Alien Geometries



""...but you start with parallel lines that intersect and you go from there.""

- Black Mage, Eight Bit Theater, #1004

A staple of Cosmic Horror Story and of Mind Screw artworks. Elder Gods, Old Ones, and other horrors tend to bend the laws of physics to suit them. Why make a triangle where the angles add up to 180 degrees, when you can make one where they add up to 200 degrees in a flat space and get some extra room? Even the very body of a particularly squamous thing may exhibit this, though more often it shows up in architecture as physically-impossible buildings—occasionally sentient themselves.

Alien Geometries are often depicted as being dangerous to the sanity of normal humans; where you have to read the Tome of Eldritch Lore for it to drive you crazy, just looking at this stuff can have an unpleasant effect on your mental stability. Or at least really hurt your eyes.

More innocuous forms may appear normal. Then you realize that it is physically impossible for something this size to fit in that, or you travel a short distance and find yourself kilometers away, or you turn left and end up to your right. Doubly fun if found in the Mobile Maze.

Eldritch Locations are a good place to find this. Sometimes it is a single wall or building that is just a little... off. See also Hyperspace Is a Scary Place, an entire alternate universe that just does not make sense. A Minus World in video games might be considered one due to unintentional programming bugs.

Compare with Bizarrchitecture and Sinister Geometry.

Anime and Manga

 * Casshern shows what appears to be a metal bolt of lightning -- or a metal construct -- striking from the sky and staying in place for several days, inciting a transfer of what we are led to believe is superdimensional energy into our dimension. This energy is visible in the form of sparkling mystical runes hovering in the air facing the observer. It's awesome.
 * The Angel Ramiel in Rebuild of Evangelion manages to pull this off with some aplomb; the very first time we see it shift shapes from its fairly mundane octahedron to... other things, we see that it is somehow impossibly deep and one piece all at the same time... and then it starts changing shape when firing beams of pure killing. The effect is enhanced by the fact that what it does is almost painfully easy to render in CGI, but to see a physical object actually do it would be skull-crackingly horrifying.
 * Leliel in the original Neon Genesis Evangelion appears out of nowhere over the city as a giant, floating sphere with black and white stripped patterns on it. But when the Evas fire at the Angel, the sphere fades out to dodge the shots, then casts a shadow which absorbs everything into it. It's then discovered that the sphere isn't its body: the angel is a 600-meter wide and 3 nanometer thick disc that is connected to a Dirac Sea. The floating object is actually a 3D shadow that appears when the Sea is opened. It gets even more mind numbing when you realize that the "shadow" is NOT intangible: it casts its own shadow and can physically interact with other objects (Unit 01 tearing its way out, in particular). It can even bleed.
 * The third season of Sailor Moon does this when one of that season's miniboss squad accidentally breaks reality, resulting in the entire house becoming a zone of warped space.
 * The Reverse World in Pokémon Giratina and The Sky Warrior is an Escher-like place where 'up' varies, but apparently only for landbound creatures (see the Pokémon Platinum note in Video Games below).

Comic Books
""Thou quantum imps and cherubs by whose dance
 * In the JLA storyline Rock of Ages the Joker nearly drives Superman and the Martian Manhunter mad by trapping them in a maze-like satellite, the structure of which is controlled by his subconscious mind.
 * In his Silver-Age Superman wrap-up, Whatever Happened to The Man of Tomorrow??, Alan Moore reveals Mr. Mxyzptlk's "true" form, described by Lois Lane as consisting of "height, length, breadth, and a couple of other things... looking at it made my head hurt." Moore likes having characters encounter and be upset by non-Euclidean phenomena; later in the same comic the room containing the Phantom Zone portal is described as eerie and unpleasant.
 * Alan Moore does this again in Tom Strong's Terrific Tales where Strong and Svetlana X find a Russian space station has become crystal-filled and Bigger On the Inside with multiple centers of gravity.
 * The crashed alien spaceship in Miracleman is probably one of the most distinct of Moore's uses of the trope, and is thus very difficult to describe. The people who board the ship all suffer from headaches and dizziness from the sheer disorientation that navigation of the craft causes.
 * In one strip of Calvin and Hobbes, the law of perspective is repealed, meaning that the sizes of objects no longer depend on how far away they are, making it impossible to tell where anything is. This is all happening in Calvin's imagination, of course.
 * In a another sequence, when Calvin was told to look at things from multiple perspectives he took literally and started seeing things as a Cubist painting, and another time when he used supposed lack of depth perception as an excuse for running into furniture.
 * This is generally how much of Galactus's technology is portrayed in Marvel comics. An alternate universe version of Reed Richards once spent decades figuring out the technology of a single room in the alien creature's massive home.
 * Galactus's aforementioned house, the Worldship Taa II also qualifies; it's a gigantic spaceship that dwarfs nearby planets without altering their gravitational fields.
 * Alpha Flight - Shaman's medicine pouch has its own laws of physics, able to hold far more than its volume, as well as responding to its owner's wishes as to what he wants. Anyone inexperienced in Shaman's magic style who looks into the pouch risks going catatonic.
 * Swamp Thing - The Demon Etrigan employed Alien Geometries in an incantation to create a path from Hell back to Earth during Alan Moore's run, when Swamp Thing rescued his beloved Abigail from Hell:

Is substance formed to shape the fields we know

Your perfect waltz that conjures form from chance

Must pause to free us from these wastes below.

By root of minus nine and circle squared

Set right and true against dimensions three

Let our ill-angled passage be prepared

Between the folds of rare geometry.""


 * During the Troll War sequence in Elf Quest Wendy Pini drew one panel with deliberately Escheresque geometry, showing a spiral staircase from the side with the characters at the top appearing much bigger than those at the bottom.
 * A scene from First Comics' Elric: The Sailor on the Seas of Fate depicts Elric in the captain's cabin, leaning down to look at a model ship. The viewpoint zooms in through the model ship's porthole, revealing Elric and the captain inside.
 * Donald Trump's hair, as portrayed in Doonesbury.

Fanfiction

 * Harry Potter and The Methods of Rationality, in line with the original Harry Potter series, has this: Hogwarts' corridors which can change when you aren't looking. The number of stairs you climb has only a passing correlation to your actual elevation when you look out the window. At least one corridor is tiled in pentagons.
 * Aeon Natum Engel and its rewrite Aeon Entelechy Evangelion are the crossover fanfics consisting of Neon Genesis Evangelion, Cthulhu Tech and First Encounter Assault Recon, with the latter adding elements of Eclipse Phase and RevelationSpace'' to the mix, so they have this in form of Heralds/Harbingers who replace the Angels. Mot, the equivalent of Ramiel, cranks this up to eleven.
 * In Fuck the Jesus Beam, there is a city that literally does not exist, as it is only a lie. Despite this, it is also a physical location. Given the name Αδιβ, when someone who can see in only three dimension looks at it, it appears normal, but in progressively higher dimensions, the architecture becomes more and more bizarre.
 * In the Pokémon fanfic Ashs Return, the doors in Glitch City manifest this way to anyone trapped inside.

Films -- Live-Action

 * An indie black-and-white short film of The Call of Cthulhu by The H.P. Lovecraft Historical Society does a particularly good job of getting this idea across, in a scene (faithfully adapted from HP Lovecraft's story) wherein a victim falls into a crevice which an optical illusion has led the audience to believe is a convex crag of rock.
 * Cube 2: Hypercube is a rare example of this trope being employed in a visual medium. The actual warped geometry shows up only a few times, due to the special effects required being rather expensive; the rest of the time it's showcased indirectly (e.g. duplicates of characters showing up).
 * The climactic scene of Labyrinth takes place in an Escher-esque landscape where 'up' varies. The scenery was based on a drawing by M.C. Escher.
 * The Red Matter-generated black holes in the new Star Trek movie. From the front, they look like your average swirling, funnel-shaped Negative Space Wedgie. Approach from the side, and you can see that it's missing its third dimension.
 * Grave Encounters plays with this quite a bit a short distance into the movie.
 * The tesseract-thingies during the "beyond the infinite" sequence from Two Thousand and One: A Space Odessey. Just look at them, and the way they move.
 * Even more impressive when you consider that they were created in the pre-digital era using 28 precisely-timed exposures.
 * The Tanz Akademie from Suspiria, an art deco nightmare from hell.
 * Dream levels in Inception are built like this deliberately by the level's "architects," in order to trap and delay the subconscious projections of the dreamers to keep them from attacking. The strange architecture can even be weaponized, as demonstrated by Arthur at one point, where he flees from a projection shooting down at him from the top of a staircase...only to have Arthur alter the stairs into a Penrose Staircase (looping stairs) and attack the projection from behind.
 * Last Year At Marienbad was shot within several castles and edited together to create a lack of continuity in the castle and a strange, disorienting effect.

Gamebooks

 * If you try to map the Citadel in the Fighting Fantasy gamebook The Citadel of Chaos, you'll quickly find its rooms are connected in contradictory ways and sometimes occupy spaces that should be beyond the outer walls. Maybe Jackson just wasn't too strict about the layout, but it is called the Citadel of Chaos...

Literature

 * The Monoliths in 2001: A Space Odyssey have the dimensions 1 by 4 by 9...
 * In the film, the proportions are a sexier 1:9:25 (1^2:3^2:5^2), or maybe 1:8:27 (1^3:2^3:3^3). In comparison, 1:4:9 looked like a brick.
 * The House of the Maker from The First Law trilogy by Joe Abercrombie. The protagonists enter about halfway up, walk around a bit inside but never ascend or descend, then exit on the roof. Most of the characters can't wait to get out of the place, even if it does involve crossing the narrow, rail-less, hundred-foot-high bridge. And there's always the possibility of leaving the place before entering it.
 * This happens a lot in Warhammer 40000 when the Powers of Chaos are involved.
 * In Graham McNeill's Warhammer 40000 Ultramarines novel Dead Sky Black Sun, the city in the Eye of Chaos features this — producing a Mobile Maze with it.
 * In Dan Abnett's Eisenhorn novel Xenos, the saruthi "tetrascapes" include regular octagons that nevertheless tessalate. Eisenhorn rescues some green soldiers from such a tetrascape, and later chooses them over experienced soldiers to go into one. Wise of him: the green soldiers had actually seen a tetrascape before, and the experienced ones hadn't. As a result, the "greens" manage to shoot and kill dozens of enemies, but the elite Deathwatch Space Marine attached to Eisenhower's squad can't hit anything thanks to the effect the twisted geometries have on ballistics.
 * In Dan Abnett's Gaunts Ghosts novel His Last Command, a Chaos warp gate throws Maggs and Mkoll into a place where stones hang in the sky and the stars are all wrong (both), as well being bitterly cold. Also, their vox units register as both within ten kilometers and out of range.
 * In Ben Counter's Horus Heresy novel Galaxy In Flames, Death's Tomb is bigger on the inside than the outside — as well as other repulsive features.
 * Alice in Wonderland - Alice, in Lewis Carroll's Through the Looking Glass set to walk to a hill and always found herself walking into the doorway of the house. Finally, when she walked away from it, she reached it.
 * Lampshaded by the Red Queen, when Alice found herself unable to run quickly. "Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!" The Queen was able to travel much faster, since she was a Queen, and can cross the width and breadth of the chessboard in a single move.
 * House of Leaves starts with a house that is 3/8ths inch Bigger On the Inside than on the outside. They are only able to measure all the way across because . This discrepancy disappearing is, believe it or not, the cue for things to get worse.
 * Thursday Next's Uncle Mycroft, among his other Mad Science projects, developed "Nextian Geometry" with his wife, said to be based on how a cylinder looks like a rectangle from the side, which allows one to use a circular cutter on dough without any left over: it makes circles tesselate.
 * Robert A. Heinlein's short story And He Built a Crooked House involves an architect who, inspired by higher-dimensional geometry and high real-estate prices, builds a house in the shape of an unfolded hypercube. Then an earthquake makes it fold in on itself into a hypercube, so to the architect's delight it's eight times roomier on the inside than on the outside. Just one small problem: the house's new topology makes it a bit difficult to leave once you're inside. When you do get outside, you may have a whole new problem.

A math-nerd resident of Second Life actually went and reproduced the Crooked House in 3d, and if it's still rezed somewhere public you can actually walk through it. Not a real hypercube of course but some excellent special effects. Here's the story with video.
 * Heinlein's Glory Road had the hero and companions invading a tower "where the architect used a pretzel for a straight-edge."
 * It's so convoluted that it took hundreds of spies decades to figure out a route to the MacGuffin.
 * The premise of The Number of the Beast, as well: we're only seeing dimensions x, y, z; but there are at least three others which can be rotated around or extended along, and which apparently can be used to travel between universes. This is a conceit to let him run through every literary universe ever, and have a massive Crossover event. The novel culminates in a party, in what is effectively the Crooked House, with every single character he created attending (plus several guests). Special mention goes to the literary critics lounge, which was shaped like a Klein bottle... once you were inside.
 * Deep Secret - the Hotel Babylon has halls where you can go around more than four right angles before coming back where you started thanks to the building being on top of a bunch of ley lines.
 * The Wheel of Time - The Aelfinn and Eelfinn ("the Finn") inhabit one or more separate dimensions described by the author as having radically different natural laws. Successive windows do not show what one might expect. That the magic system in the series is heavily geometric likely has a great deal to do with why its use is explicitly forbidden there. The doorways into their realm also resemble this in the "real world", and are described as "twisted".

Though it's less apparent, the same is true of the Ways, an artificially-constructed dimension meant for quick travel. Except in one dream sequence (which, for complicated reasons, probably reflects the reality of the Ways), the realm is extremely dark, but travelers there have noted that by the arc of the bridges they're walking on, the platform they've just arrived at should be directly beneath the last. During the dream sequence, it becomes apparent that the platform-islands extend infinitely downward—and unless you follow the bridges with your eyes, appear to be on the same plane. The doorways seem to be a description of a three-dimensional Möbius strip.
 * A rare Star Wars example can be found in Star Wars Expanded Universe novel Lando Calrissian and the Mindharp of Sharu. In it the titular character finds the titular artifact - which seems to exist in several dimensions simultaneously and as such it really hurts just to look at it. Lando then uses it to unlock the passage into the Great Pyramid of Sharu - where he is expanded in size several dozen times, while his droid companion is shrunk to the size of a louse. There are even more examples of Alien Geometries in the book: the aliens who build it were very, very alien indeed by Star Wars standards.
 * Night Shift - Inverted in Stephen King's short story I Am the Doorway. An alien lifeform sees a boy walking with a sieve under his arm: "an abominated creature that moved and respired and carried a device of wood and wire under its arm, a device constructed of geometrically impossible right angles."
 * In Stephen King's short story 1408, the titular room's door is crooked to both the left and the right. Or not at all. Maybe it can move? And it gets worse from there.
 * In Stephen King's novel From a Buick 8, the titular car is actually an interdimensional portal/device that only looks like a car. It's noted that the human eye perceives it as a car because that's the only image the mind can supply for the actual shape of the device.
 * A significant plot device in Madeleine L Engle's A Wrinkle in Time deals with folding space-time through a fourth space-dimension for teleportation.
 * C.S. Lewis used something similar in That Hideous Strength. One character is briefly imprisoned in the "Objectivity Room", where everything is slightly off—the spots on the table are arranged just short of obeying a pattern (even a broken one), the similar specks on the ceiling are almost the mirror-image of the table, and the peak of the arched entryway looks like it might be just a fraction off-center to the left. Or not. Maybe the right? And let's not start on the paintings...
 * He also appeared to use this in Out of the Silent Planet, but the room turns out be normal human geometry, just a very unusual sort.
 * HP Lovecraft loved non-Euclidean geometry:
 * The sunken city of R'lyeh
 * The Antarctic city in At The Mountains Of Madness.
 * Perhaps most explicit in The Dreams in the Witch House where a mathematics student discovers the unearthly topology of his own bedroom serves as an extra-dimensional portal. Well, he was renting it because of its reputation as being haunted. This was a bad idea.
 * In The Call of Cthulhu, Parker was swallowed up by an angle of masonry which was acute, but behaved as if it were obtuse.
 * Before that there is the famous scene where the gate that seals behind the Great Old One himself opens, and the sailors can't be even sure whether it's a vertical door or a horizontal hatch - even though one of them climbed or walked up its surface!
 * The Hounds of Tindalos, by Lovecraft's friend Frank Belknap Long, features ravenous creatures of weird geometry who travel trough time and space, and the only way to avoid them once they're on your trail is to completely avoid sharp angles (such as in a completely circular room).
 * Eon, by Greg Bear, features an asteroid hollowed out by people from ..elsewhere, with seven chambers running along its internal axis. The first six contain cities, parks, a spaceport and loading area, and power generators. The seventh chamber And then things start to get weird.
 * Discworld
 * Bloody Stupid Johnson, architect, Bungling Inventor, and general anti-genius regularly does this kind of stuff entirely by accident.

He once designed a letter-sorting machine whose central component was a wheel that had pi equal to exactly three. This causes it to sort out letters it hasn't had put in yet, among other oddities.

Empirical Crescent, a row of terraced houses where every door and window leads somewhere other than where you'd expect it to lead. At least it makes it easier to get rid of rubbish—just toss it into the garden. After all, it might not be your garden.

The reason for this corruption of dimensions occurs because the row of houses is crescent shaped on the outside only. Inside, it's supposedly laid out like a straight row. Presumably the two configurations conflict. Occupants had a tendency to leave in the middle of the night, often without stopping to pack...
 * The Colour of Magic featured a parody of Alien Geometries: the Temple of Bel-Shamharoth (itself a spoof on Cosmic Horror Stories). The most striking feature of the Temple is that its walls, ceilings and floors are composed entirely of interlocking regular 8-sided tiles. Whilst is it possible to create octagonal tiles that be fitted in a regular, interlocking way, they must be concave polygons: there is a mathematical proof that no convex polygon of seven or more faces can tessellate on a Euclidean plane.
 * The first chapter mentions one of the gods using a 7-sided (but still cube-shaped) die to cheat.
 * The buildings of the Unseen University, which have been rather strongly influenced by the vast amount of magic that has flowed through its halls over the centuries, have floors and rooms where logic says they simply could not exist. Magic is as much a part of the architecture as cement.
 * The Library of the Unseen University is a particularly strong example—the presence of so many ancient magical texts distorts space-time like an elephant on a trampoline, dimensions and gravity being twisted into the kind of topographical spaghetti that would cause even MC Escher to go for a good lie down (or sideways). That's quite apart from the fact that it serves as a gateway into L-Space, and is therefore linked to all libraries everywhere in all points of space, time and reality. Technically, it contains every book that has ever been written, every book that is ever going to be written, and every book that ever could have been written (whether it actually was written or not).

Once, the Librarian took a trip deep into the shelves, passed tribes of lost students, and ended up in the same library in the past.

In addition, Pratchett explains that any sufficiently large collection of books (magical or otherwise) can exert the same effect as the Unseen University Library; the equation goes "Knowledge = Power = Energy = Matter = Mass". And since mass warps space around it, so does a high quantity of knowledge. "After the man had fallen through every place and every time that ever he had even imagined, he began to fall through the places that his mind could not conceive. He passed into structures that did not follow geometry, saw shapes that had no edges or sides, that extended into themselves and into all directions. He saw triangles with one hundred eighty one degrees. He saw minds that had no reason or morality. He saw colors indescribable to others. He saw the true shapes of his dreams, and the ten dimensions of the earth and sky. He saw what no one saw, felt what no one felt. He heard sounds with his finger tips, and tasted with his ears. He had secrets whispered to him in a language that can't be translated."
 * In reality, however, power is equal to dW/dt, or energy acting as work divided by time, and energy is equal to m*c^2, or mass multiplied by the speed of light in a vacuum squared. However, the lack of correct physical defintions is overlooked due to the Rule of Funny.
 * Death's house is bigger on the inside than on the outside, being the size of a cottage on the outside, but the size of a small castle on the inside. This is not so much intentional, but is rather the result of a slight blindness to ordinary architecture on Death's part. Many of the rooms have the peculiar effect of being enormous at the same time as being regularly sized. Death's room in particular is stated to be about a mile wide, but most can be crossed in only a few steps.
 * The real killer is that Death himself is weirded out by this last fact. This is because it's ordinary humans (like Albert) who cross the room in a few steps, even though it's clearly a mile wide. Death's theory is that the human mind refuses to accept the true size of the room, and acts as if it were normal sized.
 * The Tooth Fairy's house in Hogfather is another example of this trope.
 * The Gnarly Ground in Lancre is a seriously bizarre landscape of crags and valleys "scrunched up" into a small area, overlapping in space; what geographical features you see there and have to deal with depend largely on your mindset. It makes a good hiding place.
 * Unseen University's mad but Good With Numbers Bursar has posited that there is an extra number between three and four, which he calls 'umpt' (as in, 'umpteen' minus ten).
 * In John C Wright's Chronicles of Chaos, Vanity's secret passages often don't add up, geometrically, with the places they go to and lead from.
 * In Time's Eye, by Arthur C. Clarke (Creator), there are spherical alien objects that apparently have a 1-to-3 ratio for their diameters and circumferences, instead of a one-to-pi ratio.
 * The Starfish structures in Blind Lake have disturbing interior geometry. Robot probes (and people) who go in too far don't come back. The deep interior seems to be entirely exempt from the usual rules of time and space.
 * In the eighth book of the Everworld series, the main characters are cast into an inverted realm where the ground they stand on is above their heads, and gravity pulls them up, with the colors of everything reversed for good measure. This naturally strongly bothers David, April, Jalil, and Christopher. Senna, however, likes it, and compares the reversed plane to fine art.
 * In Stephen Baxter's short story collection Vacuum Diagrams, the story The Eighth Room deals with something similar to Heinlein's story. However, in this case, the room was not created accidentally... it's more of a logic puzzle. I'll leave it there.
 * There's also another short story by Baxter called "Shell", set on a planet that is folded in on itself. There is no sky — people looking up see the other side of the planet curving over them, as if it's a shell. When one character uses a hot-air balloon to explore the other side, she witnesses the "shell" flatten out and then become curved normally, while the land she just left curves into a shell over the sky.
 * In Ambrose Bierce's The Damned Thing, a creature, judged by ignorant folk to be a mountain lion, is a color that the human eye cannot see and makes noises that the human ear cannot hear. This inverts this already inverted trope because the color is natural and it is humanity that has become too alien (or at least insensitive) to comprehend it.
 * In the Deathstalker series the AIs of Shub constructed a world of their own to live on. Unfortunately for humans who might visit, it exists in more dimensions than they can perceive and so is unhealthy to look at for extended periods of time. The Madness Maze, despite a relatively innocuous appearance, had convoluted, nigh-sentient path designs that would either evolve you into a higher being or tear you apart.
 * In Flatland, the two-dimensional protagonist A. Square struggles to fathom the third dimension when he is introduced to it by a travelling sphere, and it almost drives him insane. In a dream he sees that inhabitants of one-dimensional Lineland are similarly incapable of comprehending the second dimension. And let's not even get into Pointland's issues.
 * Many sequels have been written. Flatterland has even more bizarre geometry, including a hyperbolic world, a fractal world, a grid world, and so on.
 * Threshold by Caitlin Kieran contains a fossil in a shape that cannot exist, causing the heroine to black out when she looks at it too long. What is this sinister shape? A regular heptagon.
 * In the Star Trek Voyager novel The Final Fury, Captain Janeway, Tuvok, and Neelix arrive aboard a Fury planet wherein the hallways and doors meet at angles that aren't quite "right" -- literally and figuratively -- and the aliens themselves despise those who follow the "right-angle" or "right-hand path."
 * A feature of the bizarre planet in The Inverted World. Within about a dozen miles of the "optimum", everything is pretty much Earth-like. Go any farther than that, however, and things start to distort unpleasantly. Because the optimum is constantly moving, the entire City has to move after it to avoid destruction.
 * Even weirder,
 * The protagonist of Return From the Stars comes back to Earth after over a hundred years of absence. In the meanwhile, architecture has changed/evolved so much and so confusingly that when he first steps out into a spaceship depot, everything around looks to him like an abstract, shapeless muddle of pathways.
 * In the Star Trek Shatnerverse novel The Return, The Borg have built a hypercube base inside a subspace tunnel.
 * The Last Battle, the final volune of The Chronicles of Narnia, ends in a world that's essentially the opposite of reality, in that the closer you get to the center, the more there is.
 * In An Elegy for The Still Living, the main character spends a few minutes walking down a forest path before realising he is seeing the same scenerey looping over and over again. Also, this passage:


 * In George R. R. Martin's A Clash of Kings, Danaerys enters the House of the Undying Ones in Qarth. Once in the antechamber, she makes four consecutive right-hand turns without returning to her starting point.
 * In Larry Niven's Protector the Brennan Monster amuses himself by creating full scale replicas of some of Escher's art, using things like artificial gravity to make them work.

Live Action TV

 * The Doctor Who Expanded Universe speculates that this is the default setting for the interior of a TARDIS, and that the Doctor's TARDIS projects a more easily comprehended interior so as not to freak out the Doctor's human companions. She is just a sweet old thing.

In "The Lodger" the Doctor uncovers an alien time-distortion device similar to the TARDIS in the upstairs flat of a British apartment building. Amy, poring over the building plans for the address, discovered that the building didn't even have an upstairs, it was a one-story building. Perception filters kept people from noticing anything out of the ordinary.
 * The spacecraft used by the Goa'uld are relatively normal... until you notice the pyramid on top. Naturally, the entire spaceships can fold up so that their central pyramid can land on a planet-bound pyramid.
 * Not to mention how a triangular-pyramid shaped spacecraft can land on a square-pyramid.
 * The plot of the (admirably silly) Star Trek Voyager episode "Twisted."
 * Neverwhere does a very nice demonstration of this in passing. The protagonist is led down into the London Underground, then through a door, and down a stair case. This continues, always going down, until they reach a small door and step out on to the roof of a building.
 * The Hitch Hikers Guide to The Galaxy series has a few examples:
 * The interior of Slartibartfast's Bistromathics ship in Life, the Universe and Everything.
 * The new and improved Guide in Mostly Harmless shifts through a few inexplicable forms.
 * Threshold involved an alien invasion. The aliens used devices that apparently contained more that four dimensions, and cannot be fully perceived visually. Just seeing or hearing the signals originating from these 'beacons' can kill or transform the view into an alien agent, with triple DNA helix where earthlife has only contains double. The aliens themselves are usually seen in dreams; crystal forests where spider-like entities are only partially seen.
 * In the Twilight Zone episode, "Little Girl Lost", a little girl falls through a portal in the wall of her bedroom into an alternate dimension, in which space is twisted, distorted and nonsensical to ordinary human perception. Fortunately, the family dog's superior hearing and sense of smell help get the little girl back into our dimension.
 * In the Night Garden is a BBC kids' show (from the people who made Teletubbies) where the various characters often ride around the eponymous garden in the Ninky-Nonk (a train without tracks) or the Pinky-Ponk (an airship). When they're boarding, these vehicles are comfortably large enough to accomodate all of them, yet when they're actually in motion the Ninky-Nonk is small enough to run up trees and over branches, and the Pinky-Ponk is small enough to get knocked off course by a toy ball.

Paintings

 * Paintings by H. R. Giger, famous for his design of the Xenomorph in Alien, though his work tends more to towards the horror aspect than the impossible. He also likes to paint landscapes having sex with themselves. Think about that for "scenery porn."
 * MC Escher could be could be considered to make "lite" versions of Alien Geometries; notwithstanding that, his style is often used to represent them. Some of his works are geometrically accurate representations of the sorts of triangle-mangling spaces described in the intro (hyperbolic planes in the Circle Limit sketches for example). And yes, his work does have an impact on one's sanity...

Tabletop RPG

 * In Mortasheen, this is where the Xenogog lives naturally, only coming into ours with a screw up in a time travel experiment.
 * The near-universal hallmark of things made in the name of Chaos in Warhammer 40000. For example the Dark Eldar capital Commoragh has spatial anomalies, "wandering shadows that tear apart the unwary" and many other dire things. It lies deep within a nest of extra-dimensional tunnels.
 * Dungeons and Dragons
 * The Classic Dungeons & Dragons system delved deep into this trope with its boxed set for PC Immortals, redefining game-reality in terms of five spatial dimensions. Mortal creatures exist in three, Immortals in four, and Old Ones in five. Which three a mortal creature occupies can vary: Nightmare-reality creatures share only one spatial dimension with Normal-reality beings such as humans, and "nippers" from the Astral Plane overlap with dimensions of both Nightmare and Normal reality. As for how all this applies to the geometry of the planes themselves, thinking about it could make Your Head Asplode.
 * The Planes in D&D 3rd edition got like this at times, such as an infinite plane having an edge. Seasoned Planeswalkers tended to give the advice "Try not to think about it."
 * In the Queen of the Demonweb Pits module, the players ventured into The Abyss to confront Lolth, the demon queen of the spiders. Lolth's domain consisted mainly of long, open passageways hanging in space. Even though these passages pass over and below each other, they never ascend or descend in any way.
 * Less heady are the Githzerai monasteries on Limbo, which take advantage of the fact that "down" is whichever direction you want it to be, giving us some extremely Escher-esque architecture.
 * Possible example: In Dungeons and Dragons fourth edition, as a bookkeeping shortcut, space in combat is explicitly non-euclidean: you can move diagonally at the same rate you can move straight, and if you have a sphere, the shape is explicitly a cube; and yet straight lines remain straight, and a rectangular room still has only 90 degree angles. The result, when you map it out onto normal space, is frankly bizarre.

For those of you asking, "How does a world where spheres are cubes make life easier?": the world in question appears to be made largely out of 5' squares, so having a setup where you don't need to wave around Warhammer templates every time someone casts Scorching burst speeds things up quite a bit (of course, hexagons would have been a closer approximation). Height is implicitly the same as the other two dimensions at some points, and implicitly non-existent in other parts of the rulebook.
 * GURPS: Illuminati University describes a campus which teaches human students and everything else capable of paying the exorbitant university fees how to function as Mad Scientists, World-Conquering Dictators, Marketing Specialists, and other strange jobs.

The campus is a stereotypical university: the campus has an open area or "Quad" in which students and staff may pause for reflection, study, impromptu lectures and other activities from which adventures may spring. Illuminated University has The Pent, which has five sides for no particular reason; students who happen to have a protractor handy will discover that all five of the corners have 90-degree angles. One of the dorms is stated as having rather similar angles.
 * Mage: The Awakening has the Twisting Maze Zone, a localised distortion of reality caused by Abyssal forces. While it looks chaotic, constantly shifting jumble, this is actually because its directions extend into the fourth one as well. Unlike many examples, mages can use this to their advantage, using their will to walk through hidden parts of it to teleport around-in fact, they must, as the way to banish it is to walk through the areas of the Zone as they normally are-i.e., sans Twisting Maze-thus forcing them to apply to Earth laws.

Once that is done, the Zone literally Logic Bombs itself out of existence, causing anybody nearby to gain a brief glimpse into the space-time continuum. Should someone have the force of will to process it, they have an epiphany about how the world works, resulting in an Experience Point gain. If no one does anything about it though, the Zone grows so bad that it ends up rewriting history so that it-and the area it effects-ceases to exist.
 * And, as you'd expect, Call of Cthulhu and Cthulhu Tech occasionally include this for...well, we all know why.
 * The Terminal in Over the Edge. It's Al Amarja's massive airport, nine-storeys high and built like a maze. Navigating it is so difficult, people need to hire guides. Of course, the best part is when you leave the airport and see that it's built like a step-pyramid. An upside-down step-pyramid.
 * The dimensions of Primordial world bodies are often based on their moods and personalities. For the more focused and stable ones, the worlds are typically consistent and predictable. For others, you get things like spatial relationships that are constantly rearranged, being able to pass from one side of a layer to another with no obvious transition, and having a sun that is inexplicably always right above you while also being at the center of a spherical arrangement.
 * As you might imagine, this makes travel around Malfeas... interesting. Once you enter the dimension proper, you must cross Cecelyne, the Endless Desert, for five days to actually get to the Demon City. No, it doesn't matter if you're walking on foot, riding on horseback, and piloting a First Age airship. The trip always takes five days. Then you get to the Demon City, which is layers upon layers stacked on top of one another - but each layer has Ligier, the Green Sun, shining above it, no matter how deep down it is.
 * Werewolf: The Apocalypse has the Black Spiral Umbral Realm. Being a Spirit World the Umbra is pretty odd at the best of times, but normally follows at least the guidelines of the laws of physics, if only because vistors expect it to. But the Black Spiral... From the outside it looks like just a spiral pattern on the floor in black. Once you start walking it it seems longer, twistier and with entirely too many dimensions. In fact it's a path into the mind of the Wyrm. No-one's ever come out the other side sane.

Toys

 * The Transformers already skirt the trope, what with size and mass-changing and the oddness of the scales...but then we come to the Autobot Micromaster Countdown's playset. He's a deep space explorer. He has an interstellar rocket and a command base. The base is used to launch the rocket. But also fits inside the rocket: mgnaaaaa!

Video Games

 * In Temple Run, the temple was surely designed by an Eldritch Abomination. Or by MC Escher. Or by a terrain randomizer that doesn't keep track of where you have been, so that it may happily let you take seven quick 90 degree turns to the right in a row and come to a new location each time.
 * Unlike binary space partitioning-based 3D engines, portal-based 3D engines organize spaces by where they join together rather than where they are located in space. This means that games like Marathon allow multiple entities to occupy the same location without touching under certain conditions, such as a Klein Bottle-shaped level. While the Marathon series unfortunately doesn't employ it in the actual campaign outside a few Easter eggs, the multiplayer level 5-D Space provides an example of the possibilities.
 * Several third party Game Mods do this, for example one level of "Keep the Home Fires Burning" has a 720-degree circular hallway with two overlapping hallways going down the middle. And "Schmackle" in Marathon EVIL has a part where you go through a portal into an alternate version of the level occupying the same space. Sort of like the "Tier Drops" example below.
 * Duke Nukem 3D has a similar engine, and its quirks are used to full effect in some of the secret levels.
 * The level "Lunatic Fringe" is a 720-degree circular hallway around a central hub, so you have to walk around the hub twice before actually returning to where you started.
 * The level "Tier Drops" has four overlapping areas connected by a hallway around them and drop tubes inside. The guys at 3D Realms beat the level in just ten seconds.
 * A few of the game's levels actually use these quirks transparently and a number of user-made levels deliberately work to show them off or to fake architecture that's not truly possible with the game engine.
 * Descent can use similar techniques with user-made levels. One example was appropriately titled "4D".
 * A level based on these concepts exists for Americas Army. A video of it can be found here
 * The Source engine as of 2011 has a feature that allows for something like this; they mainly just used it for testing unfinished level designs. It can produce some crazy stuff if you know how to use it right. Unfortunately, the version of the engine it's in can really only make Portal 2 add-on content for now.
 * The Unreal engine is also capable of things like torus-shaped levels and endless corridors with creative application of warpzones, right from the earliest version of the engine. It's far from perfect, though: non-projectile or "hitscan" weapons can't shoot through, stacking more than four warpzones results in the engine glitching and drawing the portal surface's texture, warpzones must have the exact same dimensions at both ends or the game will crash, etc.. The level DM-Fractal even has a relatively simple "anyone falling into the floor trap falls out of a hole in the ceiling" trick.
 * A little-known 2.5D sci-fi (considering it was made in Russia, nostalgical sci-fi) first person shooter Madspace embodies this trope, complete with this feature actually being mentioned in the manual. If you decided to dig the game out, there's one thing you should consider... never ever use the ghost cheatcode.
 * Games like Asteroids, 'and the like, use an unwrapped toroidal universe—the environments have the same geometry as the surface of a donut. Super Mario War does the same thing by default, but because all the levels are custom-designed, you can modify it to take place in an enclosed space, or add kill zones at the sides of the game window Super Smash Bros-style.
 * VVVVVV takes place in a Wrap Around universe. However, the Tower level is taller than the rest of the universe, but is hard to notice since said level is on auto-scroll.
 * Portal is based all around this idea. If you shoot a portal into a wall, and then turn around and shoot another connected one into the other wall behind you, you can see your own back. And then, if you wish, run in a straight line forever without ever leaving the space you're in. That's just for starters - try, if you place a portal well, being able to see an infinite series of your own back. It's also fun to place a pair of portals in the ceiling and floor and then fall through endlessly (and there's even an achievement related to that in the first game).
 * Which is at least easy to wrap one's head around, it seems. For the truly mind-bending type, see the above example about the Source engine.
 * Pacman is a cylinder, since you can only go between left and right.
 * The eponymous Temple of Doom in La-Mulana is non-simply connected, almost certainly deliberately. If you try to make a map that shows where all the areas are in relation to each other, taking every connection into account, you'll quickly discover that it can't be done. In particular, it's not at all clear what the lowest point inside the ruins is.

It makes more sense if you ignore the backside areas, and view the temple as being more of a cylinder than a flat map. Still holds with the temple as a whole, though, as the backside areas explicitly don't make sense if mapped on a single plane. The most obvious example would be how the Shrine of the Mother is both above and below the Endless Corridor, depending on where you enter it.
 * The Polyhedron in the Russian art-house game Pathologic. Hooooo boy. From the outside, the building simply appears to be impossible.  Yes, the game is a total Mind Screw.
 * In Diablo II, the Arcane Sanctuary area contains some quite Escher-esque geometry: platforms are supported by pillars that stand on other platforms which ought to be at the same height. The game gives the option of displaying in perspective (parallel lines converge at the horizon) or isometric (parallel lines remain parallel). In Arcane Sanctuary, the perspective option is disabled, due to it being impossible to draw.
 * The Canyon of the Magi doesn't let you use perspective either for no obvious reason. Maybe an anomaly left behind from the magical battle that happened here? Or maybe I'm overthinking this and it's just a bug.
 * Giygas' final form in Earthbound is fought inside what is best described as a "pocket dimension"... except Giygas' final form is also the pocket dimension itself. Rather fitting of Giygas, the Trope Namer for You Cannot Grasp the True Form.
 * Echochrome is a puzzle game based on the works of M. C. Escher. The geometries are as weird as you might expect.
 * To elaborate, in the game, you are allowed to "cheat" the laws of perspective because only the camera angle's perspective counts as "real". If there is a beam covering up a hole, the hole then ceases to exist. This is a necessary skill to guide the main character to safety.
 * Pokémon Platinum: The Distortion World in Amazing Technicolor Battlefield form. Made into Mind Screw material thanks to the flat, unchanging background theme and the fact that there are no Pokémon in it at all except for Giratina. Not to mention the fact that holy crap the sky is upside-down.
 * There's also the Lost Cave in Fire Red and Leaf Green, in which it is possible to walk through a door, turn around, and find yourself in a completely different room than you started in.
 * The Psychic move Trick Room...well, look at the description. "The user creates a bizarre area in which slower Pokémon get to move first for five turns." The user warps space so that going slower makes them move faster. That's pretty alien.
 * That sounds like it'd be fun to experience, but would get annoying after a little bit. If you try to run or make any sudden movement, it feels like you're running through shoulder-deep water. If you move slowly, you will jerk forward somewhat erratically.
 * According to the characters, the Primeval Thaig in Dragon Age II, although we have to take their word for it, cuz that's kind of hard to program.
 * Certain areas in the Silent Hill series, usually paired with Chaos Architecture; examples include but are not limited to: the girls' bathroom in the alternate school which leads you to the second floor when you exit it, the door between the first and second floors in Nowhere, and the convoluted space-time of the alternate Lakeview Hotel in Silent Hill 2, where the doors now teleport you around the building, and you have to find the correct one that will warp you to the otherwise inaccessible east wing. And going back in the same door leads to a different door than the one you entered. Not to mention the Historical Society, where you jump down several extremely deep holes, then take an elevator even further down, but when you come out of it on the lakefront, you're only about 20 feet below where you started. It also has Escher-esque architecture at points, e.g. the room with the hole leading to the prison (doors on the floor and walls), and the rotating room in the Labyrinth.
 * The Daedric ruins in Morrowind fit this trope. They certainly weren't built by man or mer...
 * It's possible to make these in Dwarf Fortress if you're careless, insane or second-hand enough. Case in point: Headshoots had rooms that the players couldn't find without zooming on a dwarf that happened to be in them.
 * Of course, rather disappointingly, there's nothing physically impossible about it—it's just a web of extremely convoluted tunnels.
 * In Guild Wars, there is no in-game map showing the entirety of the Realm of Torment. There is a very good reason for this- direction and distance are coherent within regions of the Torment, but not between them.
 * This is the whole point of the city of Sigil in Planescape Torment: a city existing on the inside of a giant torus rotating around an endless spire. The cityscape constantly changes and every archway can potentially lead you virtually anywhere (including somewhere you really don't want to go). Why, at one moment of the game
 * Not so much a geometrical issue, don't think it fits here. Sigil is alive. The streets move because they'd rather be somewhere else. Or they're making space for newborn streets. Also, the city is a Portal Nexus. If you enter a house and find yourself in a larger-than-seems-likely space, you might not technically be in Sigil anymore, whether you just entered heaven or hell is pretty much a matter of luck.
 * Sigil also happens to be at the top of an infinitely-tall spire; something that certainly can't be done in regular geometry.
 * Said spire is also located directly in the center of a plane that stretches infinitly into all directions.
 * And manages to be infinitely tall while having both a bottom and a top, both of which can be visited.
 * However, one character talks about being transported by a hidden portal that consists of the archway that appears when approaching two trees from the right angle.
 * X: Beyond the Frontier and its sequels plays it straight with "Spacial Compression" improving your cargo capacity.
 * A wonderful example in text adventure Trinity, which contains a Klein Bottle that you can walk through. After you do, east and west are reversed everywhere else in the game. This is useful for.
 * Visions and Voices has the mirror worlds. While they aren't that extreme, they can be pretty freaky—numerous characters are evidently freaked out upon seeing them.
 * The Milkman Conspiracy level in Psychonauts.
 * Prey is based aboard a cybernetic moon size space ship where things like gravity and even space-time are not consistent. The player character occasionally remarks on this.
 * A particular player-made map for Far Cry 2 is shaped like a cube with two sides removed and tilted on its axis. Due to the inclines that a player is able to move on without sliding off or falling, the players can run on all four of the inner faces, even though they appear to be perpendicular without close examination. This leads to strange cases of a player standing on the wall of a building and firing at someone on the street ahead of them, which is going into the sky.
 * The Legend of Zelda series pulls this off a lot of times in different ways. The most common is the classic Lost Woods. Take a wrong turn and you magically end up back at the start, even when it should be normally impossible. This happens in some other areas like Ganon's Tower in Wind Waker.
 * The Fairies' Woods in Oracle of Ages, in which moving from one screen to the next and back again winds up placing you in a completely different spot than before.
 * In the Forest Temple in Ocarina of Time there is a corridor which twists 90 degrees to the side, meaning you end up walking on what was originally a wall. One puzzle in the temple involves activating a switch that twists and untwists the corridor, so you can access different sides of the room. And you're still somehow oriented the same way relative to the rest of the dungeon.
 * Majoras Mask - The Stone Tower in involves reversing gravity so you can run around in the ceilings of rooms. This is made even more mind-twisting by the fact that the horizontal orientation of the temple is preserved after it is flipped over -- i.e. a room on the right of the entrance normally would stay on the right of the entrance when flipped -- meaning the dungeon somehow inverted itself. Even the Perfect Guide writers were confused by the whole thing.
 * Super Mario Bros - Warp Pipes ignore any physics beyond Rule of Fun, but the ones in Bowser's Inside Story really take the cake. There are multiple, microscopic pipes inside of Bowser that, without actually going through Bowser at any point, lead outside of Bowser, simultaneously increasing the size of those who go through them to macroscopic. Though it's impossible in-game, there is nothing in theory to prevent Bowser from entering one of these pipes. What would happen if Bowser did use such a pipe? Decent people shouldn't think too much about that.
 * Well, remember: the reason they lead in and out of them is because they are "warp" pipes. They basically teleport the user to another pipe. There's always the possiblity that World 1-2 could be in another universe entirely...
 * Super Mario RPG seemed to parody video game wraparound: the overworld actually is donut-shaped, despite there being no real reason for it.
 * Super Mario 64 has the endless staircase. However high you climb, the bottom is only a few feet behind you when you turn around.
 * Super Paper Mario has a unique twist. Mario exists in a 2-D world (with Shout Outs galore to the first Super Mario Bros games), but the first ability Mario learns is to "flip" between dimensions. In other words, he gains access to the third dimension. Now, this isn't any problem for the player, but what's this like for Mario? ...let's just say he needs a Sanity Meter to stay in 3-D. However, the game is a sequel to the first two Paper Marios, which were in 3D. Mario also seems to handle switching between 2D and 3D just fine between games anyway.
 * There's a game in development, called Miegakure, in which the puzzle aspect involves a 4th spatial dimension. Just trying to visualize a textual description of the game mechanics is enough to cause a headache. A three-dimensional environment can be represented by multiple two-dimensional images. Imagine taking an object, and tracing its outline on a flat surface from each side. You can get a good idea of the actual shape of the object in three dimensions by putting those images together in your head. What Miegakure does is present a four-dimensional environment in a similar fashion, in a series of three-dimensional models. You can switch the "angle" from which you view the four-dimensional environment by hiding one dimension and causing another one to become visible, similar to how a flat picture of a three-dimensional object "hides" the depth dimension.
 * The Bizarre Room in the Wonderland level of Kingdom Hearts. Entering from different points of the world -- including the room itself -- leads to you stand on different dimensions of the room, i.e. the walls and roof. However, the dimensions of the areas you are entering from don't change at all.
 * Two areas in SubMachine: Subnet Exploration Project have rooms that connect in ways they should not. Appropriately, one of them houses a fan theory that the Sub Machine is looped through the fourth dimension, and the other is a series of padded cells. Another area is, for no apparent reason, sideways.
 * In The Dig, the architecture of the Cocytans shows a lot of reverence to the 5 Platonic solids, including shapes with strange symmetries; and there also the in-universe example of space-time six -- a 6 dimensional realm.
 * The Mobius Ring track in F Zero GX exhibits these; the sky is always up, as the mobius has only one side. Let's not talk about the architecture.
 * An interesting loading error in Indiana Jones and The Infernal Machine once caused a corridor to loop back around itself into the same room that the player had just left without any perceivable distortion. The player who encountered the bug thought he might be going crazy at first.
 * Minecraft on LSD, the combination of 2 mods. Youtube it. Everything is still the same but looks extremely... peculiar. A straight line looks like a coiled rope, and then you imagine that these are supposed to be blocks doing this, but curving. And then you see the distance going on the ceiling... Although it's all visual (but can often feel like you're walking on a circular world and not a flat one) for now.
 * Entering the main room of the Tremere chantry in Vampire: The Masquerade Bloodlines is easy: just walk in the front door, go straight, take a left, a right, then another left. But try to reverse those directions to leave, and you end up back at the same place you started. Any wrong turn on the way out sends you back to the main hall, and the path out is not the same as the path in.
 * Invoked in one of the epilogues in the game Primal Rage. When you play as Vertigo, the epilogue says she forced enslaved humans to build a palace whose alien gemoetries drove the human workers insane. There's a illustration with it.
 * The Game of the Ages introduces a fourth spatial dimension, which you learn to navigate.
 * Baldurs Gate: leaving aside all the bizarre dimensions and obscure dungeon twist you can encounter, Athkatla has a few doses of alien geography. Notably, no matter which way you approach the bridge from, you will always enter the district from the northern end. There are only two locations on the entire map located north of the Bridge District, and there are no other apparent ways across the river without hiring a boat.
 * There's an old russian mod of the 1990 Prince of Persia game, called 4D Prince of Persia. It takes advantage of the way room connections are programmed and creates levels where normal directions don't apply: Levels that loop and wrap around, corridors where running back doesn't take you where you came from, infinite pits... The mod doesn't do it that much actually however; it only does it on palace levels, and leaves many levels unchanged. Then there are further mods ad level packs inspired by 4D, and they take the idea much further.
 * Fez. Gomez is a 2D man suddenly gifted with the ability to shift the 2D world on the third dimension.

Webcomics

 * In Eight Bit Theater, Black Mage's face is apparently so hideous as to be non-Euclidean—the hat keeping his features in shadow prevents people from being driven insane just by looking at it, as seen here. A more recent strip implied that revealing himself would destroy the universe, but this was just an idle daydream.

When the Light Warriors enter the new-and-improved Temple of Fiends, Black Mage criticizes it's infantile sense of twisted geometry (The room is merely upside-down for no reason), claiming that to "draw out ancient and malevolent forces of the underverse" you need to "start with parallel lines that intersect".
 * Gunnerkrigg Court, In Chapter 19: Power Station, the buildings at look pretty normal from nearby, but long-distance shots reveal that they are at crazy angles relative to each other.
 * Problem Sleuth: You cannot descend into the sky because the universe is not upside-down!
 * According to Questionable Content, dildos can have alien geometries too.
 * The polygons in Triangle and Robert tend to have their own style of geometry, leading to strips like this or this.
 * The Toymania store that serves as the main setting for Tru Life Adventures is bigger on the inside than it is on the outside. Subverted, though, in that it's a fluke of how the measurements were taken.
 * In Adventurers, when Imposis is just about to leave, Ardam points out that nothing he does seems particularly impossible. Imposis gives him a Penrose triangle and continues on his way, leaving Ardam to hold it in his hands and stare at it until he gets a headache.
 * From Tales of the Questor, we have the Unseleigh castle...
 * Yet another homage to MC Escher's "Relativity".
 * The author of Xkcd drew a comic about hyperbondage (see slide 5) for a cartoon-off against Farley Katz.
 * In Homestuck, the evil planet of Derse (and presumably, its good counterpart, Prospit) has inner depths and corridors that twist upon themselves in ways that challenge the rational mind. There is also this sequence.
 * The final form that Skaia takes as well.
 * Chainsawsuit: Cthulhu gets caught using non-Euclidian dice during a game of Humans and Habitats.
 * A few arcs in Fans (notably the whole of Book 5) centered around a power-object called the 23-Sider, an RPG die with 23 identical sides. When the 23-Sider was formed in Book 5 it warped reality.
 * Pip in Sequential Art chose to prove his superiority in Cubeminer by building "Escher's Staircase". The next page shows that with a few tweaks you sometimes can build this in a 3D game.

Web Original
"Prof. █████████: We know that after 2 comes 3 and after 3 comes 4. What this formula proves is that we missed a number somewhere. Imagine if all our technology was based on the belief that after 4 came 6. We simply didn't know or conceive of 5. That is in essence what this formula proves. We missed a number. ... I don't think it "destroys" anything. I think it tries integrating itself into our system and our system can't hold it."
 * Doctor Ka's mansion in the Global Guardians PBEM Universe is effectively a tesseract, and is definitely bigger on the inside than on the out. If its layout was drawn on a set of blueprints it would feature rooms that overlapped spatially, rooms that seemed to have no exits or entrances, spherical rooms that nonetheless had corners, and rooms where the plane of gravity depending on which way you were looking.
 * SCP Foundation:' SCP 033 is a missing integer. Any surface it's written on will begin to break down.


 * Fine Structure describes universes with more dimensions than our this way.
 * The PPC Headquarters: It is unclear whether it is just a confusing maze of a building or whether it can actually move around, but thanks to the Laws of Comedy, one of the only ways to find the place you are trying to go is to distract yourself and not think about it. However, it "was" built by alien plants, and they seem to be able to navigate it just fine. Also, poorly-constructed descriptions in the Word Worlds cause some rather eye-breaking visuals for the agents when the worlds try to put them into practice.
 * In one mission, Agents tempted fate by saying "It's a wonder we're still in three dimensions."
 * Buildings frequented by the Slender Man often develop these, sometimes reaching full-blown Eldritch Location levels. Don't expect a door to lead to the same room it did two minutes again.
 * Marble Hornets: When Jay investigates Alex's abandoned house near the end of season one, he gets bounced around between rooms and eventually finds himself in the disused bathhouse from an earlier entry.
 * Everyman HYBRID: the cast is out on a hiking trip in the middle of a bright, sunny day, exploring an old abandoned house in the middle of nowhere, deep in a wooded area. A door in the basement of the house leads to a sprawling beach in the middle of the night. In a later episode, they find a crawlspace in Jeff's house that leads them to the aforementioned abandoned house miles away.
 * Appears a couple times in Sevenshot Kid. Usually it serves as a prelude to something horrible.
 * The Hell Hotel in the Halloween Episode of Tribe Twelve.
 * One Hundred Yard Stare: When Avery, Macy, and Ellie first run from Slenderman there is a good dosing of this, with them starting in a yard of some sort and ending up, after a jaunt in a building, next to a moving train.
 * The Metal Glen from Ruby Quest displays aspects from this. First there's the metal shutter in Ruby's room, which sometimes opens to a window and sometimes to a passage. Then half of the Brig turns upside-down, gravity and all. Then it gets worse.
 * The Dionaea House. All of them. The one in Boise, for example, has a second floor that is not visible from the outside. It says something that this is not the strangest thing about it.
 * Carmilla's room in the Whateley Universe. It keeps changing size and shape. Its door moves from building to building. It's possible to walk in and out of it without using any known entrance. There's a reason the staff at Whateley Academy calls it the H. P. Lovecraft Room.
 * There is no geometry but Time Cube. You are educated stupid. Time Cube is so obviously true that knowing anything except Time Cube will drive you crazy when compared to Time Cube.
 * The Empty City, also known as the City of Empty Shadows or DEVOURER. It's a living city which intentionally makes itself into Alien Geometries in order to make sure its victims stay within it until they die. Or worse.
 * Fredrik K.T. Andersson managed to invert this.
 * Bravemule, the saga of a Dwarf Fortress, gave the dwarves an utterly bizarre culture with Blue and Orange Morality. Among other things, any shape that is not rectangular is "unscrupulous". An accidentally-created octagonal room was thus treated as horrible and incomprehensible, giving dwarves Catapult Nightmares and such.

Western Animation

 * Batman the Brave And The Bold has a standard-issue Escher magical library in the Batman Cold Open of "The Eyes of Despero". Batman is largely unfazed by the shifting gravity, and actually uses it to good effect.
 * The Foghorn Leghorn cartoon "Little Boy Boo" plays this for laughs. Foghorn is playing hide and seek with a child genius and hides in the coal bin. The kid performs a few calculations and then digs Foghorn out of the lawn. A very befuddled Foghorn protests that he was in the coal bin, but the kid just shakes his head and holds up the calculations. Foghorn then goes to look inside the coal bin, but decides "No, I'd better not look. I just might be in there."
 * The titular Fosters Home for Imaginary Friends sometimes exhibits a non-malevolent version of this combined with Bigger On the Inside. In one episode, Mac is attempting to leave for dinner but he and Bloo become hopelessly lost (and begin to go crazy from hunger), when, at one point, they exit on the roof, even though they clearly went downwards.
 * Presumably alien geometries are how Scooby Doo and his gang perform their strange chase routines. Perhaps something much more sinister than anyone suspected was going on the whole time.

Real Life

 * Studies of the cosmic microwave background radiation suggest that it better fits a Poincaré dodecahedron than a sphere.

To make the Poincaré dodecahedron more clear: you are floating inside a giant dodecahedron. When you get to an outer face you Wrap Around to the opposite face. Except the faces don't exactly line up, so you also rotate one tenth of a rotation.

More aptly, it is to be observed that Euclidean geometry, mathematically speaking, is a special case: it only applies to forms in a space with zero curvature (for the two-dimensional case, a perfectly flat plane); something that is, strictly speaking, an abstract concept (in light of the fact that time and space are demonstrably curved by gravity.) Consider that you cannot, in Euclidean geometry, draw a triangle with three right angles, but it is perfectly possible on the surface of a sphere.

Sometimes made worse by the fact that a non-Euclidean two-dimensional geometry is often visualized as embedded within a three-dimensional Euclidean space (the surface of a sphere, or a saddle), which leads some people to mistakenly believe that an n-dimensional non-Euclidean space requires an unseen n+ 1 dimensional space. (It's not too difficult to imagine a two-dimensional space with positive curvature as the surface of a sphere. Now try to wrap your mind around the idea of a space where the geometry works out the same as it would on the surface of that sphere, but without any third dimension at all.)

There are five axioms used to make geometry. Change one, and you get non-euclidean geometry. Change all five, and you could very well get arithmetic. Calling it a special case is an oversimplification.

Speculations on the Topology of the Universe aside, it's clear, and even somewhat well known, as stated above, that the effects of gravity are described by the curvature of spacetime, which means that in truth, geometry is not Euclidean at all. As a famous test of this, we can see stars which should be hidden behind the Sun during a solar eclipse, due to the light following the shortest path in curved space towards us. Time is also curved, in a sense, as clocks will run slower in places where classically the gravitational potential is lower relative to clocks at greater potential. This effect too, was measured using high precision atomic clocks.

Some astronomers who like thinking outside the box suggested that one might put a satellite 550 AU away from the sun. At this point, the aforementioned curvature of the spacetime bends light just right, making it possible to use the sun itself as, essentially, the primary lens of a huge gravitational telescope. This idea is called a solar foci telescope.
 * Spherical geometry isn't just something for universe-scale models. The surface of the earth is represented as a two-dimensional non-Euclidean space every time you look at a map. As mentioned previously, a sphere can have a triangle with three right angles on it, and the earth is (sort of) a sphere.
 * An example: Start at the equator. Go to the north pole. Turn right 90 degrees. Proceed to equator. Turn right 90 degrees. You will reach your starting point. Nifty, huh?
 * Many implementations of Conway's Game of Life wrap the edges of the grid, so the cells technically live on the surface of a torus. Or in the case of a 3D implementation, a hypertorus.

Some starfield simulators do this, too. Stars that vanish off one edge of the volume of space appear at the opposite one, resulting in the stars being positioned on the 3D surface of a 4D torus.

These wraparound connections are used in the communications paths for processes or threads in some concurrent programs.
 * There are multiple projections used on pictures, most commonly the gnomonic projection. The fisheye projection is also well-known. The reason these are necessary is that people see in elliptic geometry. As a simple example, imagine that you are standing on a railroad track, facing along the track. If you look straight down, the rails will look parallel, but if you look straight forward, they will intersect. If you look halfway between, you should be able to see where they're parallel and where they intersect, despite being perfectly straight.
 * Relativistic physics border that territory at times. e.g length contraction says when moving at a high enough speed there is a visible decrease in lengths (the length decrease is always there, just you can not see the difference caused by sqrt(1-(v/c)^2)). That is still believable if you have some fantasy. The trouble is, from the other point of view the not moving system is the one shortened. Better not try it yourself.

To clear the confusion (as much as possible, anyway), if things are moving, they are shortened in the direction of their motion by a numerical factor dependent on their velocity. If you measure the length of an object at rest you will always find it is greater than the length of the same object moving at a finite speed with respect to you. Of course, in said object's reference frame, it is by definition at rest, and it is you who is moving, and therefore, shortened.

The distance that the distance one through spacetime, the spaectime interval, is of a constant length, and is determined by the formula s = sqrt(x^2 + y^2 + z^2 - (c*t)^2). That is to say, although space contracts by a factor of sqrt(1-(v/c)^2)), time expands by an equal factor, so the spacetime interval that you cross remains constant. This means that the velocity of the thing that travels along the interval is already determined for any given observer. For the layman, this means that objects do not really shrink when you travel at velocity; they are actually just rotating in four dimensions, and just appear to shrink because we can only see in three dimenions.
 * And that's just special relativity. General relativity predicts that objects in a gravitational field shrink relative to those outside of the field. Essentially, when looking from the outside in, massive objects like stars or planets are actually Bigger On the Inside. This is due to the fact that what we experience as gravity is actually the mass of planets causing space to shrink in its presence. This means that straight lines, like those of laser light, become bent in the presence of mass. This is why planets orbit other planets, and light gets bent in a graitational field: they are following the curved lines of space around the planet. The best analogy of this is placing a mass on a rubber sheet, and waching it deform the rubber around it. Letting a ball spin around the mass is rather similar to planets orbiting one another, as the balls just follow the curved paths of the rubber sheet.
 * The Bermuda Triangle, according to many theories and reports. In addition to vehicles vanishing without trace (no wreckage left), reappearing after disappearing from radar, etc. some people have reported experiencing "time warping" or "missing time" while traveling through here.

In reality, while several accidents have taken place there, they're not statistically more common than in any other area of sea with the same density of traffic. Which, in spite of the stories, is considerable.
 * The Mandelbrot Set is a two-dimensional slice of a four-dimensional object that represents the eventual fate of iterating the assignment z <- z* z + c, where z and c are complex numbers (two dimensions each). Start with z=0 and try different values of c, and you get the usual two-dimensional view of the Mandelbrot set (which is, properly, only the boundary of the usually-black region representing points that do not escape to infinity). Fix a value for c and try different starting points for z, and you get a Julia set. The complete four-dimensional object stacks all the two-dimensional Julia sets along a complex dimension for a total of four real-valued dimensions.
 * A tesseract is a "four dimensional cube". Just as a square is only one face of a cube, a cube is only one "face" of a tessaract. Think about that for a second. The vertex of the tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called the hexadecachoron and... oh no I've gone cross-eyed

Scarier still, people have built computer models Rubix Hypercubes which people have successfully solved. For your headache-inducing pleasure. A 5D version has also been made.

If you want a quick view of just how confusing tesseracts can be, see here. Lovecraft would be proud. (Or actually he'd probably be nihilistically horrified at the alien, inhuman cosmos revealed by science, but whatever.)

Tesseracts are really just the start of a really long and dark journey. Even a Hilber cube is a really simple example when compared to some of the other stuff. It is also a cuboid of countably infinite dimension. Studying maths at an uni sort gets you used to this stuff...
 * Try solving a 4D maze. Just 3*3*3*3 takes about half an hour, and that's if you've gotten used to moving around in 4D.
 * Try wrapping your head around the Moebius Loop. It's a loop with a half-twist with only one side and one edge. Similarly, the Klein bottle, which is a closed surface and only has one side, rather than having an inside and an outside. It has to go through itself at one point to make that work, though.
 * Not really, that is just an artifact of embedding it in 3d space, in proper 4d space there is no such intersection.
 * In 3d animation programs, when you make a polygon it's usually supposed to have 4 sides at the most and flat. In Maya, if you have something with more than four sides, edges that don't connect to a vertex or a vertex that's not connected to an edge, things get...weird. You end up with a shape that the program can't draw properly that will usually look like a different surface depending on where you happen to be viewing it from.

Programmers tend to do things like this a lot when they first step into the world of OpenGL and DirectX. Get the position of a vertex wrong when you're just messing around with increasingly more complex shapes, and you can easily find yourself with a headache just from looking at the shape you've made. This effect is how the Alien Geometry corridor in Legend of Zelda: Ocarina of Time was implemented. Some corridors in the Descent series are also twisted like this.
 * Higher doses of LSD can produce visions of impossible shapes.
 * Pittsburgh, Pennsylvania's downtown area is called the "Golden Triangle" and is indeed triangular. To the uninitiated, though some of the turns feel like 90 degrees as you're trying to navigate. Making three "turns" and ending up back where you started has flummoxed more than one out-of-town driver (and more than its share of locals, too).
 * This Reddit comment is a fictional yet real physics-based story that illustrates just how nefarious black holes are.
 * While the main point (the fact that all trajectories point to the singularity) is correct, an observer would *not* see the universe outside the black hole as if they had tunnel vision. This is a result of falling through the black hole making an effort not to fall, i.e., with your engines on. Then the tunnel vision would be the result of relativistic beaming, not the fall. This site contains a much more accurate account, with a video: http://jila.colorado.edu/~ajsh/insidebh/intro.html
 * Also, the article correctly notes that once inside the event horizon, all points outside lie in the past. However, it then proceeds to draw incorrect conclusions from it. You would not be blind inside the horizon, because far from not being able to see into the past, you can ONLY see into the past. When you see something, you see it not as it is at that moment, but rather what it was when the light reaching your eye was emitted from it. Furthermore, a Faster-Than-Light drive would not be prevented from escaping from a black hole, because such a drive is perfectly capable of traveling into the past as measured by certain frames of reference.
 * "Mystery spot" attractions use tricks of architecture to simulate this trope. Gravity hills do the same thing, but with natural tricks of topography.
 * Optical illusions can convey an impression of Alien Geometries, by playing tricks on how the brain's visual association area interprets perspective from proportion and shading.