Riemann's Bookworms
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Basic Information

In 1854, mathematician Bernhard Riemann gave a revolutionary speech that established the tenants of Riemannian geometry and took the luster off Euclidean Geometry. In that speech, he used this clever little analogy about bookworms (to be fair, he was following up on an analogy used by Carl Friedrich Gauss). It goes something kinda like this:

Grab a piece of paper, and draw two little cartoon worms in opposite corners. Go ahead and give them names, if that helps. Draw a straight arrow connecting them, and explain that one of the worms intends to travel along that arrow to visit the other worm. Along the way, he feels various forces, such as gravity, exert on his little wormy body, making parts of the journey more difficult. These are invisible forces, though, he can't see them. All he sees is his universe: the sheet of paper, and the other worm drawn on it. Riemann's little bookworm thought it's universe was flat. It could see in every conceivable direction, and could tell that space did not bend. It was wrong though, as there's another vantage point, from which the curved nature of the universe is obvious.

Now crumple the sheet of paper into a ball. It's best if you crumple the middle part more than the ends, so the worms are both still visible.

The force the little worm feels as gravity is actually the distortion of bent Spacetime. Climbing over a curve is harder than a flat section of space, especially when your vantage point prevents you from seeing that it's even bent.

The little worm assumed the shortest distance to his friend was the straight line you drew. But to someone with a perspective outside his limited reality, it become obvious that there's several shortcuts, if you could just learn to climb over the folded loops, or to burrow little wormholes through the mass of the paper.

Our universe is like that. We are the bookworms. Gravity and Electromagnetism are the result of dimensional folding. Once we figure out how to make wormholes, we'll be able to zip across the universe. If we can just poke our heads up off the page, we'll see that it's all bent out of shape. That better perspective will reveal profound truths about our reality. Fun, huh?

Riemann even figured out a mathematical expression, called a Metric Tensor, that defines the curvature of reality in the much the same way a Field Equation defines electromagnetic fields.

This analogy sparked a broad interest in the Fourth Dimension and Occultism, especially in England. It inspired Edwin Abbott Abbott and H.G. Wells to write stories about it. Further, it influenced and contributed to Albert Einstein's revolutionary Theory of General Relativity.


2. Book: Hyperspace by Michio Kaku

Game and Story Use

  • Hyperspace, Teleportation, Fourth Dimension, Wormhole, the analogy above can serve as a model for all kinds of magic and powers or principles of superscience.
  • Technology for semi-controlled gating, imperfect teleportation, dimensional shortcuts, etc, might take it's name or imagery from this presentation. "We're traveling through a Riemann Cut." "Our two different transportation technologies are wormholes and wormbridges, which work subtly different." "I'll activate my Riemannian Dimensional Flattener." Etc.
  • Here's a monster for your bestiary: The Riemann Wurm is a ferocious beast that occupies another reality that overlaps with, but is slightly out of sync with our own. The folds of it's reality are not the same as ours, resulting in disjointed causality and variations in spacetime. Escaping the beast is easy in the short-term: if you run twenty feet it may have to handle three curves and a mile to catch up with you, and it moves slowly like a worm. However, it's very patient and will continually show up unexpectedly via Offscreen Teleportation and Traveling At The Speed Of Plot. You have to figure out how to get it off your trail, before it eventually catches up (which it will probably do the next time you sleep). Its a bit like the Hounds Of Tindalos, but without the time travel element.
    • It's also possible the realities only adjoin in a few specific places, or few specific places in spacetime. The creature is bound purely to it's own reality, but that reality overlaps ours in 3 or 4 locations. The Wurm appears at those places, attacks, and then withdraws. Stay away from those 4 places, and you're safe, but it can take any prey in it's mouth with it back to its own reality.
    • See also: Fourth-Dimensional Lifeform for a vaguely related monster idea.
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