Position in reciprocal lattice (for Triangular real lattice) to find photonic gapmap

Hi,
To find Gap-Map, using FDTD, for square lattice of dielectric coulumns in air, I choose the 'X' position in reciprocal lattice, while varying the radius of the rods. The result is good.

But while considering the triangular lattice, I am a bit confused to choose the point in reciprocal lattice. Is it the 'M' or 'K'. Where the gap opens?

Wikipedia has a good article in Brillouin zones. You are correct that M and K are the other two lattice points (aside from the center Γ -- in two dimensions). Which one you choose, is of course up to you.

Normally one needs to show that a bandgap exists in both directions in order to have a gap in any direction along the surface.

Thank you. Your valuable comment goes with my intuition.

Is there any specific direction (either ΓM or MK) along which the gap will be maximum for a given radius?

I don't know anything about triangular unit cells, but I would guess that they're nearly identical for smaller radii.

Good luck figuring it out!