...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.
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.
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".
Probably not, but maybe a 4D space lol