Tuesday, November 29, 2005

Hollow Earth

In the comments on Bridgework as Body Piercing, the Tetrast and I got off on an interesting tangent about Moebius strips, Klein bottles, and other nonorientable surfaces. Yes, I confess, many long years ago I was a graduate student in mathematics.

Anyhow. This thread led me off onto thoughts about the hollow earth. Back when I was a kid, I loved reading Edgar Rice Burroughs. Actually, I read only one or two of Burroughs' two dozen or so Tarzan books. What really grabbed me were his lesser-known series: John Carter of Mars, Carson of Venus, and of course Burroughs' Pellucidar series.

Pellucidar was a world located inside the Earth. Yes, in Burroughs' imaginative universe, you burrow straight down into the ground 500 miles in your "iron mole"... and you come out into the world of Pellucidar. The earth is a hollow shell, with our world on the outside and Pellucidar on the inside. A world on the inside of a hollow shell, with a tiny sun floating up in the middle, at the very center of the earth!

This idea of a hollow earth really fascinated me. I read also somewhere about a fellow named Cyrus Teed, or "Koresh," who in the late 1800s and early 1900s promoted Koreshanity, which taught in its "cellular cosmogony" that we actually live inside a hollow earth. Our world only appears to be a sphere, actually it's the inside of a hollow sphere, and the surface of the earth "curvilineates upward eight inches to the mile." Seemed to me this went the Flat Earth Society one better. ;-)

Of course if you live inside a hollow earth, the question naturally arises: What's on the outside? In my years in the field of math, I toyed with an imaginative answer: Nothing much, if your hollow earth is located in real projective three-space.

You can imagine projective three-space (see also the comments in the Bridgework thread) by thinking of the region inside a sphere. Now identify each point on the surface of the sphere with the point opposite it on the other side of the sphere... in other words, if you're travelling inside the sphere and you cross the outer surface of the sphere, you will simply find yourself back inside the sphere, over on the opposite side.

So we find ourselves living in a small, closed, three-dimensional "pocket universe." Now stick your hollow Pellucidar, or Koreshanity-world, or whatever, in the middle of this "pocket universe." You are living on the inside of a hollow earth, a hollow shell. And if you dig down into the ground, starting in Pellucidar, and if you keep digging straight down far enough... you'll come out eventually on the opposite side of Pellucidar.

That's right, if your hollow earth is located inside projective three-space, then there's no way out of Pellucidar, and nothing much else outside of Pellucidar. Dig down into the ground, and you'll just come back out again somewhere else in hollow Pellucidar.

Or if Pellucidar is quite tiny in size compared to the projective three-space in which it exists, this opens up the possibility of a closed "pocket universe" dotted with many little hollow-earth Pellucidars, hollow worlds scattered throughout a solid closed universe, sort of like the chocolate chips scattered throughout a chocolate chip cookie.

Believe it or not, there was a time when I used to amuse myself endlessly with such imaginary constructs. Like I say, once upon a time I was a grad student in mathematics. And I suppose I've got to dust off those math diplomas on my office wall once in a while. ;-)



Blogger The Tetrast said...

When I was very young (3 or 4) I thought we DID live in a hollow Earth. I was told that the Earth is curved, but I got it mixed up. On the street (in NYC) my mother would point into the air, and after a little time a taxi cab would pull up. I thought that the cabbie saw her from a bit further up the curve. Who knows what confusion I might have made of Moebius strips at that time.

Tuesday, November 29, 2005 9:56:00 AM  
Blogger Paul Burgess said...

And while I'm at it, I suppose I should add that projective three-space is another one of those nonorientable manifolds. A three-dimensional nonorientable manifold. "Nonorientable" means roughly this: If you were right-handed and you started out in Pellucidar, and dug down into the ground until you came out on the opposite site of Pellucidar, and if you then traveled by land and sea back to your starting point, you would find that you were now left-handed.

Or rather, everything in the place you started from would now appear to you to be flipped over into a mirror-image of itself. The print in books. People you'd known. The landscape. Everything would now appear to you to be reversed in mirror-image. Or to say the same thing, they would appear normal to themselves, but now you would appear to them to be flipped over in mirror image. The pocket that had been on the left side of your shirt would appear to them to be now on the right side. Though that pocket would still appear to you to be on the left side. And so forth.

This stuff is a real trip, when you really start getting into it.

Tuesday, November 29, 2005 10:11:00 AM  
Blogger Paul Burgess said...

I should also mention, I'm left handed. And when I was ten, a friend of mine who was right handed dropped a conundrum on me that I've never been able to solve. He told me that actually everyone experiences the world as if they're left handed: it's just that right handed people perceive the world around them in mirror image from the way left handed people perceive it.

I know that's plumb crazy, but I've never been able to refute it. :-)

Tuesday, November 29, 2005 10:15:00 AM  
Blogger The Tetrast said...

That's a good one philosophically. And since I'm a southpaw too, I can look forward to quite a few occasions for using it.

The original Outer Limits had an episode about left-right reversal and that was probably when my parents first explained about higher-dimensional space to me. You can imagine my disappointment when it was announced some years ago that space on the larger observed scales is roughly Euclidean.

Tuesday, November 29, 2005 11:12:00 PM  

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