…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.