the ambiguous “corner” (which is a vertex in 3D geometry).
The 2D projection of the top or side face of object the article shows repeated can be tiled in exactly the same way. It has two vertexes, but those become edges in 3D.
In that example the top and one side have 2D projections which are identical except one is rotated 180º from the other. The third face, when projected on a 2D plane, however, is a square.
So these shapes have no vertexes (like a cylinder), but they all have edges (like a cylinder).
I also think the 3D equivalent - what is the minimum number of edges required for a 3D tessellation - would be interesting.
I’m not sure what you mean. The shapes in the article do not have corners by the definition I would use… they do have edges and it left me wondering if you could tile 3d space with edge-free shapes.
> corners by the definition I would use
You would speak of "vertexes". Tile a 3d space with cubes: you'll have faces, edges and vertexes. Curve the edges in smooth paths ("with continuous derivatives"): the vertexes disappear. But the equivalent of the 2d "corners" remain, as apparent in the 2d sections: no miracle happens.
The interesting part is the induction that by adding a dimension we may lose a "constraint".
>it left me wondering if you could tile 3d space with edge-free shapes.
Probably not, but maybe a 4D space lol
I felt the same way. But after a second it was again sort of interesting to me wondering if it generalizes so that a 4D space can be tiled by a shape with only 3D portions meeting at an angle.