Not sure what kind of question this is . Is it maths or physics or just idle talk? Anyway what place does Euclidean Geometry have in the "grand scheme" (if there is such a thing)? The impression I have got is that it is some kind of a poor blinkered geometry that is superceded by other geometries that describe the universe more completely. And yet it seems so incredibly important in the world we live in . So do we praise Euclidean geometry or do we "patronize" it? Does it matter ?Is it just horses for courses? Is there some really significant importance to Euclidean geometry aside from its amazing practicality or am I just looking for meaning in things that really just speak for themselves?
You seem to have the nuggets fo the answer in your question. Whether or not it has a place at the very large scale of the universe, it is certainly amazingly practical in a host of ways. Cosmology is only one of a multitude of sciences.
You might as well ask what place money has in the grand scheme of things. The ducks don't care about it.
The important status of any "first": Somebody had to get the ball rolling by introducing an ancestral conception, system, or invention in order for any later (diverse / advanced) progeny to develop. In Euclid's case it was just finally integrating a lot of scattered ideas / approaches in regard to shape and spatial organization into a formal, coherent structure reinforced with proofs. If one sets aside its abstract idealizations on paper -- the "purging of extraneous and imperfect elements" from the world as encountered -- then everyday experience might arguably conform (in a broad way) to the Euclidean flavor.[1] We don't directly "see" in regard to space any hyperbolic characteristics, extra-dimensions, etc. The latter have to be compromised even to depict them in visual imagination and illustrations / simulations. But the effects of such could still be exposed by controlled investigations[2], thus potentially substantiating their applicability, "realness", etc (whatever interpretational stripe) in _x_ theory or model. - - - - - - - - [1] When minus reflective thought / experiment, the brain's representational system (both live-sensory & daydream wise) seems confined to the neighborhood of the Old School's territory. Historically, this can correspond to Immanuel Kant's psychological version of space, as pertains to extrospective appearances / sensations being regulated by a mundane, geometrical template or intuition: "Sensibility, the form of which is the basis of geometry, is that upon which the possibility of external appearance depends. Therefore these appearances can never contain anything but what geometry prescribes to them." --Critique of Pure Reason. [2] But in terms of creative reasoning: Different, rival geometrical principles / systems can be formulated by the brain [and obviously have been]. Historically, this intellectual avenue for outrunning proposed constraints of human experience (like Euclidean geometry) was predicted by Kant, fell out of his proto-functionalist account of mind (a relationship between Sensibility and Understanding). Rather than such forbidding new developments: "In mathematics and in natural philosophy human reason admits of limits but not of bounds [...] The enlarging of our views in mathematics, and the possibility of new discoveries, are endless; and the same is the case with the discovery of new properties of nature, of new powers and laws, by continued experience and its rational combination." --Prolegomena To Any Future Metaphysics
Euclidean geometry is extremely useful because it is materially indistinguishable from the geometry of the universe in the limit of small distances. Euclidean geometry is extremely useful in the classroom because it forms a platform to discuss geometry of non-Euclidean spaces and how algebraic coordinates relate to geometry. Without Euclidean geometry we would have no Cartesian coordinates and no simple path to calculus. http://www.teach-nology.com/teachers/subject_matter/math/geometry/
Euclidean geometry as well as others are perfectly respectable branches of mathematics. For the physical world, Euclidean geometry works fine except for very large (intergalactic) distances.
Well, and the occasional planet-spanning distances. Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image!
Going back to when I was 14 - I'm sure we had triangles in circles and suchlike. I'm fairly sure that was all Euclid so I don't think he'd have had a problem with (for example) great circles.
Triangles in circles, like all triangles on flat surfaces - have angles that add up to exactly 180 degrees. Triangles on spheres have angles that add up to a non-fixed number greater than 180 degrees. They require non-Euclidean geometry. Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image!
Euclidean geometry seems to apply to any manifold over a very small local level. if I have understood correctly. Is there a special significance to that ? By the way ,(I have been unable to work this out for myself over a long time now) are all manifolds surfaces like the sphere (just different shapes and/or topologies)?
A manifold is just a system of coordinates stitched together out of small pieces of (mostly) Euclidean systems of coordinates. Euclidean geometry is very easy for us to deal with, so having very, very small pieces of Euclidean systems of coordinates makes some of the math easier.
Manifolds are smooth spaces without borders. Originally, manifolds were surfaces, or more precisely n-dimensional smooth objects embedded in higher dimensional Euclidean spaces. But methods developed in the late nineteenth century allowed one to study the geometry of the manifold without referring to the higher dimensional space, and without that context one really can't refer to them as surfaces. Since manifolds are smooth, they are locally flat, just like the graph of a smooth function looks locally like a straight line if you zoom in on a small part of it. Thus, you can will good effect assign local coordinates to it Cartesian-style and the local-Cartesian-coordinate description of short pieces of geodesics will closely resemble the description of Euclidean straight lines. But Euclidean parallel lines never diverge, while geodesics only remain at fixed separations in flat spaces.
The word "smooth" brings to the mind of a layman the idea of smooth surface. However ,if I understand correctly "smooth" here means "differentiable" . If that is right , I wonder with respect to what a quantity in spacetime might be differentiable (I think I have heard it is "static" ) Maybe spacetime is smooth but not differentiable?
Not quite - it is stronger than that. Look- if a function is continuous, we say it is of class \(C^0\). If it once differentiable (and therefore continuous) we say it is of class \(C^1\). If it is differentiable \(k\) times we say it is of class \(C^k\). And if it is infinitely differentiable we say it is of class \(C^{\infty}\) or "smooth". The coordinates that define a local "patch" of any manifold are indeed functions (I am aware that rpenner doesn't agree with me on this), so a manifold that admits of coordinate transformations that can be expressed as derivatives of class \(C^{\infty}\) is called a "smooth manifold" If spacetime is a manifold, then by the above this is an oxymoron
So what quantity can be differentiated with respect to what other quantity ? Does this differentiation give us a way to measure the curvature of spacetime? I believe the curvature of spacetime varies according to the proximity of sources of mass/energy. Apologies if I am not making sense......
Hi Confused2! I agree. But alas, that was Plane Geometry wasn't it? Life was simpler then. But even as you stated in your post, that did deal OK with great circles, or segments of arc. https://en.wikipedia.org/wiki/Great_circle
I was very tempted to claim I could work out the angles of a triangle inscribed on a sphere using only Euclidean geometry (that being the only geometry I have) but I don't have time to make good the claim. A 10 minute shufti shows it won't be easy.
Well, obviously it depends on the problem at hand. Here is a common use........ Suppose a manifold \(M\) with \( p \in M\) a point. Then this point is entitled to be described by a set of coordinates, say \(p = \{x^0,x^1,x^2,x^3\}\). Suppose further that these coordinates "extend" through some neighbourhood of \(p\). But any manifold of any interest in applications will be connected, so there will be another neighbourhood, also containing \(p\) with coordinates, say, \(\{x'^0,x'^1,x'^2,x'^3\}\). So for this to make any sense, we must have a way of relating these 2 sets of coordinates (which after all refer to the same point). I write \(dx'=\sum\nolimits_{j,k=0}^3 \frac{\partial x'^k}{\partial x^j}dx\), or for the vector \(\Psi= \sum\nolimits_{j=0}^3 A_j dx^j\) that \(A'_k=\sum\nolimits_{j,k=0}^3 \frac{\partial x'^k}{\partial x^j}A_j\) This is called the transformation law for a covector. Then assuming that the coordinate transformations have an inverse, for the "ordinary" vector \(B = \sum\nolimits_{m=0}^3\beta^m \frac{\partial}{\partial x^m}\), the transformation law \(\beta'^n=\sum\nolimits_{m,n=0}^3\frac{\partial x^m}{\partial x'^n}\beta ^m\). Some people called these respectively co-variant and contra-variant vectors
