Trouble understanding interplanar spacing using miller indices We know that two parallel planes have the same Miller indices, this implies that we can have an infinite number of parallel planes close to one another all of which have the same Miller indices.
But it's claimed that

the interplanar distance between two adjacent planes is given as $$d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}$$.

I can't understand what adjacent means here?
to me the next parallel plane to any plane is infinitely close to it, hence the $$d_{h k l }=0$$
Can someone please explain what exactly is interplanar spacing?
Thank you.
The lattice planes are a property of the lattice based on dividing up the unit cell. The cell can be divided up indefinitely to give all the Miller indices for example $(\bar{10}\;20\;0);(10\;23\;20)$ etc. Atoms do not need to be on a lattice plane to add to the diffracted signal. The structural information is in the intensity of the diffracted spots not their position. (2 2 4) is parallel to (1 1 2) etc. as shown in the figure. In the general case it is usual to use vectors to find the perpendicular distance using formulas for finding a point at a given distance from a plane.
(Edit. I added (2 2 4) to indicate that it is a different plane but parallel to (1 1 2), in crystallography it is conventional to make all parallel planes have the same Miller indices, (h k l) (as they all scatter in the same way) by removing any common factor. )