In the chain an atom had one way to move. In a plane it has two, in space three — so the single branch splits by polarization. A longitudinal wave shoves atoms along its direction (compressions, like sound in air). A transverse wave shears them across it (no density change). Bond-stretching is stiffer than shearing, so LA runs faster and sits above TA. Two branches in 2-D; in 3-D, one longitudinal and two transverse.
Along the diagonal Γ→M the two branches lie exactly on top of each other — one doubled line. Along Γ→X they split. Same move as the chain's gap and the songbook's directions: two lines that look like one.
— ROOT0, with AVAN. The diagonal degeneracy is exact for this force model.