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Weird Science is the name for the style of storytelling made famous by the Science Fiction "pulp" (named after the poor quality paper on which they got printed) magazines of the 1930s and 1940s. By the 1950s, sales fell off, probably because of the advent of television. A few of these magazines had covers far trashier than the contents, others reveled in cheese with the writers entertaining no delusions that they created great art.
Astounding Science Fiction (later Analog), Unknown (also known as Unknown Worlds), and Galaxy published generally high quality fiction. Amazing Stories, Weird Tales and Startling Stories and others occupied the middle strata while Captain Future, Planet Stories and many, many others published the more shamelessly trashy material. (Unknown and Weird Tales actually published mostly fantasy and horror, though individual Science Fiction stories and elements would get incorporated in the stories too.) They specialized in imaginative stories of Science Fiction, with the less highbrow magazines in particular having a good deal more "Boys Own Adventure" flair. The actual name Weird Science comes from EC Comics Comic Book, which tended to use the Karmic Twist Ending or, more rarely, the Cruel Twist Ending. (EC also published Weird Fantasy, actually another Science Fiction comic, and merged the two titles later.)
The stories featured exotic worlds and gee-whiz gadgetry, buxom space babes, two-fisted heroes and, most important, plenty of gee-whiz gadgetry; it is this last that can be called Weird Science. Many of the most notable names in science fiction and fantasy got their start writing in these magazines, and, as the century progressed, the standards both for storytelling and scientific plausibility increased, although an exciting story was always more important than a realistic one.
Their influence is still widely felt: any Science Fiction that involves derring-do with robots, rocketships and rayguns and doesn't worry about technical realism can be said to use Weird Science, but it is especially likely to be found in Planetary Romance.
Note also that Weird Science is not the same as Hollywood Science; the former throws realism to the wind in order to create spectacle whereas the latter comes from not bothering to get things right that could have been. They often have similar effects, but different causes; Weird Science springs from Rule of Cool, whereas Hollywood Science grows from poor research. (The extreme case, most often found in comedy, is It Runs on Nonsensoleum.)
Not to be confused with the movie, series or song of the same name, though the premise certainly is.
Weird Science tropes:
As noted above, the Trope Namer, EC's Weird Science. EC also published Weird Fantasy. They later merged the two publications into Weird Science-Fantasy (because sales had flagged). Later on they published the short-lived Incredible Science Fiction.
DC Comics' anthology titles Mystery in Space and Tales of the Unexpected.
Stargate: the movie had a lot to do with this trope, as does the series'.
The Core: the idea of taking a giant drill vehicle into the depths of the planet is this trope; unfortunately, everything else in the movie is particularly bad Hollywood Science.
The Godzilla films (and the entire tokusatsu genre, for that matter) tend to be full of this. In Godzilla vs. Megaguirus, the humans have developed a hilarious new weapon to deal with Godzilla: The "Dimension Tide" is a gun that will generate a small black hole and launch it at Godzilla. The hole will suck up Godzilla, sending him into a parallel dimension, and then vanish. (In the real world, in addition to being monstrously heavy and difficult to create, black holes are also fucking dangerous. Possibly more so than a giant lizard loose in your city. And they don't just vanish into thin air... at least not peacefully.) At one point, Godzilla tries to defend himself against the Dimension Tide by shooting the hole with his energy beam breath.
As mentioned above this was a staple of the 30s and 40s pulps. Doc Savage lived off this in particular.
Odd example, Conan the Barbarian encounters a lost civilization in "Shadows of Xuthal" which has radium lighting and an elixir that both prologed life and healing just about any wound or injury that wasn't immediately fatal.
Tarzan sometimes encoutered lost civilizations that used this.
Careful with this one... while it certainly applies to the Lens itself and the principles it uses, the straightforward physics is harder than it looks. Most of the basic concepts, although taken to spectacular extremes, are pretty sound according to the general understanding of physics and cosmology at the time the books were written. It's mainly a combination of the scale and spectacle, the way many of the ideas have been hit hard by Science Marches On, and certain persistent and oft-repeated misconceptions among the insufficiently-careful readership, which makes things look like this trope. Smith was a scientist himself, and in most of his works did mostly try to stick to what was more or less plausible at the time (with the exception of the Skylark series, where he did explicitlyabandon realism in the cause of spectacle).
Star Trek: The Original Series had an uneasy relationship with this trope, striving for something believeable, yet at the same time not having as much concern for being accurate. Gene Roddenberry's famous dictum to never actually explain any technology but just use it and let the audience accept it fits right in with the nature of Weird Science.
Weird Science and Mad Science are playable advantages in the game - the former allows you to make advanced gadgets that would be possible by modern technology while the latter covers pretty much anything you can come up with (although the GM can make a Game Breaker prohibitively expensive...)
GURPS 4e actually has a skill called "Weird Science" which lets you invent ridiculous devices. Whether you are believed or they work is entirely dependent on the setting.
GURPS 4e also has "Science!" as a Wildcard skill, which can be substituted for ANY science skill but costs a ton to buy.
Deadlands has this as one of the sets of player skills. This being an explicitly Faustian setting, the knowledge to make the gadgets comes from less than heavenly sources, and every new gadget you invent makes you a little more insane.
Adventure!, being another celebration of 20s pulp, allows any Inspired character to try their hand at super-science. Daredevils can create Advancements, devices a generation or two ahead of their time; Stalwarts and Mesmerists can also create Innovations, devices that mimic the Inspired's extraordinary abilities.
In Magic The Gathering, the blue/red Izzet League from the guild-dominated plane Ravnica run on this. Impossible is literally just a suggestion at best for them, and the things they create are weird indeed.
Fallout runs on the outrageous science of 50s sci-fi flicks. The Old World Blues DLC from Fallout: New Vegas takes this trope and runs with it. The first NPCs you meet in the expansion are a group of scientists who sound like they just came out of a 50's sci-fi flick. One of them has No Indoor Voice, claiming that people keep tampering with his volume knob. They also use Buffy Speak when trying to repeat the names of the gadgets that the others among them have invented, and the main threat is a Mad Scientist who is just as Hammy as everyone else.