Space from the inside, and outside
Andrea raises some especially pertinent questions:
I keep wondering if we would ever actually be able to tell whether or not space was curved, because from our perspective wouldn't things change with space, including our measuring instruments like Prof. G said in class today? If that is the case, then it would seem that even if space was curved, we would never really know because all our our relative reference points would be curved along with it. Wouldn't we have to somehow remove ourselves from space in order to see that it is curved, or to measure the curve?
Think about this (and the diagram should make it easier to imagine): you set out on an expedition from some point on the equator, heading exactly due north. When you get to the North Pole, you turn yourself around 90 degrees, and then start heading back down south (naturally...). When you reach the equator, you turn 90 degrees to face along the equator, towards the spot your journey began; then you head along the equator to complete your journey. Now, your trip was in the shape of a triangle (three straight lines forming a closed figure). And the angle sum of the triangle is 90+90+90=270 degrees. Your experience on the "small scale" was of Euclidean geometry; that is, any measurements you made in your immediate environment conformed to the geometry of the Elements. But the geometry of the surface you've been travelling in is quite definitely non-Euclidean, since the angle sum of a triangle was different from 180 degrees!Now, if you didn't know that the world was spherical, this result would be very puzzling. No doubt, you would question whether, perhaps, you had deviated from a straight line at any time - but would have to conclude that you had been very careful always to keep the same bearing. This addresses Andrea's question, in particular: although, on the local scale, our experience may always be Euclidean, on the larger scale (the surface of the earth, the distance between galaxies) we may discover that the measurement of triangles indicates a non-Euclidean geometry. If we wished actually to "see" the curvature directly, we would have to remove ourselves from the curved space: relatively easy to do in the case of the surface of the earth, but impossible once we start thinking about the curvature of space itself.
Keep some of these issues in mind as you think about Flatland - what kinds of knowledge could they have of the space in which they live?
posted by Robert Goulding at 10:42 AM
3 Comments:
The Professor,
I posted this with the other comments of Andrea and Byrnn and have simply copied them here. I think that your example is very helpful, but a question arises: is this phenomena of the difference bwtn Euclidean and spherical beometry true b/c of the particular scale of our planet as relatively big, or simply because we are covering a large PERCENTAGE of the sphere/globe. In other words, would there be an important difference bwtn the two geometries on a sphere with a radius of 1 foot rather then several thousand miles?
The large scale of our planet means that we have to cover a large percentage of it to notice the effect of curvature. But we could imagine, say, being on a very small asteroid, in which the curvature was apparent to the naked eye (the horizon being only, say, 200 yards away). On the other hand, you're right that the size of a sphere doesn't matter, in that the geometry of a beachball is the same as the geometry of the Earth. Our experience of the Earth would be just the same as some sentient creature scaled down proportionately (to about the size of a bacterium, I guess) has of a beachball. And if you draw the triangle I described on a beachball and measure its angles, you'll find that the angles add up to 270 degrees.
Professor G:
I did not know that a triangle on a beachball equals 270 degrees. Sphere/globe question: Does this have implications to the angle rotation for when I play beachball volleyball as opposed to regular volleyball?
Let me know if you get the chance. Thanks,
Corey
BTW: Nice curves!
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