With 1/27 the volume but only 1/9 the surface area, the smaller cube will be three times as agile as the larger cube. (Assuming they are capable of movement.)
"The bigger they are, the harder they fall."
— Joe Walcott
A scientific principle often ignored in media:
When an object undergoes a proportional increase in size, its new volume is proportional to the cube of the multiplier and its new surface area is proportional to the square of the multiplier.
For example, if you double the size (measured by edge length) of a cube, its surface area is quadrupled, and its volume is increased by eight times.
The point of this law is that with living beings, strength is (more or less) a function of area (the strength of a muscle or bone is proportional to the area of its cross-section, not to its total volume), but weight is a function of volume. And Newton's famous Second Law (the "force = mass × acceleration" one) means that if you double a critter's height while keeping it the same shape, you end up with four times the muscle power moving eight times the mass, so instead of having the same relative agility as the original, the double-sized creature actually has only half. The same goes for most machinery.
This applies to flyers as well: Double the size, and you get four times the wingpower attempting to keep eight times the weight airborne, so the creature's ability to fly has actually been cut by half. Helicopters are hit particularly hard by this law; the largest payload of a cargo helicopter versus the world's largest airplane is 20 to 275 tons. Sorry, Hotelicopter.Airships, on the other hand, benefit greatly from the square/cube law, as even small increases in size can quickly increase the volume of buoyant gas they can carry. Take the Graf Zeppelin and his successornote and the Hindenburg's sister ship the Graf Zeppelin ll for example. The original Graf was 776 feet in length. The Graf ll was a mere 30 feet larger in any direction, but carried double the volume. Because these gains came at almost no increased structural weight, the returns went entirely into making the Graf ll an even morepalatial flying cruise liner than her predecessor.
Buoyancy in general, whether it be in the air or in the water, is an easy way to minimize the limitations of the square-cube law, because buoyancy is dependent on density, not mass. Good news for the whales, then.
A full explanation for the biological aspect is a lot more complicated due to subtler factors (muscle/bone stress, required oxygen uptake, dissipating body heat, etc.), the gist of it is the same in every case: You can't just scale something up (or down) to a different size and expect it to still work the same way as it used to.
Again, the law is not limited to living creatures, but applies to anything with mass (and, well, everything has mass): A skyscraper twice as wide and tall as another will have eight times the weight, and require a far stronger support structure — wood and brick just can't hold the weight. Likewise, the humanoid Humongous Mecha needs incredibly strong legs to hold its massive frame upright (probably some sort of Unobtainium), and that's not even considering how the ground beneath it also needs to be able to support that same amount of weight without caving in, or the fact that it needs some incredibly powerful motors just to get those powerful legs and arms moving (which is why we call them Impossibly Graceful Giants).
Knowing that audiences are becoming more savvy about this as compared to the days when Attack of the 50-Foot Whatever and the Incredible Shrinking Man were safe, standard plots, many creators who knowingly break the law will try to invent some Art Major Physics to justify or Hand Wave how their creation can get away with breaking it — say, the Applied Phlebotinum didn't just change their size, but also does something else to sustain their new size and counter the Law's negative effects upon them.
See Attack of the 50-Foot Whatever and Humongous Mecha for examples of media ignoring the Square/Cube Law. Sometimes justified by the use of Required Secondary Powers. Compare Muscles Are Meaningless and Pintsized Powerhouse.
The franchise has a few characters that can grow in size, or are already gigantic compared to normal people. A Saiyan can become a massive ape monster called an Oozaru if they have their tail and see a full moon. For these characters, they already have superhuman strength, and growing larger boosts that strength somehow, Oozaru multiplies Saiyan strength 10-fold, so they can survive their size. The issue of bulky muscles becomes a plot point in the Cell saga. The Saiyans develop a Super Saiyan form that is muscular to the point they look like oversized bodybuilders (do a Google image search for Ultra Super Saiyan and you'll get an idea from the first few pictures). It grants major strength, but at the cost of mobility, making the form useless, not to mention that it is very inefficient at energy consumption and quickly wears out the user.
Lampshaded in Trunks and Cell's fight; the more Trunks powers up, the more he bulks up, but he loses speed and agility.
Given that the character by this point have enough physical strength to level mountains with a flick of the wrist, it seems odd that a slightly higher body mass would make such a huge difference. This, of course, is overcome when Super Saiyan 2 form is reached, where strength and speed seemingly increase proportionally.
Full Metal Panic!Hangs a Lampshade on this fact; although it glosses over the existence of relatively small Humongous Mecha, an extremely huge example appears on the villains' side in one episode, and accordingly a character points out that it ought to collapse under its own weight. When the particular bit of Applied Phlebotinum which prevents this is destroyed, it indeed does so. Bonus points for the mecha not simply falling over; rather, its legs rupture and collapse outwards under the weight.
Yoshiyuki Tomino originally wanted to avoid this in Mobile Suit Gundam, with all the battles taking place in space, but eventually broke down and had the middle third of the series set on Earth. His novelizations are almost entirely set in space, though.
Mobile Fighter G Gundam generally has issues with scaling weight up linearly from human to massive robot; with their given weight figures in that series, the average Gundam is made of material with the density of styrofoam.
Later Gundam shows hand wave this trope. Starting in Mobile Suit Zeta Gundam, the mechs are built out of stronger, lighter materials. In addition, the Gundam Mk. II is built using Movable Frame technology that further lightens the weight by incorporating all the internal systems into the skeletal frame of the suit. That technology becomes more common and, eventually, the standard. After "Char's Counterattack", the technology is streamlined and miniaturized, resulting in mechs that are shorter, yet far more powerful than the earlier models. This, not coincidentally, made the new model kits smaller and cheaper to produce at the same scale.
Mobile Suit Zeta Gundam also had the absolutely gigantic Psyco Gundam, which really shouldn't have been able to walk on Earth. This is lampshaded the first time it shows up, with the characters wondering how in the hell that thing doesn't crush itself under its own weight. The Psyco Gundam in fact needs it's own Minovsky Drive like the White Base to support it's own immense mass.
Mentioned briefly by Koizumi in Haruhi Suzumiya when he's showing Kyon a Celestial/Shinjin/Avatar tearing up a section of Closed Space. He notes that the giant should be unable to support its own weight, but also that the laws of physics in general simply don't seem to apply to them.
Justified in Cannon God Exaxxion, where the titular colossus has a gravity control device powered by Antimatter. It also addresses the problem of weight distribution on the feet by replacing them with invisible forcefields that distribute the machine's (still considerable, even when mitigated by gravity control) weight over a wider area, which has the unfortunate side-effect of flattening innocent bystanders who are dozens of feet away from the mech itself.
Lampshaded when Friend's cult tries to have a giant bipedal robot built, and the engineer they get shoots down most of their ideas as impossible. On the eve of the new millennium, they end up using a fake robot that was just two crawler tread "legs" supporting a zeppelin with a cover and metal frame over it.
Towards the end of the series, that same engineer manages to pull it off. Though this version is at least slightly more plausible, as it's not humanoid, but rather something that looks like a cross between a frog and a chicken with the legs connected at the sides for better weight distribution. The robot is also primarily remote controlled, since, while it does have a cockpit inside, operating it manually is rendered nearly impossible due to severe motion sickness induced by its uneven gait.
Giant Robo: Ginrei Special has one robot whose weight was 2/3 armor, and needed to use its Jet Pack just to stay standing up.
Gyo obeys the law — a zombified whale that rises onto the land immediately collapses under its own weight.
Acknowledged/lampshaded in Patlabor: Humanoid-style labors tend to be made with very large feet and small torsos. This trope is mentioned to a certain extent early in the TV series when Kanuka puts a labor through a bunch of stock action movie moves (jumping, flipping, etc). Noa asks Asuma why he's wincing, and he explains that while the new police labor model (the Ingram AVS-98) is technically capable of performing any motion that a human body can (with regards to degrees of freedom), it can't really take much more punishment than standard walking without requiring pretty serious maintenence, and implies that Kanuka's short perfomance will mean days of work and hundreds (possibly thousands) of dollars in components to bring the labor back to 100% operating capacity.
Rurouni Kenshin: Kenshin invokes this law during his match with Senkaku. While he's every bit the Lightning Bruiser, Senkaku is more than twice the size of Kenshin, and eventually his body couldn't take the strain of having all that mass keep up with Kenshin's godlike speed.
Superheroes like, say, Ant Man, usually don't even bother giving this a wave. When shrinking, Ant Man maintains the strength of a regular human despite his size, and when growing, his size is "Proportionate" (as strong as the writer needs it to be). Heck, all size changing superheroes seem to be riddled with problems. Despite their fists often being nearly the size of a pinpoint, moving with the force of a superheroic punch, they always seem to hit like a wrecking ball instead of like a knife...
However, there was at least one early story where Dr. Pym increased his size beyond a certain limit and collapsed, unable to move and needed The Wasp's help to shrink him back down to a reasonable size again.
A note should be made about the new Ant Man (Eric O'Grady): he does try to take advantage of his proportionate strength to punch out a guy (to impress a woman), but doesn't realize that his punch is more like a bullet than a hammer.
Marvel Comics heroes use "Pym particles" to grow and shrink, so there's an extra layer of Phlebotinum keeping it working. DC's The Atom, on the other hand, knows that shrinking is dangerous — it's even been made into a plot point and weapon, as seen in Justice League: The New Frontier (The Atom can alter his mass independently of his size, so it's less of an issue for him).
In Hank Pym's first appearance as Giant-Man, Marvel Comics played the Law quite accurately — after he's grown a few feet, he's no longer capable of standing and needs the Wasp's help just to get at a reversal pill. He pegs his maximum effective size as twelve feet or so, and sticks to it for a while. The limitation didn't last long, though - soon he was fighting giant monsters at Godzilla size!
The Ultimates version of Hank Pym can grow to just under sixty feet, as according to his wife, 60ft is the point at which the human skeleton can no longer support its own mass (later generic Giant-Men manage to surpass this, with no mention as to how). His ability is reverse-engineered from Jan's ability to shrink, and she mentions that she can't shrink smaller than an inch because her body automatically knows what its limits are.
Given a nod in the original Guardians of the Galaxy. Yellowjacket — whom we've only seen shrink up until then — grows to fight Blockade. She wins the fight, but the exertion puts enough strain on her heart that she loses consciousness.
Spider-man writers occasionally lampshade this, and there has been at least one Retcon that Spidey is actually stronger than the proportionate strength of a spider — or even simply possessing a spider's strength-to-weight ratio, which is likely what the creators originally meant.
He has a villain called the Walrus who claims to possess the same strength-to-weight ratio as a walrus. Super Dickery pointed out that this would make him weaker than an ordinary human. (The laws of physics were nonetheless basically told to go fuck themselves, as The Walrus indeed turned out to have Super Strength).
Discussed in Avenging Spider-Man 10 when he uses his physics knowledge to discover that Robyn Hood is going to explode.
The Hulk is known to get stronger and larger as he gets angrier (maximum height is roughly twelve feet); this might be justified, though, as his relative muscle (and presumably bone) mass increases as well as his height. Furthermore, Hulk is generally not depicted as merely scaling up; in most depictions, the cross-sections of his arms and legs increase out of proportion, which would balance things out some.
It's been implied that he draws his strength from outside of his own body, and therefore muscle mass would be irrelevant.
The size changing as he gets angrier and stronger thing is depending on the writer and the artist; some have his height stay consistent once he transforms, though this itself can be an informed ability as an artist will alter his height between panels for various reasons. Officially the Hulk's transformed height is just under eight feet tall. He'll often be shown as over ten, but that's usually stylistic or for dramatic effect.
Where Hulk comics fail to justify or avert is in that we frequently see him standing on floors that should not be able to support what his weight must be. Hard wood would splinter under him, for example, as he probably weighs about as much as a four-door car. Floors would take an even greater beating when you realize that all that weight is being concentrated on two relatively small areas.
Lampshaded in the comic, where the presence of giant ants has pretty much everyone pointing out that they should be crushed by their own weight, and the only one that doesn't say it's impossible is the guy that thinks it's covered by "imaginary physics" and "imaginary radiation", which would give them laser eyes.
A giant monster attack in Volume 4 has Robo ask "Why do we even have the Square Cube Law!?"
An interesting inversion appears with the superheroine "Micro-Might". Her power is, specifically, to take advantage of the Square-Cube Law - shrinking herself down to half height, keeping the same mass - to become stronger and tougher. (Okay, so it plays a bit fast-and-loose with the actual equations, but it's still nice to see someone USE the law instead of just ignoring it.)
When she's forced into Phlebotinum Overload, she becomes so dense that she can't move and can barely speak.
A plot point in one Fantastic Four story; Reed Richards encounters an alien able to absorb energy to grow to gigantic size and notices that its footprints aren't getting any deeper, so its weight isn't increasing, therefore its mass isn't, either. It's just puffing up like a balloon. So he manages to "overinflate" the alien by feeding it too much energy. Note that the footprints should have actually become shallower if his feet grew and the mass (and thus weight) did not increase, so we will have to assume Reed Richards deemed this bit too trivial to mention (for him to not realize this would be out of character).
The Official Handbook of the Marvel Universe tried to be scientifically accurate, so it constantly faced this problem, handwaving them away with references to anti-gravitons or similar technobabble that at least suggests that some writer is aware that there's a scientific problem.
In one issue of Nodwick, this law is specifically addressed by a potion intended to grow people to giant size; specifically, it doesn't increase mass, and as such the 'inability to support own weight' point is moot. Of course, some other problems, like how a fifty-foot tall being weighing one hundred pounds reacts when being exposed to an ambient breeze, immediately present themselves.
Mentioned in an issue of Planetary dealing with a secret government project that used '50s super-science to turn "undesirables" into, basically, the monsters of '50s horror movies. One guy got the Amazing Colossal Man treatment; he was in pain for the rest of his short existence.
Discussed in great detail in The Science Of Superheroes, with regards to superheroes that are able to make themselves larger or smaller.
Discussed in the pre-New 52 Jaime Reyes run of Blue Beetle. Since you can't scientifically increase somebody's size past a certain threshold without them falling apart into a pile of bones and blood, magic is always behind giant humans.
Discussed in the Deva Series, where it is noted that the Seed can't get much bigger without magical assistance if they want to maintain their power. And since one of their key gimmicks is Anti-Magic protection...
Lampshaded and handwaved in Nobody Dies; it's quoted in dialog between two subordinate scientists that they can never get Yui to explain why the Evas don't sink into the ground due to their weight, and the current dominant theory is that the ground is too scared of them to let them in.
Subtle aversion: Devastator, a large, gorilla-like robot made from construction equipment, is so massive that he cannot stand up straight without risking his legs caving in on themselves. It was for that very reason that the folks at Hasbro and ILM opted to go with the gorilla-walk.
The same aversion is seen with the other robots in both films. Smaller guys like Bumblebee or Barricade are pretty agile, while the medium size bots like Ironhide can move, but they aren't that swift. Moving up to Optimus Prime and Megatron, they're clearly focused on power brawling, although Optimus is a Lightning Bruiser.
In the live-action movie, in truck mode, Optimus became a conventional tractor (one with a hood) instead of his original cab-over design, to give him enough extra mass to get to 30 feet tall when in robot mode, rather than 25 feet like the other Autobots.
Giant cockroach movie Mimichand waves the Square/Cube Law during an autopsy scene, where the entomologist discovers that the Judas Breed has evolved lungs. This explains how they can breathe, but not how a six-foot cockroach with six-foot wings can fly while carrying an adult woman.
The issue is simply skipped in the Donald Wollheim story the film was based on: the mimic, a giant moth, is never seen flying and in fact may not be able to fly. Its newly-hatched offspring, the size of very large moths, fly just fine.
In Pirates of the Caribbean: At World's End, Calypso grows to gigantic size on board the Pearl, yet neither tips the ship over (again), sinks it, nor collapses the decking beneath her weight. Could potentially be handwaved by her goddess powers or something, but it's never explicitly addressed either way.
Seeing as she bursts into a bunch of crabs a few seconds later, it's safe to say that the laws of physics don't apply to her.
Arachnid actually discusses the Square-Cube Law, though its brief and hasty explanation of why the titular giant spiders aren't subject to it is unconvincing (something to do with the spiders being of extraterrestrial origin).
In Ghostbusters II, while animating the Statue of Liberty with mood slime is already telling physics to take a hike, they briefly touch upon the law when Ray notes that moving her any faster might shake her apart.
Discussed in the 90's HBO remake of Attack of the 50-Foot Woman - the doctor treating Nancy remarks that at her new size her heart is under tremendous strain due to the change in her mass, and that any undue stress may cause it to give out.
In Iron Man 2, we see North Korea's attempt at making an Iron Man suit, which appears to be 20 feet tall machine covered in guns... which immediately fell over because of how thin it's legs were to the top heavy body.
Guillermo del Toro and the effects artists admit that huge beings, robots or monsters alike, would move slowly, but the Kaiju and Jaegers move quickly to make for fun scenes.
Gipsy Danger using a civilian oil tanker as an Improvised Weapon is flatly impossible; it would have buckled under its own weight the minute they dragged it out of the water, let alone started swinging it around like a bat. Of course it looked great, though.
The dragon species created in Duumvirate has wings in its juvenile stage, but loses its ability to fly as it grows up.
Mentioned in Feet of Clay by Terry Pratchett, regarding how different sizes of animal would end up after a given fall of about two storeys—a spider wouldn't notice it, a mouse would walk away, etc.
In Guards! Guards!! almost all the characters realize right away that the dragon can't possibly fly under its own power... unless it's fueling itself with magic. The descriptions specifically point out that its flight looks completely unrealistic compared to a bird's, because it's actually magically levitating itself.
Discworld's gnomes are also terrifyingly strong, despite being six inches tall, and able to knock a man out and break bones with a headbutt or otherwise — because they have the strength of a grown man, concentrated in a very small area.Said gnomes also possess all of a grown man's bad temper concentrated into that same space, which makes the above acts of violence not only possible but also fairly probable.
Pratchett got it wrong earlier in his career; The Carpet People seem afraid of a fall of a few millimetres.
Given that the Carpet's largest city is explicitly stated to be the size of a full stop, a few millimetres to a Carpet Person would be the equivalent of a few miles to normal humans.
Made a plot point in the City of Heroes book "The Web of Arachnos". The massive army of robots are defeated when they're unable to do things the smaller bots were able to do due to their size, which leads one of the two creators to yell at the other for forgetting the Square-Cubed law and going for Rule of Cool/Intimidating over practical.
Noted in The BFG: a cook scales up a meal to the giant's scale based on his height, rather than his mass. The giant is not impressed.
There was a short story which played with this concept; among other things, a supervillain unleashes an army of giant ants on a city. The ants are all killed, but someone realizes that in order for the creatures to support their own weight, their legs would have to made of some incredible super-strong wonder-substance. They harvest the substance and are able to reproduce it. It turns out this was villain's Evil Plan all along, as it causes the global steel industry to collapse overnight. The villain ends up ruling the world. Of course, if the ant attack actually worked, he wouldn't have complained. Either way he'd win.
Jalil also point out (while watching a dragon fly) that the it can't possibly be capable of flight. He's quite put out that physics isn't working the way it's supposed to.
Animorphs: when the Animorphs run into the Helmacrons Shrink Ray, they find themselves able to toss around pebbles as big as themselves, while Tobias (to them, now a hawk the size of a human) notes that he can flap faster and climb proportionally higher, and might even be able to carry one of them. Evidently the shrink ray also reduces mass along with size.
In Gene Wolfe's Book of the New Sun, giants eventually get large enough that they have to live in the water, where they grow to truly immense sizes. The narrator is quite shocked to find out that the giant he meets (on land) is nowhere near full grown.
In Stan Lee's Riftworld!, the giants are supported by a telekinetic field, which has the side benefit of making them Immune to Bullets.
A plot point in Danny Dunn And The Smallifying Machine. In learning to walk at 1/4 inch high, the accidentally-shrunken characters have difficulty adjusting to their reduced weight; falling, they hit the (much closer) ground almost before they've realized they've tripped, but suffer no injuries due to lack of mass. Difficulty coping with the surface tension of water is also addressed.
In Orson Scott Card's Shadow series, Bean has a genetic disorder that causes his brain to continue growing as an infant's does, giving him extreme intelligence, at the cost of causing his body to continue growing as well, leading to a projected lifespan of about eighteen years. Since the problem is caused by gravity, he eventually leaves in a relativistic spacecraft with controllable gravity, so that he can possibly survive until a cure is found.
In one of the I Was A Sixth Grade Alien books the characters are shrunk to about seven inches and quickly discover that this has not affected their strength or mass after trying to get off a desk they attempt jumping down onto a open drawer and snap right through it.
In Stanislaw Lem's Fiasco, humongous mechas obey the laws of physics. Inertia, for example, is applied realistically: to stop or turn around with a giant mecha, you need a lot of space, just like with a battleship. Sudden movements would lead to great structural damage to itself, so the controls are designed in a way to limit the maximum acceleration of actuators depending on the load the appendages have to bear.
And again in Peace on Earth, where the main character, Ijon Tichy, explores the surface of the Moon with the help of remote-controlled robots, the largest of which give him an impression of being merged in some thick liquid.
The short story giANTS actively weaponized the law to deal with mutant South American army ants (specifically, stuck in the nomadic phase and heading north). Genetic Engineering Is the New Nuke indeed.
Dragon bones are specifically mentioned as a very different and much stronger material than Terran animals' bones, and larger dragons don't move much when they don't need to. Then again, they are alien enough to generate lots of concentrated phosphine without any harm.
Likewise, The Dragonlover's Guide To Pern shows that the skeletal structure of a dragon is very different from any Terran animal's. The design looks like it allows for a greater distribution of weight.
Of course, this trope suggests that their skeletal structure (and other anatomy) should also be very different from that of their native fire lizard cousins...
It was implied that there were once natural dragons on Pern, and that the fire lizards are either descendants of them or have a common ancestor, so it makes sense they would keep an unnecessarily complex body structure despite their smaller size (see the human appendix and whale leg bones).
Deliberately used in The Dresden Files novel Small Favor, where Harry is fighting a twenty-foot-tall fairie hitman with a car-sized sword, battle armor, and hefty anti-magic defenses. He manages to get said hitman chasing him over a patch of waxed floor and changes direction, causing the faerie to fall over and mangle himself in the impact. He's not overwhelmingly injured (being a faerie and thus very resistant) but it smarts like hell.
Addressed in the Harry Potter books with Rubeus Hagrid - he is a half-giant, quoted to be two times as tall as a regular man and nearly five times as wide, having the weight of 1,000 to 1,500 pounds. If you try to be as large as a Kodiak bear raised on the hind legs, having the body structure (and presumably the muscular strength) of a Kodiak bear helps a lot. Even better when your 7 days a week job is mostly physical labor around Hogwarts lands. In the movies, however, they aimed for a height of 8'6" (about 259 cm), of course towering over any human, but certainly not "twice as high." They do, however, give him an appropriate girth for his new scaled-down height.
In There Is No Darkness, the protagonist is an enhanced human — well over two meters, 180 kilograms — and when in a fight against an enhanced bear that outmasses him by a ridiculous margin, realizes square-cubes applies.
In his novella The Forgotten Planet, Murray Leinster plays this very straight in his presentation of a world in which insects have grown to enormous sizes, such as ants two feet long and spiders with yard-long legs, based on fossil records of actually giant insects, and are at the outer limit of what cube-square effects allow. But other insects, such as water striders, are no larger than normal, as gigantism would destroy them.
It's also explicitly said the gigantic size of the insects was due to a specific combination of factors which had to match exactly for them to evolve: atmosphere very humid and very rich in oxygen, thick clouds keeping constant warmth via greenhouse effect, huge quantities of nutrients available due to gigantic sizes attained by fungi. The key of the heroes survival is simply climbing to a plateau with temperate climate - the mere coolness of a temperate night renders the giant insects motionless and vulnerable.
The short story Surface Tension, by James Blish, deals with a race of microscopic humanoids, and does a good job of showing physics on such a scale — for example, the surface of the pond they live in is an impenetrable barrier.
Immanuel Velikovsky studied ancient legends and concluded that other planets were responsible for global catastrophes here on Earth. Among other notions, Venus was once a comet ejected from Jupiter responsible for the Biblical plagues of Egypt, and Earth once orbited Saturn and the Biblical flood was caused by Saturn going nova and also ejecting Earth to its current orbit. Scientists everywhere rolled their eyes and dismissed him entirely because everything he suggested violated every understanding of planetary physics, as well as conservation of energy and angular momentum. However, he did influence a portion of the American public, some of whom latched onto the notion that the only reason dinosaurs could exist was because of lessened gravity on Earth's surface due to the presence of Saturn in the sky. One author inspired by Velikovsky correctly stated that a human of saurian dimensions would collapse under his own weight and die, then incorrectly reasoning that dinosaurs couldn't possibly have survived. The book then goes on to have Jupiter eject a planet the size of Venus which causes ... well, the science was bad and the writing not much better.
In Cruel Zinc Melodies, the Big Creepy-Crawlies that were created from normal insects keep running into problems with their magically-increased size, as when a giant beetle tries to take flight from a rooftop and winds up splattered on the pavement.
In S. Andrew Swann's Dragons of the Cuyahoga, there is a bubble of magic over Cleveland thanks to a semi-permanent magic portal open in the city, allowing dragons, elves, magic spells and other "fuzzy gnome" phenomena to flourish. The novel opens with one of said dragons fooled into flying out of the bubble and into our world where the square/cube law rules supreme. The results are...messy. Murder via reality.
The book Gulliver's Travels nods to this law; the Lilliputans realise that since Gulliver is proportioned the same as they are but is 12 times the height, he is therefore 1,728 times the bulk and thus needs that much more food. However, it also ignores the fact that if the Lilliputans were really human-proportioned but 6 inches tall, they would rapidly freeze to death.
In The Name of the Wind, Kvothe and Denna discuss this as they figure out a way to stop the Draccus; Kvothe mentions that it would probably take only a fall of about ten feet to kill the thing.
The premise of H. G. Wells' The Food of the Gods is that growth goes in fits and spurts because of certain factors in the bloodstream of growing animals, or the sap of plants, which are depleted and have to be replenished before the next spurt of growth occurs. When two protagonists synthesise a Food containing the missing factors (which are the same for all species), the result is giant animals and plants (humans given the Food grow to typically forty feet high) which however suffer no loss of agility or other consequences of their giant size.
A Song of Ice and Fire averts the pitfalls of this trope with its roughly-10-foot giants, who have a body shape that is compared to bears; disproportionately large feet on the ends of short, thick legs, and wider hips than chests, bringing their centre of gravity right down.
However, the series plays this trope dead straight with the dragons, which can grow to enormous sizes and are still able to fly perfectly well. However, there has been shown to be some connection between live dragons and the magical powers in the world, so it's possible the dragons get some magic-fueled exemption from this law.
Equally lampshaded and played straight in an episode of Farscape. Most of the crew are shrunk and one complains that it shouldn't be possible. Another tells her that it's better to think of a solution than to complain that what just happened isn't possible.
Once on the show, Adam was attempting to use a toy as a scale model human to test parachutes. He calculated its weight as a proportion of height and got an unreasonably large number. He later realized his mistake and calculated as a ratio of volume. The irony is that this approach still doesn't work, because a parachute's effectiveness is based on its area.
Adam: I'm roughly 6 feet at 180 pounds. Proportionately, that's 72 inches to 180 pounds. 10 inches tall... 25 pounds. I just did the math. I need him to weigh 25 pounds.
Jamie: So what you're saying is, he needs to be made of depleted uranium.
Adam: Uh, do you have any? (looks at labeled shelves behind him) Is it under "D" or "U"?
Later used correctly in the Lead Balloon myth. Adam and Jamie's small-scale lead-foil balloon didn't float up specifically because it was too small (as they explained on the show). When they scaled the balloon up to a much larger size, the ratio of volume to surface area became large enough for the balloon to float—in fact, they actually needed to mix air with the helium to keep the foil from ripping from too much buoyancy.
Subverted in the Buffy the Vampire Slayer episode "Life Serial". In order to distract Buffy, Jonathan transforms himself into a much larger demon (that seemed to be modeled on the South ParkSatan) except that as the demon he "actually had the proportional strength of, uh...me."
Discussed on The Big Bang Theory, comparing the viability of giant ants vs. giant rabbits and mice. Notably, the production blog for the show cited this very page in explaining the problems with enlarging arthropods and mammals.
Played for laughs (naturally) in comedy series The Goodies, episode "Kitten Kong", in which (as the title implies) the three ply a kitten with a miracle growth-promoting food. In the end they manage to get it back to normal size, but then discover that they have a problem with mice — giant ones...
Animal Planet Animal Olympics played with the law. While they ignored the fact a human sized insect would be crushed by it's own weight, they did note that small animals have much more proportional strength. So, a dung beetle the size of a human could lift many metric tons, while an elephant the size of a human could barely lift a fourth of it's now much smaller weight. The only reason why the insects didn't win all the events is because they often were either too good at their events (such as a flea that cleared the entire stadium completely) or too stupid to complete the event.
André the Giant was so huge due to acromegaly, a disorder of the pituitary gland which essentially causes your body to try to violate the square-cube law. He spent the later part of his life in constant pain due to the strain on his muscles and circulatory system, and died at 47.
This trope is why all the actual giant bugs in Mortasheen are dog sized at most, and to actually get giant bug-like creatures, you have to combine them with humans to form the Brundlefly-style Arthropoids.
GURPS details the stats needed for monsters of various impossible sizes with an eye toward the Square Cube Law (how the creatures can be so big is left up to the GM).
And yet, while getting it right there, completely fails to grasp it in the monster sections. Numerous monsters in the books are given proportions that indicate the world is being overrun by very scrawny ogres made of balsa wood.
Also played somewhat straight in that a creatures lifting capacity does not typically scale directly with the creature's mass. For an 8x mass increase for an extra size category, a creature typically receives a 6x increase in carrying capacity. It gets worse when A Wizard Did It, as spells such as enlarge person increase the creatures weight by the usual eight times, but only increases their carry capacity by approximately 2.6 times.
There is an artifact in the Book Of Vile Darkness called the Despoiler of Flesh that gives you very flexible control over a creatures shape, it more or less points out that it has to be scientifically plausible or the creature will die because of an unsound anatomy.
When RoleAids released Giants, a vintage third-party D&D supplement, they took this trope into account, rationalizing giants' physiology with honeycomb-framework bones, radically different leg musculature, and super-tough hide to contain their extremely high blood pressure. Oh, and a heaping dose of A Wizard Did It (or rather The Gods Did It) for titans.
The Imperial Titans and Baneblades, and Gargants, and Squiggoths, and Hive Tyrants...thanks to 40k's Rule of Cool-powered physics, it's likely cumbersome size can always be counterbalanced by the number of guns bolted to it.
Taken to hilarious extremes with the Hierophant Biotitan who, despite being as large as a warhound, is supported by 4 relatively tiny stalks it calls legs. How it doesn't explode from its own weight is anyone's guess.
Averted in a few places, though. Space Marines are about 8 feet tall and almost 8 feet WIDE, with numerous extra organs, and a giant suit of power army to help take the strain of their larger proportions. Of all the factions to get this right, the ORKS versions of Titans (called Gargants) seem reasonable. Either they have very widely spread out feet, or tank treads to move about, and it appears most of the weight is concentrated towards the ground. Also if anyone has the right to ignore the square cube law, it would be the Orks, since their technology relies heavily on their innate Psychic Powers to work in the first place.
Werewolf: The Apocalypse fails to take this trope into account with the Garou and other shapeshifting races. A Garou's crinos form height is 150% that of their homid form height, with significantly more mass. Ironically, a Garou's dexterity increases in crinos form, when it should logically decrease.
It's a bit of a standing joke among BattleTech fans who have done the math that despite usually weighing anywhere from twenty to a hundred tons, given their physical dimensions BattleMechs must be less dense than water and should therefore logically float, especially since they're also by default environmentally sealed so they can operate in vacuum and other hostile environments. The rules, of course, still have them wading through any water deep enough to be worth depicting on the map.
The impossiblity of Humongous Mecha is repeatedly pointed out in Touhou Hisoutensoku. The fantasy-absorbing nature of Gensoukyou would actually allow for one to exist inside of it, and Sanae becomes convinced that the giant shadow she saw is one. The truth is less glamourous, and closer to respecting the laws of physics.
Played with in the Mega Man franchise. Large Robot Masters and their counterparts from the spinoff titles can jump much higher than your character and pull off some moves, even if they aren't equipped for flight. However, large bosses in the later portions have size and power, and can manage limited movement, but they can't maneuver for crap, sticking them in basic movement patterns most of the time.
In the Mega Man X series the bosses of the opening stage are generally titanic Mechaniloids with the maneuverability of a brick and are much weaker than their size would dictate. The few exceptions are Gigantic Mechaniloid CF-0 from X2, which could jump extremely high and hit pretty hard; Egregion from X4, a huge dragon Mechaniloid that could fly very fast; and Mega Scorpio from X7, a centaur-like scorpion Mechaniloid who could turn and charge fairly quickly.
Huge monsters made of metal like bronze or steel are strong enough to support their own weight (which is suitably high because it is calculated by the game based on their size and material) and are extremely durable, but that same weight also makes them relatively vulnerable to damage from falls. Very large grazing animals that have been domesticated are also almost impossible to keep because they need so much grass to eat.
In V 0.34.11, tame elephants will literally starve to death while eating. Predatory animals (such as grizzly bears or giant grizzly bears) are exempt from the feeding requirement though. And then there's the giant desert scorpion and giant cave spider, both approximately the size of a (regular) bear...
Liir are technically immortal and never stop growing, resulting in their Great Elders being very large. Eventually, they grow too big and die due to gravity. Suul'ka are Liir Great Elders who say "Forget this gravity junk" and teleport into space to continue their existence.
The Black, the leader of the Black Swimmers (Liir Space Navy), is also this, as he was originally sent to kill the Suul'ka.
Broken and lampshaded in Worlds of Ultima: The Savage Empire with the Myrmidex, a race of intelligent giant ants. As the manual says:
As such, they make hash of prevailing theory about he square-cubed laws, but they do exist, and are a formidable and savage race... much like the ants to which they appear to be related.
Broken severely in Batman: Arkham Asylum with Killer Croc. His dossier says he is 11 feet tall and 580 pounds (9 feet tall and 320 pounds in the sequel). He should realistically weigh 3 or 4 times that.
They seem to have realized this when putting him into the game proper, however. Croc moves extremely heavily, and a very powerful freight elevator struggles to slowly grind upwards with him in it. Without him, it moves much faster.
Duke Nukem: Manhattan Project has, as the end-of-stage boss of Stage 4, a cockroach about 24 feet tall and with corresponding other proportions; a real cockroach even a tenth that size would suffocate. (It also has breasts, despite being an insect rather than a mammal, but that's another trope entirely.)
Mass effect fields in Mass Effect allow one to alter the mass of an object, getting around this problem. However, it still would take too much energy to let a capital ship land on a planet, so they don't. Except for Sovereign and the other Reapers. Despite being the largest ships in the setting, they have such a massive amount of power at their disposal they can easily land on a planet, walk around, and pop back into space. They are pretty slow and wobbly on the ground though, due to having to deal with the gravity. Additional details mention that their shields are a fraction of what they can maintain in space (which can take the firepower of an entire fleet without a scratch), due to the incredible drain by their mass effect cores.
Guild Wars and it's sequel has the 9 feet tall Norn. Furthermore their figure seems to be bulkier, than the average human. That would mean, that they weight at least 650 pound (300 kg). To maintain that body weight, a similar built 6-feet man must be able to lift at least 330 pound (150 kg).
Pokémon Ruby and Sapphire introduces Wailord, the largest pokemon ever at 14.5 meters (47 ft. 7 in.) long. It weighs only 398 kg (877.4 pounds). For comparison, a male sperm whale averages 16 meters (52 feet) long and weighs 40 tonnes (45 short tons).
In fact, doing the math shows that Wailord is lighter than helium. Float Whale Pokémon, indeed.
In Ilivais X the Ilivais units are basically 80s super robots, and as such standing under their own weight would be impossible without assistance. This developed into "make them focused on flight" which then developed into "don't even bother giving them feet". As such, most of them have blade legs that end in a point, meaning that if they lose flight capability, they're utterly incapable of movement.
In the Whateley Universe story "Boston Brawl", there's an Author Tract explaining how size-warping 'giants' really work, since the Square/Cube Law and some other laws of basis physics would seem to make it impossible. The giant Matterhorn only appears to be a 40-foot giant because everyone else interfaces with his warp displacement field instead of him.
The Workshop at Whateley contains a Humongous Mecha that devisor students occasionally work on. The best it's done is take three steps before the knee came apart.
Played dead straight on occasion, too - Jimmy T's antics on Hallowe'en come to mind, as do any Shifter (as opposed to Warper) size-changers.
The Salvation War averts this to a degree. On the one hand, most of the "lesser demons" are only about 8 feet tall, basically are made of muscles on top of more muscles, and while they all have wings only two subspecies (harpies and gorgons) can actually fly, it is stated that their bodies must produce lighter than air gasses to even manage that. On the other hand, there are many much larger demons, such as Satan, that well reach over 20 feet tall, although they tend to stay on the backs of great beasts or in their throne rooms. All Angels can fly, but they also are stated to have the gas producing organs. Yahweh is stated to be HUGE, but never moves from sitting on the Eternal Throne. When Heaven is conquered, the great gates to the Eternal City are so giant they cannot be moved, so must be blasted down.
Lampshaded in this8-Bit Theater issue. Red Mage attempts to defeat a giant by pointing out all the reasons a creature of its size simply could not exist, thereby making it vanish in a Puff of Logic. Said giant then proceeds to crush Red Mage with his club.
Similarly, in #585, Vaarsuvius attempted to use his Common Sense and knowledge of the Square/Cube Law to aid in a Banishment spell, with equally futile results.
Also invoked in #326, though not in as many words. Roy uses this law to make a hydra pass out, as its blood supply couldn't keep up with the number of heads it was growing.
And played with in #754, when the "Empress of Blood" (a morbidly obese red dragon the size of a small house) is able to fly, despite her wings being maybe 1% of the size of her body. How can she do it? Because D&D rules say that dragons can fly, regardless of size.
In this strip, Raven mentions the law by name (he's a teacher), with the implication that the magic involved compensates for the violation of normal reality. This is explained in The Rant of the subsequent comic.
In Fairy Dust, tiny races are able to lift several times their weight. Larger races compensate for their weight with thicker limbs and very dense bones, making the largest ones so heavy they can't swim.
On Chip 'n Dale Rescue Rangers, the main cast — two chipmunks, two mice, and a fly — are, encounters with size-changing rays aside, roughly the same size as their real-world counterparts, yet exhibit the same fear of falling a human should, given the same heights... often a distance that would likely only daze them for a few seconds. (This being Disney, however, something always lessens the threat to our heroes and their friends anyway, so whether their fears are founded or not is never shown.)
Lampshaded on The Venture Bros. with Humonguloid, a giant with a heart condition and other severe problems. Of course, VB's science of choice being superscience, he later gets shrunk to the size of an ant and survives for decades with no major health problems and, indeed, no medical care (Although he ends up being accidentally crushed to death by a rocking chair in the season 5 finale). The same character states the "proportional strength of an ant" idea to likewise be nonsense.
In Wakfu, this is how Sallygrove defeats Rubilax. Where everytime Rubilax is hit, he grows in size and mass. However, them being in the middle of a desert, after getting too big he starts to sink into the ground. He chooses to return being trapped in a sword rather than dying suffocated in sand.
One of the classic explications of this idea, although never actually mentioning the term "square cube law", is J.B.S. Haldane's "On Being The Right Size".
Some breeds of dogs which have been bred for size are easily susceptible to numerous health issues that smaller dogs do not, from joint pain to heart problems. Similarly, leaner people tend to have fewer health issues than overweight or very muscular people, because they have less weight to carry around.
Robert Wadlow, who stood 8'11"note about 272 cm, provides a good example of what happens when people try to get that big. Namely, serious physical problems requiring him to get leg braces and walk with a cane, having little feeling in his lower body, and dying at age 22. His cause of death was closely tied to his size: a poorly sitting brace irritated the skin on his foot, causing a blister. Due to his lack of feeling he did not notice this, and it got infected which led to his death. The tallest man alive, Sultan_Kösen, who is eight inches shorter, has undergone gamma-knife therapy to stop growing, and still walks with crutches.
A sad example... Beached whales die because their own body weight on land, without any support from water, crushes their lungs, causing compressive asphyxia (i.e. they cannot breathe under their own weight). Some cases of stranding have had the animals die due to drowning - their stomachs were crushed, and the animals vomited their food.
This can also happen to the larger varieties of sharks, who are not only massive but have flexible skeletons made almost completely out of cartilage. Their bodies are simply not capable of supporting their own weight out of the water.
On a slightly lighter note, weight is the reason why the largest whales gain nutrition simply by constantly filtering plankton into their mouths while swimming around. If they had to actively hunt down food, there is no way they would gain more energy from it than they had to spend moving their massive bodies around when chasing their prey.
The Nazis were getting hammered in the tank battles on the Eastern Front, so they decided to build a scaled-up tank, with armour thick enough to shrug off enemy tank shells, and guns big enough to one-shot enemy tanks. The "Maus", as it was ironically called, weighed 200 tonnes, was 10 metres long and 3.71 metres tall. The tracks were 1.1 metres wide - more than half its 3.63 metre width - in order to try to spread the load, but it still tended to sink if the ground wasn't completely firm. The designers had a difficult job designing (and then redesigning) a suspension system strong enough to support the weight, and finding an engine big enough to drive the whole thing - in the end, more than half of the Maus was occupied by powerplant and transmission, and it still wouldn't go over 13 kph. Crossing bridges with that weight was out of the question, so it was designed to be able to ford rivers, completely submerged if necessary. It was to have a 150mm main cannon. Unfortunately, they were unable to do anything about it destroying roads and damaging nearby structures simply by its weight and vibration. In the end, only two prototypes were built. That's not all, though. Plans were on the drawing board for a 1,000 tonne, 25 metre long Landkreutzer with a 12" main gun, infirmary and toilet facility, and its big brother, the 1,500 tonne, 42 metre long "Monster" Landkreutzer with a 32" inch main gun. Those Wacky Nazis, indeed!
Another Nazi failure was the planned demolition of Berlin to build, among other things, the Volkshalle. Meant to contain 180,000 people, it was basically a scaled-up version of Hadrian's Pantheon, which Hitler greatly admired. However, it was so big that the water vapor exhaled by its occupants would have created clouds and rain-assuming the entire building hadn't already sunk into the swampy ground.
In Berlin there remain a few gigantic concrete test cylinders the Nazis cast to see if the soil could support the Volkshalle's weight. Proving real life has a sense of dramatic irony, these cylinders have been steadily (albeit slowly) sinking for sixty years.
Similarly, Joseph Stalin had several projects of similar magnitude. The most famous of these is the Palace of the Soviets, which would have been the tallest manmade free-standing structure in the world at the time, had it been completed. The top of the Palace would feature a 30-meter statue of Lenin. It was cancelled when the architects finally got over their fear of Stalin to tell him that the building would simply sink into the ground (there are underground streams in Moscow). The designs were instead used to build five scaled-down structures. The place instead was made into a very nice giant open-air swimming pool with free admittance; until finally in 1990s the pool was demolished, and the original structure occupying the spot was reconstructed – namely, the lush and astronomically expensive Church of Christ the Saviour, with a giant underground conference hall, parking lot and other amenities for heads of rich Russian Orthodox Church.
Millions of years ago, bigger creatures were able to walk the Earth thanks in part to the greater concentration of oxygen in the air. Insects in particular have an inefficient way to carry oxygen to their cells. Back when there was more oxygen, they were able to push the envelope on size - there were dragonflies with 3 foot wingspans, for example. Nowadays, the same insects would suffocate. Nowadays, as always, the insects grow to whatever size the local atmosphere supports, so all it takes for monster dragonflies and mosquitoes to return is a bit more oxygen.
Elephants often break bones just from tripping and falling over.
Square Cube law was often cited by proponents of the Tyrannosaurus Rex being a scavenger and not a predator, many people assess that should the T-Rex trip (as often happens to predators in real life), it would sustain crippling or lethal injuries. Others point that the Tyrannosaurus, while fast enough to catch up to larger prey, would not have been nearly as fast as smaller predators (like raptors), which could have lessened the damage in a fall. It may have also employed strategies (like ambushes and a poisonous saliva like a Komodo Dragon's) that may have lessened the need to to run, thus the risk of falls.
It is now discovered that the Tyrannosaurs were a lot more lightly built than previously thought, increasing their speed and efficiency as a predator. Also, it is believed that their diet changed dramatically through their lifetime. At old age they may well have been scavengers or more like specialist kill-hijackers, like male lions.
Imagine how slow sauropods would have been due to their gigantic sizes. The Walking with Dinosaurs animators found that diplodoci would have to keep three feet on the ground at all times or they would trip over while designing early models.
Early paleontologists believed that large dinosaurs lived in swamps, because they approach being too large to move under their own power without the buoyancy of water. But Science Marches On, and discoveries of new fossils or rethinking existing ones led to new schools of thought about dinosaur lifestyles, muscular and skeletal makeup, and even posture. Note that no land-dwelling dinosaurs were ever as large as certain sea creatures, though (except the near-legendary Amphicoelias fragilimus).
Dwarf species of deer and antelope, such as the muntjac or duikers, tend to have legs so skinny it's hard to believe they can stand upright, let alone run. Even species weighing over a hundred pounds can leave footprints the size of a kitten's. Compare that to the soup-dish feet of a moose or giant sable antelope.
Ants and other insects are a classic example. While actually weak compared to say humans under a strict application of the Square Cube Law, they easily trump a linear interpretation of size to strength. Ants can lift/drag as much as 50 times their weight. Fleas can jump 200 times their height. A strict application of the law would have ants lifting thousands of times their weight. This difference is attributed to the fact that evolution forces larger things to have a higher percentage of muscle, and the different scaling rates of the anatomy such as the digestive system and neurology, etc. etc.
It's not even necessary to adjust for scale. In absolute numbers, a cricket can jump much higher than an elephant, because this law works so much in its favor.
9/11 Conspiracy Theorist Richard Gage once had a demonstration as to why the Twin Towers would not collapse. He took two boxes representing the "top block" of the Towers and the bottom block. He then dropped the upper block on the bottom, to simulate the building failing in the impact zone due to fire and impact, and the top portion falling on the lower, collapsing it. Since his model did not collapse, he asserts, it meant the Towers would not have collapsed. Some "debunkers" calculated that the model he used, being a few feet tall, would be something like thousands of times proportionately stronger than the 1000+ ft. Towers, especially once you take into account that the model was made of cardboard. Curiously enough, Gage and his organizations largely act like the video doesn't exist.
The Body Mass Index formula bases your healthy weight on the square of your height, thus royally screwing tall people. A mathematician noticed this and posted a letter pointing it out. It turns out that even the inventor of the Body Mass Index knew that his formula wasn't very accurate. Squaring works well for children, but cubing works better for babies, and an index of 2.5 for adults.
This is why, until a certain point in time, horses couldn't survive a broken leg — there was simply no casting material strong enough to carry around that much weight. This improved somewhat with technology, in that it raised the chances of survival from "somewhere below absolute zero" to "possible, if the horse is very very lucky and the circumstances are right".