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Lines and planes in the mathematical sense are 1 and 2 dimensional. They don't have any height (and lines also no width). So they can't exist as a physical object made out of atoms as they are already 3 dimensional.
They only exist as a concept.
The fact that something isn't a 3d object doesn't mean it doesn't exist. Does a line of contrast between 2 colors exist? Does a movie projected at a wall exist?
Does a line of contrast between 2 colors exist?
I'd say no. And even if it did, those colours are made out a material that consists of atoms that reflect light, both of which are "fuzzy" and 3D and can't make a proper line.
Does a movie projected at a wall exist?
Sure. There is photons bouncing of a wall and the information they carry we call "the movie". I guess that counts. But the relevant bit is the wall and again it's made out of atoms and therefore is not a proper flat 2D surface.
So yeah, I'd say not being 3D does mean something can't exist in the physical world.
Then, o pedant, do straight or flat objects (thus linelike, planelike) exist?
Why ask a pedantic question and be upset by a pedantic answer?
You are merely seeing through pedantic eyes.
That's a pretty pedantic argument.
Graphene is a great example of a planar molecule, within 1 atom of tolerance. Graphene is found in naturally occurring graphite rocks.
You could get serious Van der Waals off a couple sheets of that.
Their manifestation in our world was limited, they have already gone to another plane of existence