Richardthughes
Posts: 11178
Joined: Jan. 2006
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Applause!
| Quote | 383
PiotrJuly 10, 2014 at 12:44 pm
Eric Anderson:
BTW, just as a secondary question, in spherical geometry, are you saying that a square and a circle (of appropriate size) would trace out the same exact shape, or would the circle itself trace out a different shape than we are used to seeing? If the latter, then KF’s point holds, regardless of which geometry you are using.
A sphere is a 2-dimensional manifold, which means that locally, near each point, it resembles a Euclidean plane. In other words, sufficiently small squares are practically identical with Euclidean squares as to their shape and properties. We humans are very small compared to the size of the Earth, so a square we draw on flat ground is to all intents and purposes Euclidean; the difference is negligible. But if you draw a really large square (i.e., a regular polygon with four equal angles and four equal sides), for example one whose sides have a length of 10 km each, you will discover that each angle, if measured accurately, is slightly larger than a righ angle (and the area of the square is slightly larger than 100 km², by the way). The larger the square, the larger the difference. Cartographers, land surveyors and civil engineers can’t ignore it.
If the length of each side is about 10,000 km, each angle becomes a straight angle (180°), and the square becomes a great circle of the sphere (in this example, of the Earth).
Circles in a spherical geometry are identical with Euclidean circles, as to their shape. The area surrounded by a circle and its circumference expressed as a function of the radius are locally given to a great accuracy by the normal formulas of planar geometry; but again the larger the radius, the larger the difference, since the curvature of the surface can no longer be ignored. If r = 10,000 km, the circumference equals 4r (not 2πr), and the area equals 2πr² (not πr²).
Lest you should think this is merely an exercise in equivocation, we have very real problem of the same kind simply because we happen to live in a Universe whose geometry is not perfectly Euclidean, and whose spacial dimensions are interwoven with the dimension of time into a 4-dimensional manifold, only locally approximated by a “flat” Euclidean 3-space plus time as a separable dimension. Consider the composition of velocities. It might seem “self-evident” that they simply add (Galileo and Newton wouldn’t have doubted it for a moment), but for large velocities the difference between real spacetime and its Galilean approximation begins to matter and we have to use the formulas od special relativity. If we ignored the non-classical geometry of spacetime, GPS systems would be useless. |
That's about as educational as UD has ever been / will ever be. Cue KF meltdown is 3..2...
Glasses ready, neat spirits only. Sips for:
[] Well poisoning
[] Ad hominem
[] Correction
[] Game over
[] Red herring
[] Rhetorical games
[] Bankrupt belief system
[] Must try harder / do better etc
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