why do we learn euclidean geometry when nothing the in the universe is on a flat plane?
Because it is sufficient for most situations. Why use non-euclidian when euclidian serves your purpose?
And BTW, many thing in the universe are on a flat plane.
It is a good starting point for students of geometry,
The structure it has serves as a great field to learn "the way of math" using a familiar subject.
Wouldn't only things only in the 2nd dimention be flat plane?
Any three random points in the universe makes a plane. Unless perhaps if you have to consider a relativistic gravitational field, which a very small percentage of the world's population have to do.
The 3 dimensions of an object aren't always relevant for your purpose. You don't need a building's width to calculate its height.
Are you sure of that? That would only be true in a "flat" 3 dimensional space. The theory of general relativity asserts that the curvature of space depends on the mass in the area. It is true, of course, that as long as we are really close to an enormously dense object, the curvature of space is so small that space is indistinguishable (by normal, everyday means) from flat space and Euclidean (plane or solid) geometry works nicely.
Umm, more massive objects cause more curvature, not less.
I agree, that's why I added :
My point is that in nearly all practical situations, space CAN be considered to be "flat". Few people (of more than 6 billion) have a need to consider the reality of curved space. Universe is perhaps a strong word here, I assume it contains table-top situations too. Surely math teachers don't expect all of us to become cosmologists.
I think Rasine was being rhetorical.
How would you explain "Non-euclidean geometry" without even knowing what is euclidean geometry?
because somebody out there wants you to !
Racine, you know good question but if the gravitron is found in physics we might live in a flat world. General Relativity would be invalid and you curved universe might just actually turn out to be the absolute flatness you are not wanting to learn.
Also they teach you that in school cause it's alot more visual and easy to understand, you can't learn english without the ABC's.
Because it is the most aesthetic of all the arts mankind created. Its beauty charms people.
Along the lines of what has already been said in this thread:
in any small enough region of space [assuming it's "smooth"], euclidean geometry is the best approximation. (The tangent space is Euclidean.)
(Given a smooth curve and a point on that curve, the tangent line at that point gives the best linear approximation to the curve at that point.)
super good pt rob, which i would have never thought to mention to him.
thats all spicy and nice, but if students learned a form of geometry, inwhich is not only for flat surfaces, maybe they will understand better becasue they will not have an opposing form of geometry already embetted in their heads. if you take the forumla for the area of a sphere in ecludian geo and apply it to the eath, your answer will give you exess space.
i believe that even if somthing looks flat, nothing in our universe is unless it is void of matter, which it is not. it is curved, even at a minascule amount.
Curvature at a point on a general curve is defined with a 'radius of curvature', where a portion of a curve is approximated by an arc of a circle.
But, just as there are no flat planes in the universe, there are no perfect circles in the universe either. So by your argument, we can't really define curvature!!
Hi Rasine, you didn't make a mistake ... your question is completely valid!!
In my humble opinion, I think there should be a clear distinction between
mathematics (which deals with abstract notions, such as planes and lines)
and physics which utilizes the tools of mathematics to give a quantative description of nature.
Although many subjects in mathematics find their origin in physics and
astronomy, mathematics stands on its own without necessarily having any
use other than the study of mathematical structure itself (exception is the field of applied math).
Therefore, Euclidean geometry is studied for its structure.
I will quote a passage from Einstein's book on relativity:
robphy made a wonderful point.
Remember the most useful subject in mathematics is calculus. Why is it useful? Because it is all about approximating curved geometries by flat geometries, i.e.reducing difficult geometries locally to the one easy one, the flat one.
So in a sense euclidean geometry is the simplest easiest geometry, and is useful in studying every other geometry.
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