Universe Shape...
The NY Times is reporting that data from the Wilkinson Microwave Anisotropy Probe may suggest that the Universe might be shaped like a doughnut or a cylinder: it might be possible, like in the old video game Spacewar, to drift off one 'side' of the Universe and reappear on the other."
The classic calculus example of something infinitely long but with finite volume is "Gabriel's Trumpet." Take the hyperbola y = 1/x from x=1 to infinity and rotate it about the x-axis, its asymptote. The resulting three dimensional object can be integrated to show it has finite volume but infinite surface area. As my old math professor Harry Crull used to say, you could fill it with a finite amount of paint, but no finite amount of paint could paint the surface, with a constant coat thickness.
The shape of the Universe is relative, depending on the experiment applied, how it was applied, and the results of the experiment obtained. A relative shape of the Universe can be anything or nothing or not anything absolute. Prescribing any particular shape to the Universe is neither absolutely right, nor absolutely wrong.
'Gabriel's Trumpet Volume' in 2D can be graphed using:
y = x^-1 and y = -x^-1, x \neq 0
selfAdjoint's description is probably the closest shape described, as this is the relative shape from inverting a sphere volume.
I am not certain, however I believe the it was Einstein that described the relative shape of the Universe by stating that any photon dispersed into space, without external interference (i.e. matter, gravity, energy, etc) capable of deflecting or intercepting the photon, given enough time, would result in the photon returning to its point of origin. Describing the shape of the Universe using 'lines' may be incorrect, as these 'lines' are more like 'finitely long arc lengths'
A general relative description of this type of result is that the Universe is simply a closed loop, something infinitely long but with finite volume, or a Gabriel's Trumpet Volume, in which the entire known observable Universe exists as merely a pinpoint on the surface of this volume, or exists as this volume, or both!
How a sphere, toroid, loop, Gabriel's Trumpet Volume, or other type of dimension can accomplish this result may be purely a 'relative' matter of speculation. Speculation may result from that space-time fabric itself is fundamentally composed of closed loops, or the space-time manifold geometry results in a return loop of space-time.
Note that Gabriel's Trumpet Volume is only finite if a limitation is placed on the range of the x or y asymptotic axis, however, in the classical sense, because the y value never actually crosses the intercepts, all asymptotic volumes are infinite.
Another prescription as to the shape of the Universe is a Hypersphere. Note that Gabriel's Trumpet Volume describes the space-time manifold of a Hypersphere.
Hypersphere:
In mathematics, a higher-dimensional sphere having three dimensions of space and a fourth dimension of time.
Hypersphere Volume: V_4 = \frac{\pi^2 r^4}{2}
Hypersphere Surface Area: S_3 = 2 \pi^2 r^3
A torus has the same formula 2 \pi^2 r^3 as the Hypersphere.
Reference:
http://www.hypersphere.com/hs/abouths.html
http://www.bright.net/~mrf/toc.html
http://arxiv.org/abs/physics/0402075
http://www.bright.net/~mrf/App4.html
http://www.bright.net/~mrf/hierarchy(1).html
http://www.fm/7-sphere/Sunset.htm
http://www.specularium.org/hypersphere6d.html
http://www.afn.org/~afn59513/Think4d/hsphere.html
http://www.evolutionpages.com/dodecahedral_universe.htm
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http://www.obspm.fr/actual/nouvelle/dec01/luminet.en.shtml
http://www.spacedaily.com/news/cosmology-01f.html
