What is non-Euclidean Geometry?

[chrisblue77]

A system of geometry that doesn't accept Euclid's Fifth or parallel postulate. Two types of Non-Euclidean geometry emerged in the nineteenth century: Lobachevskian (hyperbolic) and Riemannian (elliptic).

from More Annotated H.P Lovecraft
So douse that mean that the place acts like an M. C Escher drawing?

[Rob]

That's how I've always envisioned it - angles that don't make sense. Like that nice bit in the CoC movie where the sailor falls into the angle and disappears

[Genus Unknown]

I don't think it would be quite like an Escher painting, but much like quantum physics, I haven't the education to go into it or even form a vague idea of it.  It's a real branch of mathematics, though.  One of my math teachers in high school did his thesis on it. I remember because I had already read Lovecraft by the time he mentioned it, and had one of those "holy shit it's all true" moments the guys talked about in part 1.

Here's a Wikipedia page on it that I don't understand.

The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ?  and a point A, which is not on ?, there is exactly one line through A that does not intersect ?. In hyperbolic geometry, by contrast, there are infinitely  many lines through A not intersecting ?, while in elliptic geometry, any line through A intersects ? (see the entries on hyperbolic geometry, elliptic geometry, and absolute geometry for more information).

Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line:

    * In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels.
    * In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
    * In elliptic geometry the lines "curve toward" each other and eventually intersect.

For the layman, non-euclidean geometry can be understood by picturing the drawing of geometric figures on curved surfaces. For example, the surface of a sphere or the inside surface of a bowl.

If you understood any of that, you paid a lot more attention in school than I did.

[Rob]

Lol, to my limited intellectual capabilites that kinda sounds like a fish-eye lens effect...which I guess is sort of appropriate 

[Kaelestes]

It took me a couple read-throughs and a reference to another page, but I think I've got a good handle on it. In Euclidean geometry, if you have a line of infinite lenght on a 2D surface (length & width with no depth) and you also have a point on that 2D space which does not lie anywhere on the first line, there is only a single line you can draw through your point which will not intersect with the first line, and that means the second line must be parallel to the first.

Hyperbolic and Elliptical geometries are much more difficult to explain without getting into the math behind them, but I'll give it a shot. Rob is on the right track with his fish-eye lense interpretation. In non-euclidean geometry, the straight line becomes a curved line as it approaches its theoretical termination point. As you can imagine, this makes them vastly more complex. For those interested, the basic equation is: ax^2 + bx + c (the quadratic equation), where "x" is the variable. In hyperbolic geometry, these lines curve away from one another, so that an infinite number of these curved lines could be drawn through your point which would never cross the first line. Elliptical geometry is simply the opposite, wherein all lines drawn through your point will eventually cross the first because they curve toward eachother. I'm not sure why the distinction between these two types is made, but there you have it.

Here's an MC Escher painting called 'Circle Limit III' which uses a complex interworking of hyperbolic geometric shapes:


Genus Unknown, I really like the idea of a city displaying properties of quantum physics, which is leaps and bounds more complex -- bordering on bizarre -- than anything I've described above. I took an advanced Chemistry class last term that went into describing quantum properties of elementary particles in depth, and which had a similar effect on my brain as Cthulhu must have on those unfortunate enough to get a good look.

[Ruth - CthulhuChick]

This is my desktop background and one artist's conception of what a city (R'yleh, of course) of non-Euclidean geometry would look like. Not my two cents on the matter, but it does kind of bend my mind so I thought I'd share. It's like Escher, but a bit twistier. I'm a big fan of Euclidean geometry.

[Cacodaemoniacal]

So, is Cthulhu's universe non-euclidean and ours is euclidean? Seems like those other geometries are conceptions of our universe. Lovecraft is saying Cthulhu's universe is constructed differently and Cthulhu is part of both.

[Danial]

They way envision it, is that the "straight" edge of, say, a block, might look straight from where you're standing, but then as you move it might bow to the left or the right, kind of like one of those fun-house mirrors

[Cacodaemoniacal]

I guess all he really says is that it's not Euclidian. Could be any construction. Can 3 dimensional space be Euclidian/non-Euclidian?

[Kaelestes]

Yes, both Euclidean and non-Euclidean geometries exist in three dimensional space. It's simply easier to describe them on a 2D plane. I would venture to guess that Lovecraft's use of the term was designed to suggest an unearthly geometry beyond even the hyperbolic and elliptical; something so strange to the eye it would seem inconstant or unreal and defy mathematical description.

[Genus Unknown]

So "non-Euclidean" isn't exactly a specific field comparable to, say, quantum physics, but rather a catch-all term for a number of other geometries outside or in contradiction to Euclid, comparable to a term like, say, "non-Newtonian physics," right?

[Kaelestes]

That sounds like a reasonable assumption. And no, the field has nothing to do with quantum physics. That's a completely different nightmare.

[Genus Unknown]

When I say "comparable to quantum physics," I mean in the sense that quantum physics is a distinct field of study, as opposed to NEG, which seems to be a broader, "negative" term that simply refers to any geometry that isn't Euclidean. Or something. I don't know, man, this discussion is frying my brain already, and I'm not even talking about the actual mathematics of it.

[Cacodaemoniacal]

so, in 3D, the lines would become planes? They would bubble instead of be flat...so a 3D fish lens?

But then, I find myself unsure how Euclidean geometry relates to how the world looks. Do walls look flat because of Euclidean geometry or because of our senses?

[Kaelestes]

It's important to understand that these are simply means with which to describe the different types of geometries we see every day. All shapes have one dimensional, two dimensional, and three dimensional properties. The Euclidean and non-Euclidean titles are just methods of breaking down the shapes we see into simpler forms. Knowing this is beneficial both mathematically and functionally - as with architecture, or in this case literature - but bears no real value to you or me. As I mentioned before, Lovecraft was likely using the word to emphasize the unknowable. I believe the intent was to create the sense that seeing R'lyeh was unlike anything you could possibly describe because it defied conventional reality's physical laws of mass and volume. And hence it was non-Euclidean, or perhaps non-geometric.