Excellent Approximations and Lying to Children
[I]t’s true that Euclidian geometry is only a special case of the mroe general geometry of spacetime. But it’s an amazingly good approximation to any situation you will ever encounter. Which is why we teach it to children– because it’s vastly simpler, and the cases where it doesn’t work are very far from everyday experience.
The post on which this comment is based seems to propose doing things the hard way — why teach non-Euclidean geometry without having the foundation in Euclidean geometry. Do you really want to teach that kinetic energy is \((gamma -1)mc^2\) and, perhaps more importantly, do you want to derive how you got that, rather than going with the Newtonian approximation that’s going to hold as long as you are limited to everyday speeds?
Physics curricula aren’t perfect — I think e.g. the Bohr model can do more harm than good — but then again that’s not really an example of a model that’s approximately correct. The suggestion that we abandon teaching classical physics and instead we dive into quantum and relativistic topics (starting with lasers at Eight O’clock on day 1) means explaining the details while simultaneously trying to get across basic ideas like forces and energy. I think that’s a lot to ask a student to digest.
Euclid does not work on the surface of the Earth whereupon a triangle’s three interior angles can sum to 540 degrees but never to 180. The rubber sheet demo of gravitation is exactly wrong, for its triangles’ three interior angles always sum to less than 180 degrees, the opposite of gravitation deforming spacetime.
Does it really matter? Yes it does. One cannot navigate the high seas or large scale survey using Euclid. Hyperbolic spacetime would be anti-gravitation, and useful if reduced to practice. A star’s core collapse to a black hole massively inflates local spacetime as it limitlessly implodes, hinting that black hole interior singularity exotica never obtain. Approximations set up the joke. All the fun is in the footnotes.
Organic chemistry is molecular orbitals. Any instruction based on that premise will churn out dolts. 1920s’ LCAO approximation is a jury-rig and quantitative swindle, offering an easy A and a very functional grasp of the discipline. Instruction has a lecture or three on when not to use LCAO, the small case that got Woodward and Hoffmann a Nobel Prize.
The Earth can be modeled as a sphere (or as an ellipsoid) in Euclidean 3-space, and navigation is still reasonable. I think a non-Euclidean interpretation is only necessary when general relativity comes into play.
@Drew: Earth’s surface is a non-Euclidean 2-sphere (and distorted – equatorially oblate, with trace bottom bulge). A mercator projection map will not get you a Great Circle route from Osaka to Seattle. There are no parallel lines on the surface of the Earth. The ratio of a circle’s circumference to its diameter is less than pi. One mile square is not one square mile, it is more; the four interior angles add to more than 360 degrees. The circumference of a plane Euclidean circle is (pi)(diameter). A plane Euclidean ellipse adds distortion. Even that matters big time:
http://i39.photobucket.com/albums/e191/toomers/ell1.jpg
a, b are the half length axes, h = (a-b)^2/(a b)^2. Good to 5 sig figs.
The Shroud of Turin is then a trivial fraud. A face curved in two orthogonal directions cannot be projected onto a Euclidean plane (linen cloth) without distortion, or cutting, or creasing. The study of such projection is called “cartography.” Heavy white cotton cloth, flat plaster bas relief sculpture of a face, kitchen oven. Heat the slab to less than 230 C, lay on the cotton cloth until light charring. (Synthetic fiber melts, wool smells awful.) Flat to flat is no distortion, and it obtains a photographic negative, just like the “real” thing.
One must know when approximation fails. GR is not Newton, economics can only work locally, massless boson photon vacuum symmetries need not be exact for fermionic matter. When theory fails to predict observation, theory is fundamentally wrong though rigorously derived. Get better founding postulates.