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Perhaps one of the confusions here (at least for me) is that looking atoften
Bruce's link (below) it is now clear that the claim is NOT that the
gymnast/diver imparts NO angular momentum on takeoff, only that they
impart no angular momentum around the vertical axis--that is, they doimpart
AM around a horizontal axis perpendicular to their direction of motion.
That this can be converted to AM around other axes without changing the
overall vector AM is what is apparently not well understood but seems to
involve techniques like the 'tilt twist'