When commentators and football fans say "breaking the plane" they're talking about the endzone. The first and foremost rule in scoring a touchdown is that the football must break the plane or cross into the endzone some kind of way for anyone to even think about it being a touchdown.
If a team runs the ball, it wouldn't matter if the player got his whole body into the endzone. If the ball didn't break the plane as he went in it is not a touchdown.
If the ball is passed and is caught by a player who is falling forward out of the endzone and he doesn't allow the ball to break the plane it is not a touchdown.
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