Other forms: quadrilles
You may not have danced a quadrille, a complicated dance that includes at least four pairs of dancers — because it reached its height of popularity in the mid-1800s.
Your language instincts probably push you towards the idea of squareness or fourdom when you see quad, and there are good reasons for both tendencies: the Latin quattuor lies behind many English words incorporating notions of "four" and "four corners." A quadrille, originally a card game for four people, morphed eventually to also mean a square dance for four couples.