Why is triptycene aromatic?
The confusion arises because of a simplistic definition of aromaticity.

What matters for aromaticity is the planarity of the relevant part of the molecule. In triptycene there are three planar benzene groups joined together. Each is individually aromatic, but the whole molecule is not. Similarly, triphenyl-methane has three individually planar benzene groups, each of which is aromatic, but the molecule cannot be planar because of steric interactions between the groups. There are many other such examples of complex molecules containing an aromatic component attached to other things.