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Long before we were all swiping our fingers across iPads and iPhones like little children use finger paints, Tom Cruise was spying on people's bad intentions. Although the datacenter itself in Minority Report isn't necessarily iconic, the Minority Report GUI is, much to the excitement of CNN's John King and his magic touch-me board.
Although there was a recent remake by George Clooney, when we talk iconic, we mean the 1964 movie. The premise is that most of the United States defense system is automated and the movie is mostly about how the humans try to hold back the end of the world. While no individual computer is, itself, iconic in this film, the SAC (Strategic Air Command) showcases the computer's steps leading to destruction. Great fun is had by all.
While technically not a massive, government datacenter, perhaps no insane, over-the-top, sentient computer is more well-known than the HAL-9000 (Heuristically programmed ALgorithmic computer). HAL is nuts. HAL is nuts, and yet -- we will always love him.