Star Trek is one of the most beloved science fiction tv shows. The original series was wildly socially progressive for its time and addressed all sorts of social issues in a fun and exciting campy tv show. It may be a work of art but unfortunately, it got a lot of things wrong about exploring vector space. From the Alpha to Delta Quadrant, the galactic core and the Romulan Neutral zone, Vector space is one of the more cerebral and ambitious science fiction locations the crew of the enterprise. They almost got em right, but there are just some things they got totally wrong!
1. Darker than Schwartz (Schwartz Space)
Oh what a classic episode. Blacker than Schwartz, is one of the most quintessential episodes when the crew of the Enterprise gets lost in Vector space. Just after Kirk picks up Vulcan diplomats out of the Cardassian Demilitarized Zone for a special summit, they become trapped in an unknown force that pulls them far away from the galactic center into Schwartz space. There they are locked with a mysterious emanating blue and green orb which is highlighted on the bridge. Every time they try and fly away from the orb, the ship starts coming apart because of the Schwartz space orb’s higher and higher pitched screeching attack. It nearly destroyed the auxiliary power supply, fried the external sensors and it took a day for Scotty to repair it all into working order. While fans love this one, mathematicians hate it. If the orb was truly a Schwartz function, it’s attack frequency wouldn’t change that way because its’ fourier transform would have to be an automorphism. Sorry Gene Roddenberry, you should have done your homework.
2. Engine Trouble (Minkowski Space)
Minkowski Space, also known as Space-Time, is used in nearly every episode of Star Trek. Whether it’s discussed or not, the enterprise bends Minkowski Space with its warp drive in front and behind the ship and annihilates some matter with Dilithium Crystals. While most forgive the need of the massive amount of emissions from Dilithium Crystal antimatter destruction in a modern universe, the misinterpretation of Minkowski Space is totally unforgivable. In the episode ‘Engine Trouble’ Scotty goes into detail on how the warp drive’s four dimensional manifold is malfunctioning. “Aye dun know How she did it, It would’ve takin the combined power of the entire fleet Mr. Spock! It’s fused the matter compressor with the fuel rods and could blow any moment!” Fuel rods? Four Dimensional manifolds? Anti-matter? Not once do they mention a Lorentz transform or how they need an infinite amount of energy to get the enterprise to warp 1! Yeah it’s a nifty rubber science explanation to break the laws of physics but it totally doesn’t follow Minkowski space where time is imaginary and multiplied by the speed of light which makes way more sense.
3. Land of the Fréchet
Now this is one where the Prime Directive really comes into the forefront of the episode. In The People of the Fréchet, the crew of the Enterprise beams down to a distress signal on an unknown planet. It’s an earth like planet, similar to the American 1940s, in which the inhabitants are separated into an oppressed minority and the dominant majority class. In this fictional planet, the citizens of Fréchet discriminate against any citizens who aren’t locally convex for not complying to social norms. Spock, the science officer, mentions how this practice of thinking all Fréchet spaces, normally defined as locally convex metrizable topological vector space is incredibly closed minded and that some definitions of Frechet space don’t require being locally convex. Captain Kirk just calls the practice barbaric and attempts to save some of the citizens from the brutal oppression. If only they did a little research, they would have realized that humanoids can’t be locally convex and that it was a dumb concept to begin with they would have stood true to the math.
4. Hardly a Way Out (Hardy Space)
One of the best bottle episodes on the show. It basically happens in an actual bottle. While observing strange readings from a J type planet, the team beams down only to find themselves trapped. Kirk gets trapped with Sulu, Lt. Uhura, and a red shirt in what appears to be a perfectly circular white Unit Disk in Hardy Space. The red shirt freaks out and tries to escape but is soon electrocuted to death when he exceeds the r > 1 condition of the Hardy space. Even after boosting the range on their transmitters, Kirk can’t communicate back to the bridge and soon discovers that they are stuck in some prison right before the first commercial break. After observing and listening to their bleak surroundings Holomorphic visions of strange men spray painted orange and wielding Egyptian like weapons begin attacking the landing party. Sulu and Kirk barely fend them off in a fist fight but they don’t seem to go away. Uhura determines that they are the only way out of their prison and attempts to communicate and follow them out of the hardy space. This is total baloney! Even the Holomorphic functions, though complex, are still bounded by r -> 1. I guess it wouldn’t have made a good episode if they couldn’t get out.
5. Mirror Mirror (Dual Space)
In the second season episode Mirror Mirror, Kirk and his landing party of Scotty, Uhura, and Dr. McCoy return from negotiating with the Halkan council for federation rights to mine dilithium crystals on their planet. They get beamed up in the middle of a lightning storm and are sent to a Dual Space where it’s the same ship and crew members but everyone is evil. Spock even has a goatee! In the popular explanation, they crossed over to a corresponding dual space where everything maps to their old universe but it’s the evil complex conjugate. After navigating through the evil enterprise which was about to destroy the Halkan’s planet filled with mutinies and seductive femme fatales, Kirk and his crew barely survive and find their way back to the normal enterprise which had received Kirk, McCoy’s, Scotty’s and Uhura’s evil complex conjugate pairs. They really almost got this one right except that evil Spock would have Mutton chops and not a goatee if they were to form a set of basis functions.
6. Too Hot; Too Cold (Bochner Space)
The crew really came close to being goners on this one! When the ship traveled into a reverse Bochner space, heat transfers and all calculated Partial Differential Equations began working in reverse. Starting with a few oddities first observed by Scotty in the engineering deck and then studied in detail by Spock, the ships life support systems began pumping heat in all the wrong directions. It’s definitely one of the more goofy episodes where parts of the enterprise show all of the characters bundled up in thick puffy futuristic jackets in the cold while the bridge heavily features a shirtless Shatner sweating in his captains chair while he thinks of a plan to get out of there and keep his crew alive in a monologue full of unnecessary pauses. Eventually Spock states that they just need to reverse the polarity on all of their equipment which absolutely doesn’t make sense when dealing with Bochner Space partial time derivatives. Maybe they didn’t realize the show would be popular with such big nerds and they could have gotten away with it.
7. Ghost in the Machine (Fock Space)
In this more technical episode, the enterprise travels into the Fock Space gaseous Nebula of the Gamma quadrant. There they begin attempting to maneuver through flashing purple gas formations which have been emitting strange readings on the ships long range sensors. This is when Checkov appears to be having trouble operating the navigator which appears to be picking up on strange quantum signals when attempting to localize the enterprise’s location based on the star maps. Upon further investigation, the main ship computer is malfunctioning and causing all sorts of problems and only half opening doors. Checkov and Spot then investigate the computer, which is revealed to be a quantum processor of various lights blinking behind cardboard wrapped in aluminum foil. Further analysis done by an old school binary machine discovers that the Bosonic Fock space tensors used by the computer are being interfered with antisymmetric tensors in some purple gaseous nebula. More systems on the ship begin to break down and the crew gets desperate to determine a path out of purple flashy nebulae. Spock wires up a Fermionic inhibitor Ion Trapper to stop the interference and they are able to escape Fock space. This trick always bothered me because Fermionic processes are just the half spin equivalent of Bosonic integer spins and I think this solution was probably more of a cop out of an episode that just turned too cerebral.
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I laughed. I cried. I’m a better person for having read this paper.
I’m even more confused now