Non-Euclidean

[mcglothlin.13]

This will probably be the biggest egghead post of all time on the HPLLP, but here goes:

I've always been intrigued by Lovecraft's description of R'yleh as "non-Euclidean".  The adjective seems to evoke a sense of visual weirdness, especially given the sailor Parker's demise.  However, it's not really all that mysterious for an object to be non-Euclidean, nor need it imply something that's non-picturable.  Non-Euclidean geometries have been fairly common since the 19th century.  For a geometry to be non-Euclidean this merely means that, at minimum, the parallel postulate of Euclid is denied.  That postulate states: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.  But an object that exists on a plane where this postulate fails is not really difficult to imagine (see hyperbolic and elliptic geometries as examples).

Question: Given this above information, did Lovecraft know what "non-Euclidean" actually meant?  If so, does that take away any of the weirdness of the architecture of R'yleh?  If Lovecraft didn't know the meaning of "non-Euclidean", does that make his use of this adjective juvenile or amateur?

[Barrucadu]

I think he was just using it to convey a sense of weirdness — really, most architecture (that I've seen) follows Euclidean geometry, so a city where it's almost all non-Euclidean would be pretty weird to behold.

[Genus Unknown]

Well, what word would YOU choose to describe the brain-melting geometry of R'lyeh?  As I understand it, calling a geometry "non-Euclidean" is sort of like describing some newly-discovered physical law as "non-Newtonian."  It's a negative term, telling us what this geometry is not, not what it is.  We can point out non-Newtonian physics, like quantum physics, that can be understood, while allowing (at least in fiction) for some different, alien set of laws that are infinitely weirder, and which also contradict Newtonian physics.  By that token, while there are certainly non-Euclidean geometries, like elliptic and hyperbolic geometries, that are common and not hard to imagine, there are also (at least in fiction), other, much more bizarre geometries, like that of R'lyeh, that also deny various postulates of Euclid, and thus could be accurately termed "non-Euclidean."

Also, there's already a discussion of this here.

[mcglothlin.13]

Genus Unknown: I'm not being critical of Lovecraft's use of the adjective; my point was that once you understand what the bare minimum definition of non-Euclidean means, it doesn't quite evoke (at least for me) the same visual weirdness that it once did.

I don't think the Newtonian/Quantum distinction is analogous to the Euclidean/non-Euclidean distinction.  Basic logical laws that operate in the Newtonian universe seem (the key word there is "seem", we may not know enough yet) to not hold in the quantum universe.  That's not the case for the geometrical distinction though.  The same logical laws hold in the non-Euclidean realm as they do in the Euclidean one, there's just (at least) one less axiom (i.e. the parallel postulate).

You do give a good suggestion though: there are many kinds of non-Euclidean geometries that are completely inconceivable (including for Escher!).  Perhaps Lovecraft had one of these incredibly bizarre geometries in mind that deny many Euclidean notions and not just the parallel postulate.  However (and this is just nitpicking at Lovecraft at this point), it doesn't seem R'lyeh can be too non-Euclidean when you have a bunch of Euclidean sailors walking around on it!

Thanks for the post link.  I was unaware that a discussion of this had already taken place.

[Cacodaemoniacal]

One way I think of it (besides as a cubist or futurist painting) is if you could see an angle of a building as it intersects a plane of time--or sound--or some other concept we might have a hard time understanding visually without an explanation.

It seems to conflate experience and perception with underlying geometry or physics; but it works. 

Perhaps it's a description that is probably way more compelling to people with less rather than more math/physics under their belt.

[Rob]

Saw a preivew ad for a film last night - can't remember the name but it was something to do with controlling people's dreams. Anyhoo there's one bit with people in a city street and the rest of the city kind of..bends around them..it's like they are looking down on it instead of aideways at it. I thought that looked a bit non-Euclidean!

Ah it's called Inception   0.20 seconds

http://www.youtube.com/watch?v=S3XzUYd6nrU

[mcglothlin.13]

Perhaps it's a description that is probably way more compelling to people with less rather than more math/physics under their belt.

This may be hero worship but it seems to me that would be a very un-Lovecraftian thing of H.P. to do.  I have a hard time believing that he would just "hack" some technical adjective purely for some emotive purposes. 

But . . . perhaps he did just that!

[Cacodaemoniacal]

When you are trying to faithfully capture the meaning in art , it's seriously crazy making.

But honestly, non-Euclidean gets the idea across really effectively with an economy of words. The old fashioned style of writing that Lovecraft uses can seem unnecessarily verbose. But when done well,  it can create a musically complex cascade of information. I think it's part of the genius of Lovecraft.

I guess if you were to be really, really precise, it might be more correct to say a city was built according to non-Euclidean geometry. But this works.

I'm really curious about what different types of readers take away from his word use though.

[lorddoomicus]

wonder if I could use Bryce or Pov-ray to make some nice examples of "Non-Euclidean" geometry?  Might be easier in Pov-ray because it does recursion.

- Derrik

[Genus Unknown]

You might get the appearance of regular, Earthly non-Euclidean geometry as mathematicians know it, but you couldn't get the crazily-angled alien non-Euclidean (and non- a damn sight many other things, I'd say) geometry of R'lyeh, because that kind of geometry doesn't exist and can't describe a real, physical thing.  It's written specifically so as to be impossible and unimaginable.

[mcglothlin.13]

It's written specifically so as to be impossible and unimaginable.

How so?  If it's completely impossible how did these (Euclidean) sailors walk on it?  This may be splitting hairs but, when you try to think about it, I think it shows that Lovecraft's use of the adjective "non-Euclidean" is meant mainly to evoke an alien character about the architecture of R'lyeh.  It shouldn't be taken as a literal mathematical notion.

[Kaelestes]

It's written specifically so as to be impossible and unimaginable.

How so?  If it's completely impossible how did these (Euclidean) sailors walk on it?  This may be splitting hairs but, when you try to think about it, I think it shows that Lovecraft's use of the adjective "non-Euclidean" is meant mainly to evoke an alien character about the architecture of R'lyeh.  It shouldn't be taken as a literal mathematical notion.

I disagree. Lovecraft spent his entire writing career trying to come up with methods of hinting at ideas, and sights, and sounds that would be so foreign to you and me that we might just go crazy for having experienced them. He wanted to defy the imagination's ability to even conjure them up. The sailors walked on R'lyeh because that's how Lovecraft wrote it. Therefore, it's plausible to believe that the sailors could in fact interact with these creatures and places even if they follow none of our physical laws and have an adverse effect on our minds and bodies as a result. Foreign physics and foreign matter - ol' Cthulhu himself was made of something 'else' - mean foreign mathematics. These are the concepts that make Lovecraft's work so incredible and frightening. Without them the material would be thoroughly blasé.

EDIT: The sailors aren't just Euclidean. You and I aren't just Euclidean either. The lines and curves of our bodies intersect and rebound and run every which way. Our geometries are very complex.

[Genus Unknown]

Yeah, I think of the geometry of R'lyeh as something equivalent to the Colour Out of Space - something literally unimaginable, something impossible, something that can't exist but just goes on existing in Lovecraft's world anyway.  What he describes in the story - angles that seem to make concepts like "horizontal" and "vertical" meaningless, angles that are convex at first glance and concave the second, angles that are acute but "behave" (whatever that means) as though they are obtuse - these things have no real-world counterparts.  They don't describe any kind of geometry, Euclidean or not, that exists in the real world.  They just serve to create the impression of something stupefyingly alien, something that could drive you crazy if you stare at it long enough.  It's not supposed to make sense.

[lorddoomicus]

Maybe I could write a raytracer that uses some kind of non-linear dynamics or chaos system to make up strange looking worlds ... mmmmm... got me thinking now ...

- Derrik

[Kaelestes]

Maybe I could write a raytracer that uses some kind of non-linear dynamics or chaos system to make up strange looking worlds ... mmmmm... got me thinking now ...

- Derrik

Is that some kind of a graphics generator?