"...but you start with parallel lines that
intersect and you go from there."
A staple of Cosmic Horror Story
and of Mind Screw
artworks. Elder Gods, Old Ones,
the Reality Warper
, The Omnipotent
and other cosmic entities
tend to bend the laws of physics to suit them. Why make a triangle where the angles add up to 180 degrees, when you can make one where they add up to 200 degrees in a flat space and get some extra room? Even the very body
of a particularly squamous thing may exhibit this, though more often it shows up in architecture as physically-impossible buildings—occasionally sentient themselves
are often depicted as being dangerous to the sanity of normal humans
; where you have to read
the Tome of Eldritch Lore
for it to drive you crazy, just looking
at this stuff can have an unpleasant effect
on your mental stability. Or at least really give you a headache.
More innocuous forms may appear normal. Then you realize that it is physically impossible for something this size to fit in that
, or you travel a short distance and find yourself kilometers away, or you turn left and end up to your right. Doubly fun if found in the Mobile Maze
are a good place to find this. Sometimes it is a single wall or building that is just a little... off
. See also Hyper Space Is A Scary Place
, an entire alternate universe that just does not make sense. A Minus World
in video games might be considered one due to unintentional programming bugs.
Compare with Bizarrchitecture
, Sinister Geometry
, and Scooby-Dooby Doors
(when done for comedy).
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Anime and Manga
- Neo Human Casshern shows what appears to be a metal bolt of lightning — or a metal construct — striking from the sky and staying in place for several days, inciting a transfer of what we are led to believe is superdimensional energy into our dimension. This energy is visible in the form of sparkling mystical runes hovering in the air facing the observer. It's awesome.
- The Reverse World in Pokémon: Giratina and the Sky Warrior is an Escher-like place where "up" varies, but apparently only for landbound creatures (see the Pokémon Platinum note in Video Games below).
- Mononoke in the Zashiki-Warashi arc, where the cast is trapped in infinite, identical copies of the same room.
- The Angel Ramiel in Rebuild of Evangelion manages to pull this off with some aplomb; its internal facets constantly shift as it moves, and the very first time we see it shift shapes from its fairly mundane octahedron to... other things, we see that it is somehow impossibly deep and one piece all at the same time... and then it starts changing shape when firing beams of pure killing. The effect is enhanced by the fact that what it does is almost painfully easy to render in CGI, but to see a physical object actually do it would be skull-crackingly horrifying.
- Leliel in the original Neon Genesis Evangelion appears out of nowhere over the city as a giant, floating sphere with black and white stripped patterns on it. But when the Evas fire at the Angel, the sphere fades out to dodge the shots, then casts a shadow which absorbs everything into it. It's then discovered that the sphere isn't its body: the angel is a 600-meter wide and 3 nanometer thick disc that is connected to a Dirac Sea. The floating object is actually a 3D shadow that appears when the Sea is opened. It gets even more mind numbing when you realize that the "shadow" is NOT intangible: it casts its own shadow and can physically interact with other objects (Unit 01 tearing its way out, in particular). It can even bleed.
- Pretty much every Angel in both the television and Rebuild universes seems to exercise this trope. Special points go to the Rebuild version of Zeruel, for being apparently solid and hollow at the same time, and being full of blood while also able to unravel itself into razor-sharp ribbons.
- Witches in Puella Magi Madoka Magica live in sprawling horror-filled labyrinths. Their alien nature is highlighted by Russian- and Czech-inspired animation, whose flat cut-out geometry is in stark contrast to the cutesy Japanese animation in the rest of the show.
- The third season of Sailor Moon does this when one of that season's miniboss squad accidentally breaks reality, resulting in the entire house becoming a zone of warped space.
- The Maze card in Card Captor Sakura creates a maze that isn't bound by normal spatial physics.
- Harry Potter and the Methods of Rationality, in line with the original Harry Potter series, has this: Hogwarts' corridors which can change when you aren't looking. The number of stairs you climb has only a passing correlation to your actual elevation when you look out the window. At least one corridor is tiled in pentagons.
- A newer chapter describes a stairway in Ravenclaw Tower as being straight when viewed from the inside and actually climbing it, but viewed from the outside, logically only a spiral staircase could fit.
- A Boy A Girl And A Dog The Leithian Script: This The Silmarillion fanfic provides two examples:
- Angband -the equivalent to Hell in this story- was erected and designed by Morgoth, and it is built in very weird shapes that make no sense to men or elves. People who has been inside barely can describe the place (beyond vague words such like “ugly”, ”burnt”, ”angled” “deep”, “rough” and “paths and overpasses winding up and down and you never know what direction you are going towards”) or they do NOT want to remember what it is like. The ex-thrall confirms that nothing of its design suggests that it was built by elves, and Luthien said that none of the architecture seemed designed with people in mind at all, but it was actually designed to NOT seem homely.
- The Halls of Mandos are a minor example. They are an underground network of halls, tunnels and corridors dug under the Aman’s Western ranges. The stonework and architectural style is plainly different of anything built by men, walls and rooms can be rearranged with enough will force and distance and direction work in strange ways, but at last you can understand the designs.
- A Different Medius has The Dead Sea, which resembles a cross between an M.C. Escher painting, and Cthulhu's domain.
- In Fuck The Jesus Beam, there is a city that literally does not exist, as it is only a lie. Despite this, it is also a physical location. Given the name Αδιβ, when someone who can see in only three dimensions looks at it, it appears normal, but in progressively higher dimensions, the architecture becomes more and more bizarre.
- In the Pokémon fanfic Ash's Return, the doors in Glitch City manifest this way to anyone trapped inside.
- The Pony POV Series has Discord's castle in the Dark World. It constantly shifts both interior and exterior — seemingly as much of its own will as Discord's — and for bonus points, even contains a void that's a portal to his cousin Ponythulhu's domain.
- In Fallout: Equestria - Project Horizons at one point there is an attack spell that appears to be this. It appears as some sort of distortion, and shatters bone without damaging flesh. Light would be distorted if the space it moved through was. Flesh is stretchy and can survive the fact that the angles on a triangle no longer add up to 180 degrees. Bone, not so much.
- In the climax of Sonic X: Dark Chaos, the Dark Chaos Planet is made up of both this and hefty doses of Mind Screw. Tails lands on the constantly-shifting surface and jumps into a pit that leads into the core; he lands and enters the core after a few seconds of falling sideways.
- Scootamom: Princess Celestia's first attempt at knitting goes very wrong, and the resulting... something manages to warp time and space around itself. It makes ordinary ponies ill just looking at it.
Films — Live-Action
- The Overlook Hotel in The Shining has hallways that lead into walls, windows that can only be seen from inside, hotel rooms that seem to overlap the same space, and doors located too close together to lead to separate rooms.
- An indie black-and-white short film of The Call of Cthulhu by The H.P. Lovecraft Historical Society does a particularly good job of getting this idea across, in a scene (faithfully adapted from HP Lovecraft's story) wherein a victim falls into a crevice which an optical illusion has led the audience to believe is a convex crag of rock.
- Cube 2: Hypercube is a rare example of this trope being employed in a visual medium. The actual warped geometry shows up only a few times due to the special effects required being rather expensive; the rest of the time it's showcased indirectly (e.g. duplicates of characters showing up).
- The climactic scene of Labyrinth takes place in an Escher-esque landscape where 'up' varies. The scenery was based on a drawing by M.C. Escher.
- The Red Matter-generated black holes in the new Star Trek movie. From the front, they look like your average swirling, funnel-shaped Negative Space Wedgie. Approach from the side, and you can see that it's missing its third dimension.
- Grave Encounters plays with this quite a bit a short distance into the movie. the doors that should lead outside now lead to more hallways, corridors are either entirely blocked off or in wrong locations, and time seems to pass at its own rate inside the demonic hospital which makes it impossible to tell what time it is without a watch or cellphone.
- The tesseract-thingies during the "beyond the infinite" sequence from 2001: A Space Odyssey. Just look at them, and the way they move.
- Even more impressive when you consider that they were created in the pre-digital era using 28 precisely-timed exposures.
- The Tanz Akademie from Suspiria, an art deco nightmare from hell.
- Dream levels in Inception are built like this deliberately by the level's "architects," in order to trap and delay the subconscious projections of the dreamers to keep them from attacking. The strange architecture can even be weaponized, as demonstrated by Arthur at one point, where he flees from a projection shooting down at him from the top of a staircase...only to have Arthur alter the stairs into a Penrose Staircase (looping stairs) and attack the projection from behind.
- Last Year at Marienbad was shot within several castles and edited together to create a lack of continuity in the castle and a strange, disorienting effect.
- If you try to map the Citadel in the Fighting Fantasy gamebook The Citadel of Chaos, you'll quickly find its rooms are connected in contradictory ways and sometimes occupy spaces that should be beyond the outer walls. Maybe Jackson just wasn't too strict about the layout, but it is called the Citadel of Chaos...
- The Monoliths in 2001: A Space Odyssey have the dimensions 1 by 4 by 9... "And how naive 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.
- This happens a lot in Warhammer 40,000 when the Powers of Chaos are involved.
- In Graham McNeill's Warhammer 40000 Ultramarines novel Dead Sky Black Sun, the city in the Eye of Chaos features this — producing a Mobile Maze with it.
- In Dan Abnett's Eisenhorn novel Xenos, the saruthi "tetrascapes" include regular octagons that nevertheless tessalate. Eisenhorn rescues some green soldiers from such a tetrascape, and later chooses them over experienced soldiers to go into one. Wise of him: the green soldiers had actually seen a tetrascape before, and the experienced ones hadn't. As a result, the "greens" manage to shoot and kill dozens of enemies, but the elite Deathwatch Space Marine attached to Eisenhower's squad can't hit anything thanks to the effect the twisted geometries have on ballistics.
- In Dan Abnett's Gaunt's 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 as being bitterly cold. Also, their vox units register as both within ten kilometers and out of range.
- In Ben Counter's Horus Heresy novel Galaxy In Flames, Death's Tomb is bigger on the inside than the outside — as well as other repulsive features.
- Alice in Wonderland - Alice, in Lewis Carroll's Through the Looking Glass set to walk to a hill and always found herself walking into the doorway of the house. Finally, when she walked away from it, she reached it.
- Lampshaded by the Red Queen, when Alice found herself unable to run quickly. "Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!" The Queen was able to travel much faster, since she was a Queen, and can cross the width and breadth of the chessboard in a single move.
- House of Leaves starts with a house that is 3/8ths inch Bigger on the Inside than on the outside. They are only able to measure all the way across because a closet mysteriously appeared in the house when they left for a week. They also get slightly different measurements with every method they try until confirming the final number with a very accurate method — you'd normally think this is because of measurement deficiencies, but in retrospect... Also, this discrepancy disappearing is, believe it or not, the cue for things to get worse.
- Thursday Next's Uncle Mycroft, among his other Mad Science projects, developed "Nextian Geometry" with his wife, said to be based on how a cylinder looks like a rectangle from the side, which allows one to use a circular cutter on dough without any left over: it makes circles tesselate.
- Robert A. Heinlein:
- The short story And He Built a Crooked House involves an architect who, inspired by higher-dimensional geometry and high real-estate prices, builds a house in the shape of an unfolded hypercube. Then an earthquake makes it fold in on itself into a hypercube, so to the architect's delight it's eight times roomier on the inside than on the outside. Just one small problem: the house's new topology makes it a bit difficult to leave once you're inside. When you do get outside, you may have a whole new problem.
A math-nerd resident of Second Life actually went and reproduced the Crooked House in 3d, and if it's still rezed somewhere public you can actually walk through it. Not a real hypercube of course but some excellent special effects. Here's the story with video.
- The short story By His Bootstraps involves a time-travel machine, constructed by aliens, housed in a building which is described in these terms
- Glory Road had the hero and companions invading a tower "where the architect used a pretzel for a straight-edge." It's so convoluted that it took hundreds of spies decades to figure out a route to the MacGuffin.
- The Number Of The Beast, as well: we're only seeing dimensions x, y, z; but there are at least three others which can be rotated around or extended along, and which apparently can be used to travel between universes. This is a conceit to let him run through every literary universe ever, and have a massive Cross Over event. The novel culminates in a party, in what is effectively the Crooked House, with every single character he created attending (plus several guests). Special mention goes to the literary critics lounge, which was shaped like a Klein bottle... once you were inside.
- Deep Secret - the Hotel Babylon has halls where you can go around more than four right angles before coming back where you started thanks to the building being on top of a bunch of ley lines.
- The Wheel of Time - The Aelfinn and Eelfinn ("the Finn") inhabit one or more separate dimensions described by the author as having radically different natural laws. Successive windows do not show what one might expect. That the magic system in the series is heavily geometric likely has a great deal to do with why its use is explicitly forbidden there. The doorways into their realm also resemble this in the "real world", and are described as "twisted".
Though it's less apparent, the same is true of the Ways, an artificially-constructed dimension meant for quick travel. Except in one dream sequence (which, for complicated reasons, probably reflects the reality of the Ways), the realm is extremely dark, but travelers there have noted that by the arc of the bridges they're walking on, the platform they've just arrived at should be directly beneath the last. During the dream sequence, it becomes apparent that the platform-islands extend infinitely downward—and unless you follow the bridges with your eyes, appear to be on the same plane. The doorways seem to be a description of a three-dimensional Möbius strip.
- A rare Star Wars example can be found in Star Wars Expanded Universe novel Lando Calrissian and the Mindharp of Sharu. In it the titular character finds the titular artifact - which seems to exist in several dimensions simultaneously and as such it really hurts just to look at it. Lando then uses it to unlock the passage into the Great Pyramid of Sharu - where he is expanded in size several dozen times, while his droid companion is shrunk to the size of a louse. There are even more examples of Alien Geometries in the book: the aliens who built it were very, very alien indeed by Star Wars standards.
- Stephen King:
- Night Shift - Inverted in the short story "I Am the Doorway". An alien lifeform sees a boy walking with a sieve under his arm: "an abominated creature that moved and respired and carried a device of wood and wire under its arm, a device constructed of geometrically impossible right angles."
- In Stephen King's short story Fourteen Oh Eight, 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...
- Justified Trope: The room was specifically built this way to drive people crazy so they'd be suitable hosts for the demonic powers.
- He also appeared to use this in Out of the Silent Planet, but the room turns out be normal human geometry, just a very unusual sort.
- HP Lovecraft loved non-Euclidean geometry:
- The sunken city of R'lyeh. In The Call of Cthulhu, a unfortunate human visitor to this locale is swallowed up by an angle of masonry which is acute, but behaves as if it were obtuse.
- Before that there is the famous scene where the gate that seals the Great Old One himself opens, and the sailors can't even be sure whether it's a vertical door or a horizontal hatch - even though one of them climbed or walked up its surface!
- The Antarctic city in At The Mountains Of Madness.
- Perhaps most explicit in The Dreams in the Witch House where a mathematics student discovers the unearthly topology of his own bedroom serves as an extra-dimensional portal. Well, he was renting it because of its reputation as being haunted. This was a bad idea.
- The Hounds of Tindalos, by Lovecraft's friend Frank Belknap Long, features ravenous creatures of weird geometry who travel trough time and space, and the only way to avoid them once they're on your trail is to completely avoid sharp angles (such as in a completely circular room).
- Eon, by Greg Bear, features an asteroid hollowed out by people from ..elsewhere, with seven chambers running along its internal axis. The first six contain cities, parks, a spaceport and loading area, and power generators. The seventh chamber goes on forever, contains objects made from redistributing probability over space, and a mathematical singularity running along its centre. And then things start to get weird.
- Bloody Stupid Johnson, architect, Bungling Inventor, and general anti-genius regularly does this kind of stuff entirely by accident.
He once designed a letter-sorting machine whose central component was a wheel that had π equal to exactly three (he did this because he thought that π = 3 point whatever was "a bit untidy"). This causes it to sort out letters it hasn't had put in yet, among other oddities.
Empirical Crescent, a row of terraced houses where every door and window leads somewhere other than where you'd expect it to lead. At least it makes it easier to get rid of rubbish—just toss it into the garden. After all, it might not be your garden.
The reason for this corruption of dimensions occurs because the row of houses is crescent-shaped on the outside only. Inside, it's supposedly laid out like a straight row. Presumably the two configurations conflict. Occupants had a tendency to leave in the middle of the night, often without stopping to pack...
It's also stated in Thud! that he invented the 13-inch foot and a triangle with three right angles.
- Circles with π = 3 and triangles with three right angles both do in fact appear in non-Eulidean spherical geometry.
- The Colour of Magic featured a parody of Alien Geometries: the Temple of Bel-Shamharoth (itself a spoof on Cosmic Horror Stories). The most striking feature of the Temple is that its walls, ceilings and floors are composed entirely of interlocking regular 8-sided tiles. Whilst it is possible to create octagonal tiles that can be fitted in a regular, interlocking way, they must be concave polygons: there is a mathematical proof that no convex polygon of seven or more faces can tessellate on a Euclidean plane.
- The first chapter mentions one of the gods using a 7-sided (but still cube-shaped) die to cheat.
- The buildings of the Unseen University, which have been rather strongly influenced by the vast amount of magic that has flowed through its halls over the centuries, have floors and rooms where logic says they simply could not exist. Magic is as much a part of the architecture as cement.
It is specifically noted that there are rooms in which gravity changes direction through the day and windows that only exist on one side of their walls.
- 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 (or sideways). That's quite apart from the fact that it serves as a gateway into L-Space, and is therefore linked to all libraries everywhere in all points of space, time and reality. Technically, it contains every book that has ever been written, every book that is ever going to be written, and every book that ever could have been written (whether it actually was written or not).
Once, the Librarian took a trip deep into the shelves, passed tribes of lost students, and ended up in the same library in the past.
In addition, Pratchett explains that any sufficiently large collection of books (magical or otherwise) can exert the same effect as the Unseen University Library; the equation goes "Knowledge = Power = Energy = Matter = Mass". And since mass warps space around it, so does a high quantity of knowledge.
- 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 (he forgot that things were not supposed to be bigger on the inside when he made the place and can't quite manage to make them fit now). Many of the rooms have the peculiar effect of being enormous at the same time as being regularly sized. Death's room in particular is stated to be about a mile wide, but most can be crossed in only a few steps.
- The real killer is that Death himself is weirded out by this last fact. This is because it's ordinary humans (like Albert) who cross the room in a few steps, even though it's clearly a mile wide. Death's theory is that the human mind refuses to accept the true size of the room, and acts as if it were normal sized. And for humans, it seems like it's Death that acts weird, by either moving through very solid walls (he cannot even see) or suddenly appearing from thin air when they were in a small, empty room just a second ago (when he moves through that mile wide space and not the short, straight path tje humans usually take). Death's realm is weird.
- The Tooth Fairy's house in Hogfather is another example of this trope.
- The exterior portion of the Tooth Fairy's domain are also a rather interesting take on this trope as the Alien Geometries present are actually based off the poor understanding of size, proportion and three dimensional imagery present in a child's drawing. The massive white gap where there ought to be a horizon is particular unsettling.
- The Gnarly Ground in Lancre is a seriously bizarre landscape of crags and valleys "scrunched up" into a small area, overlapping in space; what geographical features you see there and have to deal with depend largely on your mindset. It makes a good hiding place.
It is also known that how you perceive features of the Gnarly Ground depends a great deal on your own outlook. What one person sees as a shallow stream at the bottom of a ditch bridged by a large slab of rock can appear to others as a roaring torrent pounding over boulders at the bottom of a deep gorge bridged by a narrow, shaky bridge of balanced rocks. The most disturbing aspect of this is that how one perceives the features controls how one interacts with them, so one person may see you hanging by your fingertips while another sees you standing in the stream.
- Unseen University's mad but Good with Numbers Bursar has posited that there is an extra number between three and four, which he calls 'umpt' (as in, 'umpteen' minus ten).
- The Bugarup University (the UU's counterpart in Fourecks) has a tower which is only thirty feet tall at the bottom, but half a mile tall at the top, making it both significantly taller and significantly shorter than the Unseen University's famous (and surprisingly Euclidean) 800-foot tall Tower of Art, depending on where you're standing.
- 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 (from its leavings, since the creature itself is never seen), is a color that the human eye cannot see and makes noises that the human ear cannot hear. This inverts this already inverted trope because the color is natural and it is humanity that has become too alien (or at least insensitive) to comprehend it.
- In the Deathstalker series the AIs of Shub constructed a world of their own to live on. Unfortunately for humans who might visit, it exists in more dimensions than they can perceive and so is unhealthy to look at for extended periods of time. The Madness Maze, despite a relatively innocuous appearance, had convoluted, nigh-sentient path designs that would either evolve you into a higher being or tear you apart.
- In Flatland, the two-dimensional protagonist A. Square struggles to fathom the third dimension when he is introduced to it by a travelling sphere, and it almost drives him insane. In a dream he sees that inhabitants of one-dimensional Lineland are similarly incapable of comprehending the second dimension. And let's not even get into Pointland's issues.
- Many sequels have been written. Flatterland (Ian Stewart) has even more bizarre geometry, including a hyperbolic world, a fractal world, a grid world, and so on.
- Threshold by Caitlin Kieran contains a fossil in a shape that cannot exist, causing the heroine to black out when she looks at it too long. What is this sinister shape? A regular heptagon.
- In the Star Trek: Voyager novel The Final Fury, Captain Janeway, Tuvok, and Neelix arrive aboard a Fury planet wherein the hallways and doors meet at angles that aren't quite "right" — literally and figuratively — and the aliens themselves despise those who follow the "right-angle" or "right-hand path."
- A feature of the bizarre planet in The Inverted World. Within about a dozen miles of the "optimum", everything is pretty much Earth-like. Go any farther than that, however, and things start to distort unpleasantly. Because the optimum is constantly moving, the entire City has to move after it to avoid destruction.
- Even weirder, in the novel version, it turns out that the Inverted World is actually EARTH - the inhabitants of the City only perceive it the way they do because their perceptions (and possibly their physical reality) have been altered.
- The protagonist of Return from the Stars comes back to Earth after over a hundred years of absence. In the meanwhile, architecture has changed/evolved so much and so confusingly that when he first steps out into a spaceship depot, everything around looks to him like an abstract, shapeless muddle of pathways.
- In the Star Trek Shatnerverse novel The Return, The Borg have built a hypercube base inside a subspace tunnel.
- The Last Battle, the final volume of The Chronicles of Narnia, ends in a world that's essentially the opposite of reality, in that the closer you get to the center, the more there is.
- In An Elegy for the Still-living, the main character spends a few minutes walking down a forest path before realising he is seeing the same scenery looping over and over again. Also, this passage:
After the man had fallen through every place and every time that ever he had even imagined, he began to fall through the places that his mind could not conceive. He passed into structures that did not follow geometry, saw shapes that had no edges or sides, that extended into themselves and into all directions. He saw triangles with one hundred eighty one degrees. He saw minds that had no reason or morality. He saw colors indescribable to others. He saw the true shapes of his dreams, and the ten dimensions of the earth and sky. He saw what no one saw, felt what no one felt. He heard sounds with his finger tips, and tasted with his ears. He had secrets whispered to him in a language that can't be translated.
- In George R. R. Martin's A Clash of Kings, Danaerys enters the House of the Undying Ones in Qarth. Once in the antechamber, she makes four consecutive right-hand turns without returning to her starting point.
- In Larry Niven's Protector the Brennan Monster amuses himself by creating full scale replicas of some of Escher's art, using things like artificial gravity to make them work.
- The Hitchhiker's Guide to the Galaxy:
- In the Venus Prime series, the interior of the Amalthean world-ship is described as being made up of nested spiral shells. The diagrams at the back of the last book don't help to make it any easier to comprehend.
- The intricate geometric designs that Cryptics use instead of heads in The Stormlight Archive invoke this trope. And in ''Words Of Radiance, a Cryptic named Pattern manifests in the Physical Realm as a complex geometric pattern, implied to be based in quantum uncertainty.
- Greg Egan's Orthogonal trilogy features a(n oxymoronically) straightforward example: Time is fundamentally the same as space, meaning that there are technically four spatial dimensions and no such thing as time. The trope is even discussed in-universe when Yalda hypothesizes "four-space" and Giorgio points out how batshit insane it sounds.
So according to your theory, an object could have a trajectory entirely orthogonal
to our own? Yalda:
It could move with infinite velocity
Yes, that's how we'd describe it. But that's no stranger than saying that a vertical pole has an 'infinite slope': unlike a mountain road, it gets where it's going vertically without bothering to go anywhere horizontally. An object that gets where it's going without bothering to move across what we call time isn't doing anything pathological; in reality, there's nothing 'infinite' about it.
- The Third Policeman has several different forms of this. The most prominent example would be the police barracks, which are two-dimensional on the outside and seemingly three-dimensional on the inside. There's also Eternity, which loops, and the inside of Mather's walls.
- In the Spiral Arm series, there is the Ouroboros Circuit, an artifact of a race of Precursors known as the people of sand and iron. At first glance, it looks like a wreath of tangled wires; but if you try and trace the wires with your eyes, and you'll find yourself staring into hyperspace.
Live Action TV
- The Doctor Who Expanded Universe speculates that this is the default setting for the interior of a TARDIS in Doctor Who, and that the Doctor's TARDIS projects a more easily comprehended interior so as not to freak out the Doctor's human companions. She is just a sweet old thing.
In "The Lodger" the Doctor uncovers an alien time-distortion device similar to the TARDIS in the upstairs flat of a British apartment building. Amy, poring over the building plans for the address, discovered that the building didn't even have an upstairs, it was a one-story building. Perception filters kept people from noticing anything out of the ordinary.
- Stargate SG-1: 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," where the ship becomes a maze where no door or hallway leads the same place twice due to a Negative Space Wedgie.
- 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.
- Threshold involved an alien invasion. The aliens used devices that apparently contained more that four dimensions, and cannot be fully perceived visually. Just seeing or hearing the signals originating from these 'beacons' can kill or transform the view into an alien agent, with triple DNA helix where earthlife has only contains double. The aliens themselves are usually seen in dreams; crystal forests where spider-like entities are only partially seen.
- In the Twilight Zone episode, "Little Girl Lost", a little girl falls through a portal in the wall of her bedroom into an alternate dimension, in which space is twisted, distorted and nonsensical to ordinary human perception. Fortunately, the family dog's superior hearing and sense of smell help get the little girl back into our dimension before the portal closed forever.
- In the Night Garden is a BBC kids' show (from the people who made Teletubbies) where the various characters often ride around the eponymous garden in the Ninky-Nonk (a train without tracks) or the Pinky-Ponk (an airship). When they're boarding, these vehicles are comfortably large enough to accomodate all of them, yet when they're actually in motion the Ninky-Nonk is small enough to run up trees and over branches, and the Pinky-Ponk is small enough to get knocked off course by a toy ball.
- In Warehouse 13, the personal effects of permanent prisoners of the Warehouse are stored in the Escher Vault, which is basically a three-dimensional M. C. Escher painting. Authorized personnel use special goggles to follow along with the vault's ever-shifting perspective. Unauthorized personnel are never seen again unless they have Super Speed.
- In Rose Red, the titular mansion is like this. Sometimes. It was built to perfectly normal standards, but after a series of incidents it went from "just" haunted to something more, and may in fact have been sentient. Features include staircases leading into ceilings, dead-end hallways that screw with perspective, rooms that weren't there a minute ago (or were there but aren't any more), and other hilarities. About the only guaranteed stable locations are the entryway, the attic and the arboretum, and even then the things in them often are moved around or fully animate.
- SCP Foundation
- SCP-001 ("Dr. Mann's Proposal"). The SCP is a gravel path in a wooded area. When traveled counter-clockwise the path is continuously uphill regardless of how far the traveler goes.
He found the path did not conform to the pure geometry of Euclid.
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.
- SCP-184 ("The Architect"). SCP-184 increases the interior size of buildings. After it's been working for a while things inside the expanded areas start to get so strange that it drives people insane.
- Paintings by H. R. Giger, famous for his design of the Xenomorph in Alien, though his work tends more to towards the horror aspect than the impossible. He also likes to paint landscapes having sex with themselves. Think about that for "scenery porn."
- M. C. Escher 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...
- Some of his works are what have been called lampoons, because they work by violating normal conventions of art (like things that are behind other things being blocked from view by the things they are behind). The effect of violating rules of art that are present to mimic reality is to make the image look strange for reasons that are not always obvious.
- The LP sleeves of the first two Blue Öyster Cult records (Blue Oyster Cult and Tyranny and Mutation) depict strangely alien geometries and stuctures under strange skies on strange otherworlds. While nothing violates perspective rules, they still look eye-wateringly odd.
- William Hogarth's print False Perspective◊ is meant to be a parody of mistakes made by bad artists, but ends up looking very much like a prototype of Escher's style.
- In Christian eschatology, the end of the world is accompanied by the sky rolling up like a scroll. If this wasn't Mind Screw enough, it also involves everyone on earth witnessing the return of Jesus. At once. Even though the earth is a sphere.
- Remember that during biblical times many nations still thought the Earth was flat, the sky was assumed to be a bowl-shaped dome, and the stars just points of light fixed to it. With that in mind, makes sense.
- A bit of interpolation between these two, and with modern theoretical physics, and we have a possible Mind Screwdriver: The end of the world splits the fabric of space across all of the 3-dimensional universe, revealing Eternity, from which Jesus emerges and destroys the wicked with the brightness of his coming, which may as well mean that their heads a'splode from the Mind Screw of it all.
- In Mortasheen, this is where the Xenogog lives naturally, only coming into ours with a screw up in a time travel experiment.
- The near-universal hallmark of things made in the name of Chaos in Warhammer 40,000. For example the Dark Eldar capital Commoragh has spatial anomalies, "wandering shadows that tear apart the unwary" and many other dire things. It lies deep within a nest of extra-dimensional tunnels.
- The Webway in general seems to follow alien geometries. In particular, gravity seems to always point towards the tunnel floor/walls. In Path of the Outcast there is a city built into a large spherical chamber in the Webway where buildings cover the entirity of the chamber walls and a person on the streets would see the city curve up and into the "sky". Commorragh is even worse since the place is filled with portals, meaning that walking down the street might actually involve trevalling the distance of several lightyears.
- Dungeons & Dragons
- The Classic Dungeons & Dragons system delved deep into this trope with its boxed set for PC Immortals, redefining game-reality in terms of five spatial dimensions. Mortal creatures exist in three, Immortals in four, and Old Ones in five. Which three a mortal creature occupies can vary: Nightmare-reality creatures share only one spatial dimension with Normal-reality beings such as humans, and "nippers" from the Astral Plane overlap with dimensions of both Nightmare and Normal reality. As for how all this applies to the geometry of the planes themselves, thinking about it could make Your Head Asplode.
- The Planes in D&D 3rd edition got like this at times, such as an infinite plane having an edge. Seasoned Planeswalkers tended to give the advice "Try not to think about it."
- In the Queen of the Demonweb Pits module, the players ventured into The Abyss to confront Lolth, the demon queen of the spiders. Lolth's domain consisted mainly of long, open passageways hanging in space. Even though these passages pass over and below each other, they never ascend or descend in any way.
- Less heady are the Githzerai monasteries on Limbo, which take advantage of the fact that "down" is whichever direction you want it to be, giving us some extremely Escher-esque architecture.
- Possible example: In Dungeons & 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.
- Another good example from Dungeons & Dragons is Sigil, a city that exists on its own plane, connected to others only by portals in the forms of doors. The city resembles the inside of a tire; it's a tube that wraps around on itself, so you can look up and see buildings in the sky, walk straight for hours and end up in the same place, and open any door and end up somewhere else. Gravity seems to work for whatever ground your standing on right now and light is just sort of there. To top it all off, it's floating on the top of an infinitely-tall spire in the middle of a plane that is both infinite and finite. The best part, though, is that, since Sigil exists completely separate from any other plane, there is a chance that it has no outer surface.
- GURPS: Illuminati University describes a campus which teaches human students and everything else capable of paying the exorbitant university fees how to function as Mad Scientists, World-Conquering Dictators, Marketing Specialists, and other strange jobs.
The campus is a stereotypical university: the campus has an open area or "Quad" in which students and staff may pause for reflection, study, impromptu lectures and other activities from which adventures may spring. Illuminated University has The Pent, which has five sides for no particular reason; students who happen to have a protractor handy will discover that all five of the corners have 90-degree angles. One of the dorms is stated as having rather similar angles.
- Mage: The Awakening has the Twisting Maze Zone, a localised distortion of reality caused by Abyssal forces. While it looks chaotic, a constantly shifting jumble, this is actually because its directions extend into the fourth one as well. Unlike many examples, mages can use this to their advantage, using their will to walk through hidden parts of it to teleport around-in fact, they must, as the way to banish it is to walk through the areas of the Zone as they normally are-i.e., sans Twisting Maze-thus forcing them to apply to Earth laws.
Once that is done, the Zone literally Logic Bombs itself out of existence, causing anybody nearby to gain a brief glimpse into the space-time continuum. Should someone have the force of will to process it, they have an epiphany about how the world works, resulting in an Experience Point gain. If no one does anything about it though, the Zone grows so bad that it ends up rewriting history so that it-and the area it affects-ceases to exist.
- And, as you'd expect, Call of Cthulhu and Cthulhu Tech occasionally include this for...well, we all know why.
- The Terminal in Over the Edge. It's Al Amarja's massive airport, nine-storeys high and built like a maze. Navigating it is so difficult, people need to hire guides. Of course, the best part is when you leave the airport and see that it's built like a step-pyramid. An upside-down step-pyramid.
- The dimensions of Primordial world bodies are often based on their moods and personalities. For the more focused and stable ones, the worlds are typically consistent and predictable. For others, you get things like spatial relationships that are constantly rearranged, being able to pass from one side of a layer to another with no obvious transition, and having a sun that is inexplicably always right above you while also being at the center of a spherical arrangement.
- As you might imagine, this makes travel around Malfeas... interesting. Once you enter the dimension proper, you must cross Cecelyne, the Endless Desert, for five days to actually get to the Demon City. No, it doesn't matter if you're walking on foot, riding on horseback, or piloting a First Age airship. The trip always takes five days. Then you get to the Demon City, which is layers upon layers stacked on top of one another - but each layer has Ligier, the Green Sun, shining above it, no matter how deep down it is.
- Werewolf: The Apocalypse has the Black Spiral Umbral Realm. Being a Spirit World the Umbra is pretty odd at the best of times, but normally follows at least the guidelines of the laws of physics, if only because visitors expect it to. But the Black Spiral... From the outside it looks like just a spiral pattern on the floor in black. Once you start walking it it seems longer, twistier and with entirely too many dimensions. In fact it's a path into the mind of the Wyrm. No-one's ever come out the other side sane.
- In the 2300 AD module Bayern, explorers who penetrate the cluster nodes at the heart of the Pleiades end up in the four-dimensional realm of the AGRA Intelligence. They may discover that the Pleiades is just one part of a galaxy-spanning megastructure, go insane, end up dimensionally flipped, or all of the above before they leave.
- The Transformers already skirt the trope, what with size and mass-changing and the oddness of the scales...but then we come to the Autobot Micromaster Countdown's playset. He's a deep space explorer. He has an interstellar rocket and a command base. The base is used to launch the rocket. But also fits inside the rocket: mgnaaaaa!
- A minor example in Dangan Ronpa: The swimming pool occupies a space one story above the floor of a multi-leveled gymnasium. This trope is implied, though it may be due to lack of consideration on the creator's part.
- Demonbane, being derived from the Cthulhu Mythos, has this at several points. The characters describe the Deep Ones' artwork as "unpleasant" and headache-inducing, though they cannot say why, exactly. Later, the towers of R'lyeh are described as being "twisted in straight lines", with the protagonist Lampshading how that doesn't make any sense.
- 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 it's infantile sense of twisted geometry (The room is merely upside-down for no reason), claiming that to "draw out ancient and malevolent forces of the underverse" you need to "start with parallel lines that intersect".
- 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.
- Chain Saw Suit: Cthulhu gets caught using non-Euclidian dice during a game of Humans and Habitats.
- A few arcs in Fans! (notably the whole of Book 5) centered around a power-object called the 23-Sider, an RPG die with 23 identical sides. When the 23-Sider was formed in Book 5 it warped reality.
- Gunnerkrigg Court, In Chapter 19: Power Station, the buildings at Zimmingham look pretty normal from nearby, but long-distance shots reveal that they are at crazy angles relative to each other.
- 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, as shown in this sequence. It is clearly not just a bunch of buildings built around a central point. The core of its moon is hollow, and there it can be seen that the moon is somehow held together by chains that are loose and just float there.
- 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.
- Pip in Sequential Art chose to prove his superiority in Cubeminer by building "Escher's Staircase". The next page shows that with a few tweaks you sometimes can build this in a 3D game. But there's no guarantee that the physics engine will survive an attempt to process it.
- From Tales of the Questor, we have the Unseleigh castle...
- The polygons in Triangle and Robert tend to have their own style of geometry, leading to strips like this or this.
- The Toymania store that serves as the main setting for TRU Life Adventures is bigger on the inside than it is on the outside. Subverted, though, in that it's a fluke of how the measurements were taken.
- The author of xkcd drew a comic about hyperbondage (see slide 5) for a cartoon-off against Farley Katz.
- In The Inexplicable Adventures of Bob! the alien Nemesites know how to make buildings that are bigger on the inside. Molly describes such a building as "all tesseracty and Whovian!" This becomes a plot point when that building is destroyed, and all of its chunks of debris expand outward and fly away from each other as they "drop into normal spacetime.'' This saves a character who was trapped inside from being crushed.
- Doctor Ka's mansion in the Global Guardians PBEM Universe is effectively a tesseract, and is definitely bigger on the inside than on the out. If its layout was drawn on a set of blueprints it would feature rooms that overlapped spatially, rooms that seemed to have no exits or entrances, spherical rooms that nonetheless had corners, and rooms where the plane of gravity depending on which way you were looking.
- Fine Structure describes universes with more dimensions than our this way.
- The Knight Shift contains an upstairs hallway with doors that only lead out of the door at the end of the hallway. When that door is sealed shut they lead outside.
- Protectors of the Plot Continuum
- The PPC Headquarters. It is unclear whether it is just a confusing maze of a building or whether it can actually move around, but thanks to the Laws of Comedy, one of the only ways to find the place you are trying to go is to distract yourself and not think about it. However, it "was" built by alien plants, and they seem to be able to navigate it just fine.
- Also, poorly-constructed descriptions in the Word Worlds cause some rather eye-breaking visuals for the agents when the worlds try to put them into practice. In one mission, Agents tempted fate by saying "It's a wonder we're still in three dimensions."
- Buildings frequented by the Slender Man often develop these, sometimes reaching full-blown Eldritch Location levels. Don't expect a door to lead to the same room it did two minutes again.
- Marble Hornets: When Jay investigates Alex's abandoned house near the end of season one, he gets bounced around between rooms and eventually finds himself in the disused bathhouse from an earlier entry. The next entry shows something similar happening in his house.
- Everyman Hybrid: the cast is out on a hiking trip in the middle of a bright, sunny day, exploring an old abandoned house in the middle of nowhere, deep in a wooded area. A door in the basement of the house leads to a sprawling beach in the middle of the night. In a later episode, they find a crawlspace in Jeff's house that leads them to the aforementioned abandoned house miles away. Later still, Vinny finds a house that either warps him to different parts of the cast's homes, or is an equally weird mishmash-ed replica. That last one gets even weirder later on, when Vinny suggests he and his friends are fictional constructs and their "houses" have always been like that.
- Appears a couple times in Sevenshot Kid. Usually it serves as a prelude to something horrible.
- The Hell Hotel in the Halloween Episode of Tribe Twelve.
- One Hundred Yard Stare: When Avery, Macy, and Ellie first run from the Slender Man there is a good dosing of this, with them starting in a yard of some sort and ending up, after a jaunt in a building, next to a moving train.
- The Metal Glen from Ruby Quest displays aspects from this. First there's the metal shutter in Ruby's room, which sometimes opens to a window and sometimes to a passage. Then half of the Brig turns upside-down, gravity and all. Then it gets weirder.
- The Dionaea House. All of them. The one in Boise, for example, has a second floor that is not visible from the outside. It says something that this is not the strangest thing about it.
- Carmilla's room in the Whateley Universe. It keeps changing size and shape. Its door moves from building to building. It's possible to walk in and out of it without using any known entrance. There's a reason the staff at Whateley Academy calls it the H. P. Lovecraft Room.
- There is no geometry but Time Cube. You are educated stupid. Time Cube is so obviously true that knowing anything except Time Cube will drive you crazy when compared to Time Cube.
- The Fear Mythos has The Empty City, also known as the City of Empty Shadows or DEVOURER. It's a living city which intentionally makes itself into Alien Geometries in order to make sure its victims stay within it until they die. Or worse.
- Fredrik K.T. Andersson managed to invert this.
- Bravemule, the saga of a Dwarf Fortress, gave the dwarves an utterly bizarre culture with Blue and Orange Morality. Among other things, any shape that is not rectangular is "unscrupulous". An accidentally-created octagonal room was thus treated as horrible and incomprehensible, giving dwarves Catapult Nightmares and such.
- TV Tropes: The Inverted Trope example on PlayingWith.Mile Long Ship. How exactly do you get a ship whose length is a negative number?
- In Welcome to Night Vale the station management room is, as far as they can tell, too large to fit in the radio station building.
- Intern Dana ends up stuck in the forbidden dog park (Do not take your dog to the dog park. Do not go to the dog park. Do not look at the dog park. Do not think about the dog park.), and notes that although it looks like it's the size of a city block when you're standing still, she was able to walk along one of the black walls that surround it for two weeks without reaching a corner.
- When Dana gets stuck in the desert other-world, she gets trapped in a geographic loop; no matter how long she walks, or in which direction, she always finds herself heading towards the mountain with a blinking red light. Incidently, children in Night Vale are taught in school how to deal with geographic loops, using a simple memory device: knife.
- Don't forget the House That Doesn't Exist. It seems like it exists, like it's just right there when you look at it, and it's between two other identical houses, so it would make more sense for it to be there than not, but...
- Batman: The Brave and the Bold has a standard-issue Escher magical library in the Batman Cold Open of "The Eyes of Despero". Batman is largely unfazed by the shifting gravity, and actually uses it to good effect.
- The Foghorn Leghorn cartoon "Little Boy Boo" plays this for laughs. Foghorn is playing hide and seek with a child genius and hides in the coal bin. The kid performs a few calculations and then digs Foghorn out of the lawn. A very befuddled Foghorn protests that he was in the coal bin, but the kid just shakes his head and holds up the calculations. Foghorn then goes to look inside the coal bin, but decides "No, I'd better not look. I just might be in there."
- The titular Foster's Home for Imaginary Friends sometimes exhibits a non-malevolent version of this combined with Bigger on the Inside. In one episode, Mac is attempting to leave for dinner from the houses roof, but they go down a flight of stairs and through a door to end up back on the roof (prompting Mac to confusedly remark "But we went down."). They later fall down a trap door from somewhere in the middle of the house, sending them back to the roof again, and Mac declares "This is downright unnatural."
- Comic example: the Flanders's rebuilt house in The Simpsons episode "Hurricane Neddy" features many impossible features.
- In the Adventure Time episode "The Real You", Finn uses his magically-granted super intelligence to create a fourth-dimensional bubble. It looks like a cube wire-frame constantly inverting itself (a tesseract) before it collapses into a black hole.
- Soviet short animated film Pereval featured graphics by practicing topologist A. Fomenko.
- 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 (a 12-sided polyhedron). When you get to an outer face you Wrap Around to the opposite face. Except the faces don't exactly line up, so you also rotate one tenth of a rotation.
More aptly, it is to be observed that Euclidean geometry, mathematically speaking, is a special case: it only applies to forms in a space with zero curvature (for the two-dimensional case, a perfectly flat plane); something that is, strictly speaking, an abstract concept (in light of the fact that time and space are demonstrably curved by gravity.) Consider that you cannot, in Euclidean geometry, draw a triangle with three right angles, but it is perfectly possible on the surface of a sphere.
Sometimes made worse by the fact that a non-Euclidean two-dimensional geometry is often visualized as embedded within a three-dimensional Euclidean space (the surface of a sphere, or a saddle), which leads some people to mistakenly believe that an n-dimensional non-Euclidean space requires an unseen n+ 1 dimensional space. (It's not too difficult to imagine a two-dimensional space with positive curvature as the surface of a sphere. Now try to wrap your mind around the idea of a space where the geometry works out the same as it would on the surface of that sphere, but without any third dimension at all.)
There are five axioms used to make geometry. Change one, and you get non-euclidean geometry. Change all five, and you could very well get arithmetic. Calling it a special case is an oversimplification.
Speculations on the Topology of the Universe aside, it's clear, and even somewhat well known, as stated above, that the effects of gravity are described by the curvature of spacetime, which means that in truth, geometry is not Euclidean at all. As a famous test of this, we can see stars which should be hidden behind the Sun during a solar eclipse, due to the light following the shortest path in curved space towards us. Time is also curved, in a sense, as clocks will run slower in places where classically the gravitational potential is lower relative to clocks at greater potential. This effect too, was measured using high precision atomic clocks.
Some astronomers who like thinking outside the box suggested that one might put a satellite 550 AU away from the sun. At this point, the aforementioned curvature of the spacetime bends light just right, making it possible to use the sun itself as, essentially, the primary lens of a huge gravitational telescope. This idea is called a solar foci telescope.
- Spherical geometry isn't just something for universe-scale models. The surface of the earth is represented as a two-dimensional non-Euclidean space every time you look at a map. As mentioned previously, a sphere can have a triangle with three right angles on it, and the earth is (approximately) a sphere.
- An example: Start at the north pole. Go to the equator. Turn right 90 degrees. Continue along the equator. Turn right 90 degrees. You will reach your starting point. Nifty, huh?◊
- Right angles aren't really anything special. The issue is that a 90-degree angle only looks like two perpendicular lines intersecting at a point on a flat two-dimensional plane. Angles have a problem in 3-D space...they ignore the z-axis (depth). Lines are only perpendicular on the same plane.
- 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 perceived contraction of length is connected to the relativity of simultaneity, the fact that events which are simultaneous for one observer (person) can be not simultaneous for other observer. To measure length you have to mark where beginning and end of said length are at the same time; for observer moving with relativistic speed it would look like one first marked beginning, then end (after it moved).
The distance that the distance one through spacetime, the spacetime interval, is of a constant length, and is determined by the formula s = sqrt(x^2 + y^2 + z^2 - (c*t)^2). That is to say, although space contracts by a factor of sqrt(1-(v/c)^2)), time expands by an equal factor, so the spacetime interval that you cross remains constant. This means that the velocity of the thing that travels along the interval is already determined for any given observer. For the layman, this means that objects do not really shrink when you travel at velocity; they are actually just rotating in four dimensions, and just appear to shrink because we can only see in three dimensions.
- And that's just special relativity. General relativity predicts that objects in a gravitational field shrink relative to those outside of the field. Essentially, when looking from the outside in, massive objects like stars or planets are actually Bigger on the Inside. This is due to the fact that what we experience as gravity is actually the mass of planets causing space to shrink in its presence. This means that straight lines, like those of laser light, become bent in the presence of mass. This is why planets orbit other planets, and light gets bent in a gravitational field: they are following the curved lines of space around the planet. The best analogy of this is placing a mass on a rubber sheet, and watching it deform the rubber around it. Letting a ball spin around the mass is rather similar to planets orbiting one another, as the balls just follow the curved paths of the rubber sheet.
- Gaze upon the optical effects of special relativity. Drugs wish they could do this.
- The Bermuda Triangle, according to many theories and reports. In addition to vehicles vanishing without trace (no wreckage left), reappearing after disappearing from radar, etc. some people have reported experiencing "time warping" or "missing time" while traveling through here.
In reality, while several accidents have taken place there, they're not statistically more common than in any other area of sea with the same density of traffic. Which, in spite of the stories, is considerable.
- The Mandelbrot Set is a two-dimensional slice of a four-dimensional object that represents the eventual fate of iterating the assignment z <- z* z + c, where z and c are complex numbers (two dimensions each). Start with z=0 and try different values of c, and you get the usual two-dimensional view of the Mandelbrot set (which is, properly, only the boundary of the usually-black region representing points that do not escape to infinity). Fix a value for c and try different starting points for z, and you get a Julia set. The complete four-dimensional object stacks all the two-dimensional Julia sets along a complex dimension for a total of four real-valued dimensions.
- A tesseract is a "four dimensional cube". Just as a square is only one face of a cube, a cube is only one "face" of a tessaract. Think about that for a second. The vertex of the tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called the hexadecachoron and... oh no I've gone cross-eyed.
Scarier still, people have built computer models Rubix Hypercubes which people have successfully solved. For your headache-inducing pleasure. A 5D version has also been made. And then a guy got to seven...
If you want a quick view of just how confusing tesseracts can be, see here. Lovecraft would be proud. (Or actually he'd probably be nihilistically horrified at the alien, inhuman cosmos revealed by science, but whatever.)
Tesseracts are really just the start of a really long and dark journey. Even a Hilber cube is a really simple example when compared to some of the other stuff. It is also a cuboid of countably infinite dimension. Studying maths at uni sort gets you used to this stuff...
- Try solving a 4D maze. Just 3*3*3*3 takes about half an hour, and that's if you've gotten used to moving around in 4D.
- Try wrapping your head around the Moebius Loop. It's a loop with a half-twist with only one side and one edge. Similarly, the Klein bottle, which is a closed surface and only has one side, rather than having an inside and an outside. It has to go through itself at one point to make that work, though.
- Not really, that is just an artifact of embedding it in 3d space, in proper 4d space there is no such intersection.
- In 3d animation programs, when you make a polygon it's usually supposed to have 4 sides at the most and flat. In Maya, if you have something with more than four sides, edges that don't connect to a vertex or a vertex that's not connected to an edge, things get...weird. You end up with a shape that the program can't draw properly that will usually look like a different surface depending on where you happen to be viewing it from.
Programmers tend to do things like this a lot when they first step into the world of OpenGL and DirectX. Get the position of a vertex wrong when you're just messing around with increasingly more complex shapes, and you can easily find yourself with a headache just from looking at the shape you've made. This effect is how the Alien Geometry corridor in Legend of Zelda: Ocarina of Time was implemented. Some corridors in the Descent series are also twisted like this.
- Higher doses of LSD can produce visions of impossible shapes.
- Pittsburgh, Pennsylvania's downtown area is called the "Golden Triangle" and is indeed triangular. To the uninitiated, though some of the turns feel like 90 degrees as you're trying to navigate. Making three "turns" and ending up back where you started has flummoxed more than one out-of-town driver (and more than its share of locals, too).
- One Pittsburgh native suggests that in order to truly understand the city's geometry, eleven-dimensional string theory may be necessary.
- Though Pittsburgh's geography pales in comparison to the tangled skein of one-way and weirdly contorted roads across The Pond that are known as London. Indeed, The Knowledge is widely agreed to be the most demanding test of mental prowess ever devised, anywhere.
- This Reddit comment is a fictional yet real physics-based story that illustrates just how nefarious black holes are.
- While the main point (the fact that all trajectories point to the singularity) is correct, an observer would *not* see the universe outside the black hole as if they had tunnel vision. This is a result of falling through the black hole making an effort not to fall, i.e., with your engines on. Then the tunnel vision would be the result of relativistic beaming, not the fall. This site contains a much more accurate account, with a video: http://jila.colorado.edu/~ajsh/insidebh/intro.html
- Also, the article correctly notes that once inside the event horizon, all points outside lie in the past. However, it then proceeds to draw incorrect conclusions from it. You would not be blind inside the horizon, because far from not being able to see into the past, you can ONLY see into the past. When you see something, you see it not as it is at that moment, but rather what it was when the light reaching your eye was emitted from it. Furthermore, a Faster-Than-Light drive would not be prevented from escaping from a black hole, because such a drive is perfectly capable of traveling into the past as measured by certain frames of reference.
- "Mystery spot" attractions use tricks of architecture to simulate this trope. Gravity hills do the same thing, but with natural tricks of topography.
- Optical illusions can convey an impression of Alien Geometries, by playing tricks on how the brain's visual association area interprets perspective from proportion and shading.
- How to turn a sphere inside out.
- How about an alien colour? Pink is a purely mental perception of a mix of red and violet light that doesn't actually exist on the visible spectrum. To an alien or whatever, it might look as though all of our little girls are engaged in some creepy cult!
- To clarify this, the color wheel isn't accurate. The visible spectrum goes from infrared into red, and from violet into ultra violet. Because they are at opposite ends of the spectrum, red and violet don't actually cross the way the color wheel suggests they do. That fade between them that we see as a specific color (pink) is all in your head.
- Pink is what's called a non-spectral color. Other non-spectral colors include the grayscale colors and mixtures of such with other colors, and what's called the line of purples. As the name implies, non-spectral colors have no unique frequencies in the visible spectrum, and exist as a perception of multiple frequencies.
- Imaginary colors are even stranger. These are colors we could see, but no light can ever make.
- Impossible colors (not to be confused with the above) are some sort of inversion of imaginary colors. They are specific combinations of existing hues that would mix to something that isn't a "full" mix of the two when viewed by a standard human. An example would be blue-yellow. When mixing paints of the two, this makes green. When mixing lights, they combine into white. However, when using some kind of mind trick such as crossing your eyes on a certain image (which is available in the page for impossible colors), it is possible to mentally see a color that looks part-blue and part-yellow.
- Both Google Earth and Apple Maps provide a 3D representation of the Earth's surface, made up of a combination of 2D aerial photos and 3D topography information. Sometimes disagreements between this data result in images of melting trees, spaghetti-like roads, bizarrely deformed buildings and more. See: Postcards From Google Earth, Glitches in Apple Maps.
- Gabriel's Horn is a geometric figure that has a finite volume, but infinte surface area.
- In ultrametric geometry, all triangles have two sides of the same length, repeatedly moving a short distance doesn't result in moving a longer distance (no matter which direction you move), and every point inside a ball is its center.
- Anecdotal accounts of dreams suggests that the parts of our brain in charge of three dimensional space and short-term memory don't work all that well when we're asleep, resulting in passages to nowhere, doors that weren't there before (or the "there was a hole here, it's gone now" effect), or sudden jumps in physical location.
- Curved Spaces is a program that shows off quotient spaces of manifolds of constant curvature. One of the simpler ones is a dodecahedron where each of the pentagons is made entirely of right angles.