Alien Geometries - Television Tropes & Idioms

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 can be visually rendered on a computer, but if done in reality would be skull-crackingly horrifying.
- Leliel in the original Neon Genesis Evangelion is a two-dimensional being that appears to be a sphere from every angle, and casts the shadow of a circle, which is actually its body. In other words: It's a circle of impossible thinness that casts its shadow IN THE AIR.
  - It actually has four dimensions; since humans have three, and cast two-dimensional shadows, it makes Insane Troll Sense that a four-dimensional creature casts three-dimensional shadows. The fact that only two of its dimensions matter to us is unimportant. It has the other two.
  - It makes normal sense, too, sorta. A 4D being sitting off to the "side" of our universe really could cast a 3D "shadow" into the air of our 3D universe, as long as whatever it was casting the "shadow" in (the "shadow" sure isn't being cast in sunlight) interacted with either our air or the fabric of the our universe itself. Of course, this doesn't explain how the shadow could bleed. That's why it only sorta makes sense. Sufficient Fan Wankery can justify that, too, but you have to draw the line somewhere.
  - Not really, the whole dimensional stuff is a massive lack of understanding of physical dimensions. First of all, normal objects cast three dimensional shadows, it's simply that shadows are only really visible when they intersect with a (two dimensional) surface. Second, if you assume that the fourth dimension is spatial, a four dimensional object is still three dimensional within a particular universe.
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.

Close Anime and Manga

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.
- Moore does this again in Tom Strong's Terrific Tales where Strong and Svetlana X find a Russian space station has become a crystal-filled Clown Car Base with multiple centers of gravity. The whole thing was caused by a chance encounter with a higher-dimensional cosmic particle.
- 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.
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.

Close Comic Books

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◊.
- "If they designed it with any common sense, I should be transporting you straight to the cargo hold. Won't be a soul around."
- Also, the Red Matter-generated black holes. 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.
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.

Close Film

2001: A Space Odyssey: The Monoliths, 1 by 4 by 9... "And how naïve to have imagined that the series ended at this point, in only three dimensions!"
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.
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.
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.
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.
In Ben Counter's Warhammer 40000 Horus Heresy novel Galaxy In Flames, Death's Tomb is bigger on the inside than the outside — as well as other repulsive features.
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...
- At least, the architect *thinks* it's merely a hypercube...
  - 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 excelent 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."
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'sThe 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.
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.
  - It should be stated that 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.
  - It doesn't say regular octagons, just 8-sided. The T piece in Tetris has 8 sides without being a regular octagon, for example, and can be used as regular tiles.
  - "Regular" implies that they are regular polygons; hence, regular octagons. Best guess is, there are some spaces.
- 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 M C 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).
  - 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 The 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.
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.
In Time's Eye, by Arthur C. Clarke, there are spherical alien objects that apparently have a 1-to-3 ratio for their dimaters and circumfurences, instead of a one-to-pi ratio. Think about it.

Close Literature

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!
  - Yeah, well there's weird, and then there's weird.
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.

Close Live Action TV

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...

Close Paintings

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.
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 if 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.

Close Tabletop RPG

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. This troper has seen one example that 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.
- Super Mario RPG seemed to parody this somewhat - the overworld actually is donut-shaped, despite there being no real reason for it.
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.
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.
You cannot grasp the true form of Giygas' attack!
Echochrome. Just... Echochrome.
- For those who haven't played Echochrome, it 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.
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.
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.
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.
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.

Close Video Games

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.
- When the Light Warriors enter the new-and-improved Temple of Fiends, Black Mage criticizes the thought of it having twisted geometry, claiming that to "draw out ancient and malevolent forces of the underverse" you need to "start with parallel lines that intersect".
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.
Ms Paint Adventures. Just..Ms Paint Adventures.
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.

Close Webcomics

SCP Foundation:' SCP 033 is a missing integer. Any surface it's written on will begin to break down.
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.
You cannot descend into the sky because the universe is not upside-down!

Close Web Original

Possible Truth in Television, as there's some indication that the visible universe is the 3-dimensional surface of a 4-dimensional trumpet/funnel.
- Some theories suggest the universe is a hyper-Dodecahedron, the normal variant being a platonic solid with 12 sides. Travel some billion light years in one direction you'd eventually arrive at your original location, but turned a few degrees. Trippy.
  - Some theories? Which theories? This scientifically-educated troper is skeptical.
  - 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 [1] 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.)
- 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.
Rotate a macroscopic (everyday) object 360 degrees and it's facing where it started. To return an electron to the quantum state it started in, you have to rotate it 720 degrees.
- To be fair, "Quantum physics" is just a fancy way of saying when you get down to the basics, the Universe doesn't make any goddamn sense.
- Well, and that's why you shouldn't think of Spin as a rotation in ordinary space. It's significance is more abstract, as it refers to an internal degree of freedom of the particle, that is associated with angular momentum, even though the particle is not really spinning, and couldn't possibly be, as it's described as a mathematical point. Even at classical electron radius, the electron surface velocity if it where really spinning would exceed the speed of light, contradicting Relativity.
- That's not really the same thing—electron spin isn't really a rotation, but if you DO physically turn an electron 360 degrees, it will be in a different state. Food for thought: if YOU turn around 360 degrees, it's pretty much the same from your point of view as if you rotated all the electrons not in you by 360 degrees, thus changing their state. Turn around again and everything's back to normal.
Does Russell's paradox count? It certainly gave this troper nightmares.
- I call your Russell's paradox and raise Banach–Tarski paradox, according to which it is possible to decompose a solid body into a finite number of parts, and re-assemble them into a body with a different volume (e.g. a sphere into two spheres of the same size, a single sphere twice that big, or a cube of the same dimension as the original sphere's diameter).
  - But we can sleep easy about that problem as it only applies to mathematically perfect objects that can be divided into infinitely small pieces. In the real world things can only be divided into really really tiny pieces.
  - Or that we would be dealing with an object comprised of infinitely many pieces in general. Taking infinity and dividing by two is still infinity.
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 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 not 2D, but is in fact 4D. The equation evaluated is z = z*z + c, where z and c are complex numbers. It's usually computed using c = (screen_x + screen_y*i) and z0 = (0 + 0*i), thus eliminating two dimensions. By putting the values for the real and imaginary components of z0 on two more axes, you get the full four dimensions of the Mandelbrot Set.
- A little more complex as the equation is lim_i->infinity z_i=z_i-1*z_i-1+c. Plus not all z are members, only z that remain finite as i goes to infinity (they generally fall into a stable point, or alternate between two stable points). The normal 2D graph of the set actaully showcases points not in the set through colorization. Typically points in the set are black, and points outside the set are color coded by how long it took them to escape (which means the value of i where the absolute value of z_i was greater than 2).
ANYTHING involving the higher 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.
  - For your headache-inducing pleasure. A 5D version has also been made (link available on that page).
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.

Close Real Life

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.

Close Western Animation

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
- Or it's hollow, consisting of collapsible components to greatly reduce its size. Or it is composed of nanomachines capable of shifting their shape and compressing into a smaller space. Or it is simply very, very compressed.
- That still doesn't explain ho the base can get inside the rocket after it has been launched.
  - It's tied on by a string.

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