### Non-Euclidean Geometry, or Even Cthulhu Has an Angle

*…He had said that the geometry** of the dream-place he saw was abnormal and loathsomely redolent of spheres and dimensions apart from ours…*

*…twisted menace and suspense lurked leeringly in those crazily elusive angles of carven rock where a second glance shewed concavity after the first shewed convexity…*

*…The rest followed him, and looked curiously at the immense carved door with the now familiar squid-dragon bas-relief…..they could not decide whether it lay flat like a trap door or slantwise like an outside cellar door…*

*..it moved anomalously in a diagonal way, so that all the rules of matter and perspective seemed upset…*

*…Johansen swears he was swallowed up by an angle of masonry that shouldn’t have been there; an angle which was acute, but behaved as if it were obtuse…*

** –****Miscellaneous quotes from “The Call of Cthulhu” by H. P. Lovecraft.**

When I first read those words. a million years ago, I wondered what the heck they meant. I have a pretty good visual spatial ability, I can picture things in my head pretty accurately, but I could not quite get a grip on an obtuse angle that behaved as if it were acute. Also, I’d had enough trouble with *Euclidian* geometry that my self-preservation instinct kicked in and told me it was something Lovecraft had just made up. Worked for years.

Later in life, I found that Euclidian geometry, like most of the other things we learned up through middle school was, at best, not the whole story. Editors hoping for a spot at *Reader’s Digest* had reduced the real world to the 1960s equivalents of sound bites, i.e, Columbus discovered America, everyone in America was for the 1776 revolution except the traitors, no Eastern country had ever contribute anything worth knowing besides the *Rubaiyat of Omar Khayam* (English translation). There actually *was* something called non-Euclidian Geometry. But I had enough interests already, and didn’t pursue the subject.

Re-reading “The Call of Cthulhu” a couple of weeks back, it struck me that here was a ready-made subject for a blog post! I’d learn something, you’d learn something, and our appreciation of HPL would be thereby enhanced. I had Wikipedia for research, plenty of time to write the blog. I wouldn’t have to get deeply into the math, just explain the general purpose and nature of non-Euclidian geometry.

I’ll pause for a moment to let you bring your laughter under control.

Some infinities are larger than others –the set of whole numbers and the set of odd numbers are both infinite, but there are more whole numbers than odd– but the Wiki article on non-Euclidian geometry may be largest infinite of all. Every third word is a link to a page of definitions with its own links to pages with their own links and so on and so on. The only way to get back to your starting point is the hope that Wikipedian space is curved, and eventually you will arrive back where you began.

But I did try. I ‘m including the link for non-Euclidian geometry, but I feel honor bound to talk about what I *think* I understood.

Like all incredibly complex issues in the world, non-Euclidian geometry came about because some crank decided to solve a problem the majority of us never even knew was there. Euclid had decreed five postulates which became the bases for geometry as I remember it from the eighth grade:

1. Between any two points is a straight line

2. A straight line can be extended infinitely in either direction

3. Any center and any radius can describe a circle

4. All right angles are equal

5. Any two straight lines equidistant from one another at two points are infinitely parallel

This last one is the problem. Euclid stated it a little more complexly as “*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.*”

Either way, from the very first, mathematicians found the parallel postulate too complicated to hang out with the previous four. As early as the 11th century, Arab and Persian mathematicians were noodling with it, and discovered elements that would later become part of non-Euclidian geometries. The notion of alternate geometries really caught fire in the 19th century, essentially establishing two broad alternate forms, hyperbolic geometry and elliptic geometry. If I’ve waded through all the descriptions of manifolds, tensors and nonnegative Ricci curvatures correctly, then there are multiple flavors of each of those alternate forms.

The basic difference between the three is this:

In hyperbolic space, lines move away from each other, in elliptic space, the curve toward each other. In hyperbolic space, there are many (though apparently not infinite) lines that can pass through a point without intersecting; in elliptic space, all lines intersect. Plus some other stuff.

Elliptic geometry is easier for me to grasp, and I can see its relevance to relativity (like I understand* that*) in that curved objects, even on s scale we can deal with, like a globe, can mess with Euclid’s notions. A large triangle drawn on your handy earth globe will yield angles that add up to slightly more than 180 degrees. And this has something to do, I believe, with a statement made by my math teacher in seventh grade, just before I transferred to a different school:”at infinite, points form a circle so large that it is a straight line.” Forty some years later, I still want to know what the heck that means. If I knew his name, I’d call him right now and threaten him with a horror not even Lovecraft could have imagined : recitations of songs from the Teletubbies until he explained it to my satisfaction. (Now that I think about it, the guy kinda *looked* like Lovecraft…)

Hyperbolic geometry is beyond me, as is Riemannian geometry, wherein the ‘curvature’ of a line varies from point to point, i.e., space is flat except where it is

distorted and curved by a mass. I think.

At any rate, after all this headache-inducing study, I still can’t quite picture the *‘angle which was acute (less than 90 degrees), but behaved as if it were obtuse (more than 90 degrees) ‘ *or the* ‘** angles of carven rock where a second glance shewed concavity after the first shewed convexity.’ *

If things look one way, then another, does that mean R’yleh is actually sliding between two *different* spaces? Even in a manifold, a particular point doesn’t have two contradictory shapes.

Finally, how does an angle ‘behave?’ Does Cthulhu ever have to say, “Bad angle! Shame on you.” ?

Sorry. Non-Euclidian geometry is confusing to me. I am usually so acute, but now I’m feeling….obtuse.

September 9th, 2009 at 6:31 am

Very interesting – thanks for posting this. A minor but hopefully interesting point which I feel I should mention is that the infinity of whole numbers (integers) and the infinity of odd numbers are actually the same: both sets are what mathematicians call “countable” or “countably infinite” (more technically, they’re said to have cardinality “aleph zero”, which is the first in the hierarchy of infinities). The reason for this is that one can define a one-to-one correspondence between the set of integers and the set of odd numbers (map each odd number 2n+1 to the integer n and vice-versa).

More weirdly, it turns out that the rational numbers (those which can be expressed as fractions p/q) are also countably infinite, and hence there are the same number of them as there are integers (there’s a nice proof of this rather counterintuitive result due to the 19th century German mathematician Georg Cantor).

If we look at the real numbers, though — those which can be expressed as (possibly infinite) decimal expansions, including the integers and the rationals, but also irrational numbers like pi and the square root of 2 — then there’s another nice argument (also due to Cantor) which says that these are “uncountably infinite”. That is, we can’t set up a one-to-one correspondence with the set of integers. The reals are said to have cardinality “aleph one”.

This is, of course, deeply strange: by our usual intuition for arithmetic, clearly there should be twice as many integers as there are odd numbers. I guess the moral of all this is that, just as we have to be careful applying our usual, Euclidean intuition to non-Euclidean geometry, we have to be careful applying our arithmetic intuition to weird concepts like infinity. (Actually, Cantor suffered a number of psychological problems while figuring all this out, although at least some of that was due to some ongoing academic politics that he found himself on the wrong end of.)

September 9th, 2009 at 6:36 am

Uh…ok. do you always have thoughts like these this early in the morning?

I actually took that statement of some infinities being larger than others from a book on a ‘solution’ to

Fermat’s Last Theorempublished five or six years ago. Sigh. You just can’t trust anyone anymore.September 9th, 2009 at 7:41 am

Interesting, thanks.

The descriptions you quoted from HPL always reminded me of optical illusions, such as the one of the triangle made of beams that are at right angles, or the way a line drawing of a cube can appear to be facing different directions. From this, I assumed he was trying to convey the presence or influence of a higher “dimensionality” than humans can readily perceive. After reading Flatland, I imagined that some of HPL’s horrors were like “hypershapes” come to pop us regular shapes out of our comfortable ignorance.

September 9th, 2009 at 7:54 am

Nice noodling, there. You mention something here that I’ll take a stab at:

“And this has something to do, I believe, with a statement made by my math teacher in seventh grade, just before I transferred to a different school:”at infinite, points form a circle so large that it is a straight line.” Forty some years later, I still want to know what the heck that means.”

I believe what your teacher was describing is just the limit (in Euclidean space).

Start with a piece of paper and cut a square hole in it (let’s say one inch square).

Take a small circle, say a coffee cup. Lay a pencil across it so that it passes through the center of the circle. Then line up one edge of the hole in the paper with the pencil and move it until the top corner of the hole reaches the top of the circle. Now look at the arc of the circle that shows through the hole. It has a noticeable curve to it: the circle hits the other side of the hole quite far from the top of the hole.

Now try with a larger circle. A pizza pan, say. The curve is less noticeable. It hits the other side closer to the top.

Continue with larger and larger circles. The bit you see through the one-inch-square hole is less and less curved.

When you get to a circle with radius large enough, your eye won’t be able to discern the curve at all.

And when you get to infinity, there will actually be no curve at all: the part of the circle that passes through the hole will be a straight line, as will every other part of the circle.

September 9th, 2009 at 8:07 am

Back to the main point of your post:

BEGIN QUOTE

At any rate, after all this headache-inducing study, I still can’t quite picture the ‘angle which was acute (less than 90 degrees), but behaved as if it were obtuse (more than 90 degrees) ‘ or the ‘ angles of carven rock where a second glance shewed concavity after the first shewed convexity.’

If things look one way, then another, does that mean R’yleh is actually sliding between two different spaces? Even in a manifold, a particular point doesn’t have two contradictory shapes.

END QUOTE

Yes, I think so. Something like the space manifold sliding from hyperbolic to elliptic (at least, locally).

Or if you want to get fancy, you could try to express it as a static area of space-time that is cruved in such a way that as you move along it in the time direction, the space directions you perceive are curved in different ways.

BEGIN QUOTE

Finally, how does an angle ‘behave?’ Does Cthulhu ever have to say, “Bad angle! Shame on you.” ?

END QUOTE

Here Lovecraft may “just” be describing warped space (non-Euclidean space). A triangle in hyperbolic space would have angles that sum to less than 180 degrees. If you construct it with two angles less than 45 degrees but large enough so that the other angle, which should be obtuse in Euclidean space, is acute, then you get an angle which is acute but behaves as if it were obtuse (to someone who interprets his perceptions only in Euclidean terms).

September 9th, 2009 at 9:47 am

Thanks, honestly. I’m impressed that so many math geniuses (or is it geniii?) read LIM. I actually do find this stuff fascinating, I am just not educated enough in mathematics and geometry to fully appreciate it.

I think you guys, Scott and Nicholas, should seriously take a crack at rewriting the Wikipedia article on this subject. As now, it’s written for people who already understand a large part of it rather than a lay person –a fault of many technical articles on Wiki– and you two have a real knack for explaining concepts clearly, at least for me.

As for the straight line at infinity, I had dope out something similar myself, though I could not have provided such a clear example. It’s still a neat thing to think about, and considering I was only in that class about two weeks before we moved, I doubt that any other teacher, even the best of them, said anything I could quote over forty years later.

And the explanation for the angles is great. I was sure that it was something Lovecraft made up, and probably, it was; but there’s nothing that says fiction can’t be true. At least in Euclidian space.

September 9th, 2009 at 11:11 am

Wait, if I pull on the Omar Khayyam thread will this whole post unravel to reveal the gaping maw of interstellar space?

Jantar Mantar Marw-i-Mawarannahr…Ulughbeg, ugh… Uraniborg.Ack, kha. Sorry, what was I saying?September 9th, 2009 at 11:50 am

We live in a non-Euclidean world.

The Earth is a sphere: as a result, two meridians crossing the equator at a 90° angle will converge at the poles… Making a triangle with two right angles! ^__^

And space is curved by the mass of stars and galaxies.

Makes us wonder why Euclid made his postulates in the first place. I can’t seem to find any natural example of truly Euclidean geometry… Everything is made of spheres or curves…

September 9th, 2009 at 12:41 pm

Don’t pull that thread!!!Seriously, very interesting links, thanks.

September 9th, 2009 at 1:20 pm

Possibly your math teacher was describing “clircles”. Look at the Moebius transformations on the complex plane

f(z)= (az+b)/(cz+d) where a*d-b*c=1 and a,b,c,d,z are complex numbers

They all take lines and circles to lines and circles. If you add infinity to the complex plane to make the Riemann sphere, the lines just end up being circles which close up at infinity.

September 9th, 2009 at 1:33 pm

“Uh…ok. do you always have thoughts like these this early in the morning?”

Sometimes, yes, but not today – I’m in the UK so it was about lunchtime when I posted that

September 9th, 2009 at 2:49 pm

Well, that certainly clears

THATup.Actually, I have no idea what you just said…but it’s no less entertaining for that.

September 9th, 2009 at 7:55 pm

That clarifies a lot, actually. Should be helpful when next I embark upon Lovecraftian art. It sent me on a fun little wikipedia “see also” loop of stuff I should have learned when dating a mathematician.

September 10th, 2009 at 4:01 am

Bless you, sir. May the peculiar cats from Saturn keep all saddening manifestations from your door.

I mainly want to say:

(1) Good starting take on non-Euclidean geometry.

(2) The reason for the labyrinthine Wikipedia style is that almost all maths has been arranged in that style since the thirties, in consequence of an information explosion caused by too many mathematicians. It’s regrettable. There are good readable textbooks, but you have to spend months finding them.

(3) In normal space, we can depend on light-rays to be straight, and to tell us reliably whether lines are straight or curved, how big things are when seen in the distance, and so forth. In a curved space, light-rays will actually do the kind of thing we only think we see in optical illusions. So it’s not nonsense to imagine that what looked like an acute angle could turn out to be obtuse once you got up close. That could actually work consistently. But there are some illusions, like the “impossible cube”, which could be built, but would only seem to be complete figures from a few special angles.

(4) The following splendidly Lovecraftian writing turns up in Douglas Hofstadter’s

Godel, Escher, Bach: An Eternal Golden Braid. The Hungarian mathematician Janos Bolyai was one of the two independent discoverers that hyperbolic geometry was in fact a consistent logical alternative to Euclid. People had been trying to prove the Fifth Postulate for years, and one of them was Janos’ father, Farkas Bolyai — who tried to persuade his son to give up on the wretched quest:You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy from my life. I entreat you, leave the science of parallels alone … I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. For here it is true thatsi paullum a summo discessit, vergit ad imum. I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind … I have travelled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness —aut Caesar aut nihil.September 10th, 2009 at 4:55 pm

Tsk. I’m sad now. Here was I looking to explain how infinitely large circles are straight (not to mention point out that there are as many whole numbers as odd numbers), and people have beaten me to it! For shame ;D

But it’s nice to see someone trying to address the interpretation of Lovecraft’s strange geometries, rather than just glossing over them.

September 10th, 2009 at 6:22 pm

Blind beat me to it, but yes, lines on a globe are non-Euclidean.

More fun is that if you draw an equilateral triangle on a sphere, you get three right angles.

September 10th, 2009 at 6:52 pm

Godel,Escher,Bachhas been on my shelf for thirty years. I think you’ve just given me the incentive to read it.The quote also reminds me of the old mathematician in PI (keyboard symbol doesn’t work in WP evidently.)

September 10th, 2009 at 6:57 pm

I mentioned the lines on a globe in my blog, mainly because it was one of the few things I’d read that I could understand. But just you don’t feel left out, tell me what the difference is between a globe and sphere, other than maps.

September 11th, 2009 at 10:34 am

Your example images for the geometries aren’t quite right…They should be more like (excuse crude ASCII art):

————-

| |

| |

————-

Since the fifth postulate says “Any two straight lines equidistant from one another at *TWO* points are infinitely parallel” (emphasis added).

Not sure how you’d draw the other two…I certainly won’t attempt them in ASCII art.

Another interesting book that is at least tangentially related to the subject is “Hyperspace” by Michio Kaku. He talks about how the dissemination of mathematical discoveries of more than three spatial dimensions led to works of popular fiction like Flatland. I always thought Lovecraft’s writing reflected some of the confusion that people were experiencing as a result of the dissemination of really non-intuitive discoveries in science and math (e.g., relativity). (I probably read that somewhere, but I don’t know where, so I can’t cite it properly, sorry; I don’t think that idea is original to me, though.)

Also, I’d be remiss if I didn’t mention Terry Pratchett’s take on Lovecraft in The Color of Magic. There’s a scene where the characters are in some sort of temple where the geometry isn’t quite right, including having regular octagons tiling in the plane…

I’m loving LIM. LOVING it – keep up the great work!

-Matt

September 11th, 2009 at 10:48 am

Matt, the art never came through, but thanks for the comment and compliment just the same. In fairness, I do want to say that the phrasing of the fifth postulate you cited was my own, or at least how I remember it from eighth grade. Euclid’s postulate is stated a little more complexly, but it comes across to me like “Not not having an option is not having an option” so I have to simplify it to even remember it. Geometrically-speaking, I’m still a 14 year-old; must be a hyperbolic thing.

September 11th, 2009 at 8:49 pm

In Heinlein’s Stranger in a Strange Land, an object moves in a fourth dimension. To each observer, it appears to move straight away from them, regardless of where they’re sitting. I’ve always thought R’yleh had some kooky interdimensional stuff going on where 3D isn’t all there is. One of my math teachers once tried to explain how a third dimensional entity could really mess with the inhabitants of a 2D world by moving in “impossible” directions. Messing with piteously feeble and limited fools… that’s what HPL is all about!

September 12th, 2009 at 2:24 am

They are naive indeed, who suppose that R’lyeh “rises” only along the vertical determined by a planet’s feeble curvature!