I was trolling (in the conventional sense) through Wikipedia today, reading about the Oort cloud, cosmology, and various superstructures and -voids in the Universe.

I also read about the shape of the Universe. For anyone without a math or science background, this is should be as baffling as I found it to be at first. More easily grasped is Gauss' Theorema Egregrium, which has to to do with curved surfaces, for example the surface of the Earth. The Theorema describes a characteristic of flat maps that anyone can observe: when displaying geographical features, flat maps distort shapes, distances, or both. If you've ever noticed that Greenland appears larger than Canada on some maps, you can appreciate what this means.

More precisely, Gauss's results mean that the curvature of the Earth (or any other surface) can be determined by measuring some angles and distances on the surface, and that this determination is unique in some sense. If you draw a triangle on a curved surface, you can see that its angles do not add up to 180°. Now, the crucial point is this: the same principle applies to space. Space has a curvature which determines the relation of distances and angles within it.

I find it's best to just accept that without trying too hard to picture it.

Anyway, the 'Shape of the Universe' article discusses the ramifications of the space curvature of the Universe. If a parameter Ω is not exactly 1, then space is curved and some interesting things happen: the Pythagorean theorem (a²+b²=c²) becomes false, and the value of π varies.

This brought to mind the galactic architects of Carl Sagan's Contact and the concept of a message from God in π. Interestingly enough, that article states:

Any intelligence, working in any universe—no matter what the characteristics of its particular "space-time fabric"—must deduce the same value of π, presuming they are able to think of numbers at all, and that logic is not a property of the Cosmos.

If we're very particular, we can argue that this statement is false without the qualifier, "when Ω=1."

I seem to be pushing my usual word count, so I'll continue my thoughts in another post.


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