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666-4850--- A normal-looking house that, on closer inspection, has no right angles.
"....but you start with parallel lines that intersect and you go from there."
A sub-trope of Cosmic Horror, mainly found in print works, due to being difficult-to-impossible to show visually.
Elder Gods, Old Ones, and other horrors tend to bend the laws of physics to suit them. Why make a triangle where the angles add up to 180 degrees, when you can make one where they add up to 200 degrees and get some extra space? (This can happen, in a relatively mundane scenario; on a perfect sphere, you can have triangles where the angles sum to 270 degrees since the lines are curved. Now imagine being able to unfold that sphere into a flat surface and leaving the 270-degree triangle intact, and you get some idea of what we're talking about.) Even the very body of a particularly squamous thing may exhibit this, though more often it shows up in architecture as physically-impossible buildings-- occasionally sentient themselves.
Alien Geometries are often depicted as being dangerous to the sanity of normal humans; where you have to read the Tome Of Eldritch Lore for it to drive you crazy, just looking at this stuff can have an unpleasant effect on your mental stability. Or at least really hurt your eyes.
See also Hyper Space Is A Scary Place, wherein it's less a single wall or building that's just a little... off and more an entire alternate universe that just... doesn't make... sense.
Examples:
- 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.
- An indie short film of The Call of Cthulhu does a particularly good job of getting this idea across, in a scene (faithfully adapted from the story) wherein a victim falls into a crevice which an optical illusion has led the audience to believe is a convex crag of rock.
- An extreme example occurs in The Colour Out of Space (Lovecraft always insisted on the Anglo-Canadian spelling), where a horror from space causes an impossible colour (that everyone can actually see) to appear on nearby objects.
- 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.
- 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 Cirle Limit sketches for example). And yes, his work does have an impact on one's sanity...
- In the game 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.
- 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.
- 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.
- 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...
- Mark Z. Danielewski's House Of Leaves starts with a house that is one-quarter inch larger on its inside than on its outside, and things just get worse.
- 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 . . .
- 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.
- Also, The Colour Of Magic featured a parody of Alien Geometries : the Temple of Bel-Shamarroth (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 octagonal tiles. Try and make a drawing of it. Go on. We're not going anywhere.
- The near-universal hallmark of things made in the name of Chaos in Warhammer 40000.
- The Aelfinn and Eelfinn ("the Finn") from The Wheel Of Time inhabit one or more separate dimensions described by the author as having radically different natural laws. Successive windows do not show what one might expect. That the magic system in the series is heavily geometric likely has a great deal to do with why its use is explicitly forbidden there.
- There's a Dungeons And Dragons module entitled Queen of the Demonweb Pits that involved the players venturing 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.
- There are objects, important in modern physics, called spinors
 (and if that link seems incomprehensible to you, be glad I didn't link the Wikipedia page), that do not return to their original position after being spun around 360 degrees but instead require a 720-degree rotation. And in quantum physics, they are used to describe just about anything that can be described as "matter"--protons, neutrons, electrons, etc. I Am Not Making This Up.
- 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.
- 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...
- GURPS: Illuminated 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.
- The movie 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).
- 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.
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