> I will admit I don't know much about the geometry of our galaxy, but I was 
> under the impression that the disk is hundreds of billions of light years 
> "thick",

It's much smaller than you think:

http://en.wikipedia.org/wiki/Milky_way

> So... about 36 seconds of arc. Man, that's pretty small.

It's still one of the very few objects in the sky that allows you to 
actually see its shape with a good pair of binoculars or a small telescope 
though.

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