Chemistry - Why is the lone pair of pyridine's nitrogen atom not part of the aromatic ring?

Solution 1:

As follows from our discussion in the comments to the question, pyridine uses only a single electron in the p-orbital in order to comply with Huckel's rule. Kudos to Orthocresol:

Huckel's rule demands that there be 4n+2 pi electrons in the ring. Each carbon contributes one, making for a total of 5 from the carbons. If your nitrogen lone pair is in the ring, then you will have seven pi electrons. $7≠4n+27≠4n+2$ for $n ∈Z$

Of course, electrons know nothing of Huckel and his rule, but apparently they position themselves into the most energetically suitable configuration. I am not yet qualified to explain why exactly it is the most suitable one.

Solution 2:

It has the wrong geometry

The other P orbital can't participate in the aromatic ring because it has the wrong geometry: it points in a direction orthogonal to the orbitals involved in the aromatic part of the ring. The picture in the question shows this clearly.

There is already a P-orbital on nitrogen involved in the aromatic system of the ring (the one that points "up"). The orbital coloured yellow-ish in the picture doesn't overlap with the aromatic system at all as it points "out" from the ring and (if you did calculations) would have no overlap with the other orbitals in the the aromatic system. No overlap means it can't be involved in aromaticity. It is, ultimately, simple geometry and no other explanations are required.