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666-4850—A normal-looking house that, on closer inspection, has no right angles.
"....but you start with parallel lines that intersect and you go from there."
A sub-trope of Cosmic Horror Story, and sometimes of Mind Screw, mainly found in print works, due to being difficult-to-impossible to show visually.
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 and get some extra space? (This can happen, in a relatively mundane scenario; on a perfect sphere, you can have triangles where the angles sum to 270 degrees since the lines are curved. Now imagine being able to unfold that sphere into a flat surface and leaving the 270-degree triangle intact, and you get some idea of what we're talking about.) 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.
This is common in eldritch and paranormal fiction where theoretical non-Euclidean geometry is law, except here it's no theory.
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 (the mathematical implications of non-Euclidean geometry tend to hurt even math majors' brains). 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 Living Labyrinth or Mobile Maze.
See also Eldritch Location, a good place to find this, and Hyper Space Is A Scary Place, wherein it's less a single wall or building that's just a little... off and more an entire alternate universe that just... doesn't make... sense.
Compare Sinister Geometry.
Examples
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Anime and Manga
- Casshern shows what appears to be lightning bolt-shaped metal construct (or a metal bolt of lightning) 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 it's 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 it's own shadow and can physically interact with other objects (Unit 01 tearing it's 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.
- Gurren Lagann: The Anti-Spiral's Mugens look vaguely like tripods with halos. Most of the body is black, with colored outlines. They're also Conspicuous CG. This even extends to their theme, which sounds like a synthesizer having a seizure. Later on, we encounter their spaceships proper, which look like rocks with faces all over. The fighters are basically hands and feet with lasers. And eyes.
- The Reverse World in Pokemon is an Escher-like place where 'up' varies, but apparently only for landbound creatures.
Comic Books
Fanfiction
Film
- 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.
- It should be noted that the scenery for that was actively based on a drawing by M.C. Escher.
- The villain's ship in the new Star Trek movie is, um, well... here's a side view◊.
- The tesseract-thingies during the "beyond the infinite" sequence from 2001: A Space Odyssey. Just look at them, and the way they move.
- The Tanz Akademie from Suspiria, an art deco nightmare from hell.
- Dreamworlds in Inception are built like this deliberately by the dreamworld'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 and attack the projection from behind.
Literature
- 2001: A Space Odyssey: The Monoliths, 1 by 4 by 9... "And how naive to have imagined that the series ended at this point, in only three dimensions!"
- Oh, c'mon. That's a pretty petty point. Although, if you want to play that way, nobody ever mentioned anything except spatial dimensions. With no temporal dimension it wouldn't have been perceivable.
- In the film, the proportions are a sexier 1 by 8-10 by 30 or so. 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.
- It's safe to say that this happens a lot when the Powers of Chaos are involved. A few examples:
- In Dan Abnett's Eisenhorn novel Xenos, the saruthi "tetrascapes" include irregular 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.
- Are you sure it wasn't regular octagons? Irregular ones can tessellate fine.
- 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 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.
- Mark Z. Danielewski's House Of Leaves starts with a house that is one-quarter inch Bigger On The Inside than on the outside. This discrepancy disappearing is, believe it or not, the cue for things to get worse.
- I think the house-owner's brother decides that it's actually 3/8ths of an inch, not to mention that they're only able to measure all the way across because a closet mysteriously appeared in the house when they left for a week.
- 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 (e.g., to make 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...
- And then when you get outside, you may have a whole new problem..
- A math-nerd resident of Second Life actualy went and reproduced the Crooked House in 3d, and if it's still rezed somewhere public you can actualy 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."
- The premise of Number of the Beast, as well: we're only seeing dimensions x, y, z; but there are at least 3 others which can be rotated around or extended along, and which apparently can be used to travel between universes (essentially 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...
- Referenced in Shirley Jackson's The Haunting of Hill House. Every angle in the house's construction is off, if only by a few degrees; a few degrees is enough, however, for the human eye to notice that something (in fact, everything) is wrong.
- Strictly speaking, this is not an example of "true" alien geometries, as it is described as being quirky but physically possible. The effect is much the same, however.
- The hotel in Diana Wynne Jones' Deep Secret has halls where you can go around more than four right angles (and a variable number of them) before coming back where you started.
- The Aelfinn and Eelfinn ("the Finn") from Robert Jordan's The Wheel Of Time 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.
- And the doorways seem to be a description of a three-dimensional Möbius strip.
- Inverted in Stephen King's short story I Am the Doorway (appears in the collection Night Shift). 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...
- Naturally, common in the works of H. P. Lovecraft. Two good examples are the sunken city of R'lyeh, and the Antarctic city in At The Mountains Of Madness. Lovecraft loved non-Euclidean geometry.
- An extreme example occurs in The Colour Out of Space (Lovecraft always insisted on the non-US spelling), where a horror from space causes an impossible colour (that everyone can actually see) to appear on nearby objects.
- An impossible colour is also a plot device of some importance in the surreal novel The Third Policeman by Brian O'Nolan.
- This is mocked by Terry Pratchett in his Discworld books, where octarine—the eldritch color of magic itself, which usually indicates that reality is crumbling at the seams—is described as "greenish-yellow purple".
- The Hounds of Tindalos 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).
- 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 rep as being haunted. This was a bad idea.
- In Graham Mc Neill's Warhammer 40000 Ultramarines novel Dead Sky Black Sun, the city in the Eye of Chaos features this — producing a Mobile Maze with it.
- Bloody Stupid Johnson, architect, Bungling Inventor, and general anti-genius from Terry Pratchett's Discworld, 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.
- Probably the best example of this trope in his work is 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...
- Also, The Colour Of Magic featured a parody of Alien Geometries: the Temple of Bel-Shamharoth (itself a spoof on Lovecraftian Cosmic Horror). The most striking feature of the Temple is that its walls, ceilings and floors are composed entirely of interlocking regular 8-sided tiles. Try and make a drawing of it. Go on. We're not going anywhere.
- 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. 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, passed the library at Hogwarts, 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.
- 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 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. It makes a good hiding place.
- 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, 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.
- 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 [1] 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."
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's just a sweet old thing, isn't she?
- That is nice of her, but doesn't change the fact that she's still BIGGER ON THE INSIDE!
- In the recent episode "The Lodger" the Doctor uncovers an alien time-distortion device similar to the TARDIS in the upstairs flat of a British apartment building. Amy, pouring 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 Hitchhikers 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.
- The plot of the cancelled show Threshold involved an alien invasion. The aliens used devices that apparently contained more that four demensions, 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.
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 it.
- M. C. 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
- The near-universal hallmark of things made in the name of Chaos in Warhammer 40000.
- Pictured is the Dark Eldar capital of Commoragh ("He must have died while carving it..."), notable for being made up of alien geometries in two senses: it's a xeno city constructed in a network of trans-dimensional tunnels known as the Webway, where things like "physics" and "gravity" are guidelines at best.
- In the Dungeons And Dragons module entitled Queen of the Demonweb Pits, 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.
- GURPS: Illuminati University describes a campus which teaches students (humans 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 itself combines elements of stereotypical universities (with a necessary American bent, as the primary authors and most of the contributors are from that part of the world), with particular reference to a traditional open area in more or less the middle of the academic part of the campus, in which students and staff may pause for reflection, study, impromptu lectures and other activities from which adventures may spring. In normal campuses, this area is referred to as the Quad, as it is usually some kind of rectangle. Illuminated University, however, 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.
- For that matter, 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's 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.
- 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.
- 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.
Video Games
- 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".
- The games Asteroids, Pac Man, and the like, use an unwrapped toroidal universe—the environments have the same geometry as the surface of a donut.
- Pac Man is technically a cylinder, since you can only go between left and right. There are some hacks with vertical tunnels, though.
- Super Mario RPG seemed to parody this somewhat—the overworld actually is donut-shaped, despite there being no real reason for it.
- For the uninitiated: any map where, when you cross over one of the edges, you pop up at the opposite side of the map (as in the above-listed games, as well as most Final Fantasy games and some Ultima games) is toroidal. Such maps are usually meant to simulate a sphere— with the effect that if you travel in a straight line in any of the cardinal directions, you will end up back where you started. Because spheres can not be exactly translated to a flat surface without distortion, this simulacrum fails (On a real map of a globe, if you want to travel in a straight line between two points, the path you follow on the map looks curved).
- The titular 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.
- The Polyhedron in the Russian art-house game Pathologic. Hooooooo boy, and how.
- From the outside, the building simply appears to be impossible. The inside is implied to more or less be another dimension, inhabited by hundreds of children suspended in some kind of weird dream world. Yes, the game is a total Mind Screw, thanks for asking.
- 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.
- Diablo II 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.
- You cannot grasp the true form of Giygas' attack!
- Echochrome is a puzzle game based on the works of M. C. Escher. The geometries are as weird as you might expect.
- The Distortion World in Pokemon Platinum definitely qualifies as this in Amazing Technicolor Battlefield form. Made Nightmare Fuel thanks to the flat, ungiving background theme and the fact that there are no Pokemon in it at all except for Giratina.
- Not to mention the fact that holy crap the sky is upside-down.
- The eleventh movie of the anime, Giratina And The Sky Warrior, gets as close as it can to this part of Platinum with the Reverse World. Platinum came out in Japan a few months after this movie, so it was likely intentional.
- 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.
- Certain areas in the Silent Hill series, going with the Chaos Architecture, e.g. the alternate school bathroom 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 hotel" in Silent Hill 2, where the doors that led to rooms before now lead out of other doors in the hallways, 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 then 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, eg 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. The last survivor is holed up in one of 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 an distance are coherent within regions of the Torment, but not between them.
- Warp pipes in general 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. Thought 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.
- Bowser, Bowser Bowser Bowser. Bowser.
- Of course, Bowser is kind of big for the pipe to even transport, and he'd have to jump up there, which he's unable to do that well. Consider this lucky.
- 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 you even help an alley give birth.
- 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.
- X: Beyond the Frontier and it's 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 turning a clockwise screw into a counterclockwise screw.
- Visions & 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 consistant. The player character occsoionaly 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.
- A variation occurs 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. There may or may not be a pattern to it, but this Troper hasn't found it.
- In the Forest Temple in Ocarina Of Time there is a coridor 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.
- And the gimmick of the Stone Tower in Majoras Mask 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's 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 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, thegame 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 point of the world - including the room itself - lead to you stand on different dimensions of the room, ie the walls and roof. However, the dimensions of the areas you are entering from don't change at all.
- Two areas in Sub Machine: Subnet Exploration Project have rooms that connect in ways they shouldn't. (Appopriately, 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.
Webcomics
- In 8-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.
- In the otherworld from Ch 19 of Gunnerkrigg Court, the buildings look pretty normal from nearby, but long-distance shots reveal that they're 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 & 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...
- 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.
Web Original
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.
- 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.
- There's a way to measure distance regardless of reference frame. It's sqrt(x^2+ y^2+ z^2-t^2). If you get an imaginary number, it's instead a measure of time.
- 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, despite 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.
- Not geometry per se, but since The Colour Out Of Space has already been mentioned: Impossible colors really exist, for a given value of "existence". The eye has three different kinds of receptors which correspond broadly to red, green, and blue, and their sensitivities overlap. No light source will stimulate only one type, so if something does, you'll see a color that does not and cannot physically exist.
- If you want to see an imaginary color, look at this for 30 seconds. Green isn't normally that bright.
- 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.
- A very minor (and subtle) real-life example of this is downtown Juneau, Alaska; due to the buildings being constructed on old mining claims, there are almost no buildings with more than two right angles, and a significant percentage with none at all, giving the downtown a very Haunting of Hill House effect.
- A tesseract is a 4 dimensional cube. It's a cubed cube. Think about that for a second.
- See, 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.
- 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.
- Forget Kramer's problems with First and First: at one point in the West Village it's possible to stand on the intersection of Fourth and Tenth Streets, two streets that theoretically ought to be parallel lines half a mile apart. (The West Village is, however, an entirely human geometry and only damages the sanity when you are trying to navigate through it in a hurry.)
- 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.
- I swear there's a building somewhere in Atlanta where one end of a walkway running horizontally through a building is five floors higher up from the ground than the other end. It's just built on a hill that steep, I guess.
- And most of all, The Mind itself. After all, it was the Mind that created Mind Screw on the first place.
- The Experience Music Project Building in Seattle has walls with extremely strange curves, and a few other odd bits of geometry. Here is a picture of it.◊
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.
- There's a Foghorn Leghorn cartoon that 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 than 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 "I'd better not. I just might, I say, I just might be in there."
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 MGNAAAAAH
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