Trivia: The cast really are very good friends. At LeVar Burton's wedding in 1992, the best man was Brent Spiner and the ushers were Patrick Stewart, Jonathan Frakes and Michael Dorn. And when Brent Spiner recorded an album (Ol' Yellow Eyes is Back), the backing groups listed as The Sunspots are again the male members of the bridge crew.
Trivia: Near the end of the episode, Picard and Crusher walk out of the cargo bay. As the doors close behind them, you can see Tasha Yar waving to the camera from inside the bay. Her character is killed in the next episode, but this one was filmed second. The actress was waving goodbye to all of her fans.
Trivia: Scotty visits the bridge from "his" Enterprise (from the original 1966 Star Trek series) on the Holodeck. As the original set had long since been destroyed, the captain's chair and center console used in the episode was donated by a fan who had recreated a life size version of the bridge. The turbolift alcove and one of the stations was built on-set (and re-used in the DS9 episode "Trials and Tribbleations"). Everything else was made up of footage from the original series that was looped.
Trivia: The transporter was first created by Gene Roddenberry in 1966 for the original Star Trek, as an easier (and cheaper) alternative to get members of the Enterprise crew onto a planet's surface, instead of having the ship land on the planet each time. The same holds true here. Even the original version was based on a similar effect in the movie Forbidden Planet.
Trivia: In this episode, Picard is studying Fermat's Great Theorem, and says it has remained unsolved for 800 years. Five years after the episode was made the theorem was proven, by Andrew Wiles and Richard Taylor from Princeton University (their proof is not the same as Fermat's though, as they used modern methods Fermat did not know of). In the Star Trek universe, this was referred to in an episode of Deep Space Nine, and is considered as a subtle correction for Picard's statements.
Trivia: Stephen Hawking guest stars in this episode, playing himself as a holodeck character (The "Poker with Einstein" program). This came about when he visited Paramount Pictures to promote "A Brief History of Time", and then told the Paramount people that he had always wanted to visit the Enterprise. He not only got to visit, but the writers added this special scene just so professor Hawking could appear on screen in a Star Trek episode. Hawking also reportedly stopped by the warp engine, smiled and said "I'm working on that".
Trivia: The set designers added in small jokes everywhere. On the cross section of the ship near the tactical station, there are small objects in the picture that do not show up on television. (There are mice in the halls, cars in the cargo bay, a duck in a hall and a hypodermic needle in sickbay). In the halls, the red strips labeling the compartments say things like "3 hundred thousand kilometers per second, it's not just a good idea, it's the law, your mileage may vary of course", "Objects in mirror are closer than they appear to be", "A stitch in time saves nine", "In space, no one can hear you scream", and "Don't step on Superman's cape."