## Thursday, September 16, 2010

### The Structure of Amber

"Amber is the one real world, casting infinite reflections of itself -- shadow worlds that can be manipulated by those of royal Amber blood. But the royal family is torn apart by jealousies and suspicion; the disappearance of the Patriarch Oberon has intensified the internal conflict by leaving the throne apparently up for grabs.

"In a hospital on the shadow Earth, a young man is recovering from a freak car accident; amnesia has robbed him of all his memory, even the fact that he is Corwin, Crown Prince of Amber, rightful heir to the throne -- and he is in deadly peril."

Hence "The Chronicles of Amber" by Roger Zelazny. The princes can move between the shadow worlds, such as the one you and I live in, and possibly - with difficulty - find their way back to the true reality of Amber.

Suppose each world was a one dimensional line (say the Real line) and the infinity of shadow worlds was modelled by stacking them side by side to make a two-dimensional plane. At first sight this looks like R2 but that doesn't capture the fact that inhabitants of each shadow world are forced to remain within it (unless they're a prince of Amber of course).

So rather than R2 the universe of 1D-Amber is better modelled as:

I x R

where I is an index set 'counting' the worlds. We could arbitrarily say that 0 ∈ I indexes the true world of 1D-Amber itself.

While ordinary shadow people have to stay within their one-dimensional world-line the princes of Amber also have freedom to move within Index space, I, to visit other shadow worlds and Amber itself.

OK, now let's move up to 4D. Any shadow world (and Amber itself) we may now assume to be normal space-time. What about the Index set? Zelazny offers few clues: a throwaway remark about the MWI of QM doesn't really count - fantasy's homage to SF.

So let's make the Index set infinite, open, bounded and maximally symmetric (and 3D while we're about it to improve navigability). Amber is in the centre.

A suitable candidate is:

I = {(r, θ, φ) such that r < positive-constant}.

That's a ball without a boundary.

The universe of Amber is than I x R4. Perhaps hard to visualise and it raises questions as the princes move around I. Does an infinitesimal move between neighbouring shadow worlds result in infinitesimal moves in time and space, assuming there is a similarity relation between the worlds?

Zelazny seems to suggest that shadow worlds that are 'close' to each other are very similar, but that in 'far away ones' time and geography can behave very strangely.

Enough universe-building for today. Alex has given me his cold.

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PS. If I'd made I six dimensional we'd have a novel interpretation of String Theory. How prescient would that make Roger Zelazny?