The image in the article is of hazelnuts (I originally wrote "stones" then quickly edited it), and it's not a 3-4-5 triangle.
3-4-5 describes the length of each side - if you count the lengths of the triangle drawn in the image (the lines of chalk visible between the nuts on each side), it's only 2-3-4. To get 3-4-5 you're counting the number of nuts on each side, but those aren't lengths - those are the number of points marking the start/end of each unit length.
I see, I think you are referring to the unequal spacing of the nuts on each side, i.e. the side with 5 nuts has them closer together than the other sides.
I thought there was some point being made about the use of nuts vs. some other arbitrary item. Why does it matter they are hazelnuts and not something else?
That diagram represents a length of 2, not a length of 3, see? Here's three:
X--X--X--X
0 1 2 3
It's not that the hazelnuts are somehow imperfectly laid out or are an imperfect representation. It's wrong in principle, not practice (I mean it's wrong in practice too but every representation is).
You didn't miss it. You were focusing on the lattice edges, and PP was focusing on the lattice points. You're both right (except for PP's "No!" which should be "Yes!").
3-4-5 describes the length of each side - if you count the lengths of the triangle drawn in the image (the lines of chalk visible between the nuts on each side), it's only 2-3-4. To get 3-4-5 you're counting the number of nuts on each side, but those aren't lengths - those are the number of points marking the start/end of each unit length.