From: Rafal Smigrodzki (rms2g@virginia.edu)
Date: Tue Aug 20 2002 - 17:17:37 MDT
Anders Sandberg wrote:
On the other hand, you could have a closed universe where the volume is
infinite. Think about the globe analogy and imagine a earth map wrapped
twice around the globe: when you move across the zero meridian from the
east you don't cross over to the normal Atlantic but a new fantasy
atlantic. Continuing westwards you pass fantasy America, the fantasy
Pacific, the fantasy Asia and Europe until you again pass the zero
meridian and end up in the real Atlantic. This world would be a
2D-sphere with constant positive curvature but twice the area of the
normal sphere. One could imagine this wrapping continuing even further
and in all directions - when you circle the Earth you end up in more and
more remote versions of the geography. In the same way our universe
could be a 3D version of this, with constant positive curvature but
nowhere repeating.
### I am not sure you could call this a closed universe - after all, this
would be the result of connecting of an infinite number of units (each equal
to an Earth surface) but with different contents. Even if the local topology
was that of a sphere with a finite radius, the structure as a whole would be
infinite, that is, its inhabitants would have infinite lines of sight, which
would be the opposite of closed.
But, on the other hand, assuming that the number of discernible combinations
of elements (atoms, Flatlanders) within each unit was finite, then the
inhabitants would have a finite number of mind-states available to perceive
their environment. On a higher level, a meta-observer would be able to
observe the combinations of the units (very large numbers of various
versions of the Earth in neighboring lattice positions, interacting to
produce higher-order patterns). In effect, this would be equivalent to the
observation of the evolution of a cellular automaton (with Earth-sized
cells). So maybe you could call it "infinite volume universe" but only from
the meta-observers point of view.
For a closed n-dimensional universe with an infinite volume I could suggest
an n-dimensional surface folding as a fractal in an n+1 space. I don't have
the oomph to think this through.
Rafal
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