Euclidean Geometry = Worthless?

[Nick]

I was doing my Geo. homework when something occured to me: isn't all of spacetime curved? For example, take a table sitting in front of you. You may place a ruler on it and find it perfectly straight, but lo and behold: all fout legs are touching the Earth, and the Earth is curved. Therefore, the table must be curved.

Due to the curvature of spacetime, all lines in space, no matter how big or small are curved towards an object with large mass. So, doesn't that make Euclidean Geometry, which relies on the fact that lines and planes are straight, worthless?

There are 5 postulates that require Euclidean Geometry to be true; Here are the ones that are proved false by the curvature of spacetime:

1) For any two points, there is exactly one line that contains them.
If lines are curved in spacetime, then two points have an infinite number of curved lines containing them.

2)For any three non-collinear points, there is exactly one plane that contains them
Same as above, except replace two with three and lines with planes.

3)If two points are in a plane, then the line containing them is in the plane
This can be true, but only if the line and the plane are equally curved.

4)If two planes intersect, then they intersect at exactly one line
If the two planes were curved, then they could intersect at two lines.

Tell me what you think .

P.S. The fifth postulate is "A line, a plane, and space each contain an infinite number of points. Some points in a plane are noncolliniear. Some points in space are noncoplanar".

noncollinear = not on the same line
noncoplanar = not on the same plane.

[houserichichi]

Clasically the universe is Euclidean, that is, distances are measured via the Euclidean metric (you know how to take distances between points). The reality of things is, however, that curved spacetime requires us to change our thinking a little - relativity dictates that the metric of spacetime isn't Euclidean at all, though for all intents and purposes we're able to "pretend" because it fits most everyday experiences.

So on paper yes, you're right, but for the sake of argument you'll only really ever need to assume Euclidean through most of your life.

[jinydu]

Yes, irockyou, you're right that all of spacetime is curved by the prescence of matter/energy in the universe, since no part of the universe (as far as we know) is an infinite distance from the nearest planet (or proton, for that matter).

However, unless you're close to a black hole or neutron star or you want extremely accurate measurements, Euclidean geometry is usually a very good approximation. Also, Euclidean geometry is much simpler to work with than the general relativity equations in their full glory; so Euclidean geometry is useful.

[Nick]

So, Euclidean Geometry isn't worthless, but its not exact either?

I can live with that.. when I'm done Geometry Honors... :P

[houserichichi]

Think of it this way - space is locally Euclidean - that is, if you were to zoom in (up to a certain point) space can be treated essentially as if it were flat. Same idea as the surface of an apple or the planet...sure it's round on huge scales but when you zoom in (to the size of people walking on the planet Earth) things appear flat...same goes for the universe.

[pat]

For example, take a table sitting in front of you. You may place a ruler on it and find it perfectly straight, but lo and behold: all fout legs are touching the Earth, and the Earth is curved. Therefore, the table must be curved.

This logic doesn't quite work out here. Start with a flat surface tangent to a sphere. Move the flat surface radially away from the sphere. Drop four legs perpendicular to the flat surface down to the sphere. All four legs will touch the Earth. Provided you had the tabletop centered at the tangent point, all legs will be the same length.

In addition, even with a surprising amount of variation in the flatness of the floor, rectangular tables can be made to sit stably (without wobbles) just by rotating the table around its center point.

[moonlord]

I see Euclidean as an aproximation to real geometry, just like classic and relativistic mechanics. For the sake of calculus, we use the simpler ones.

[bo198214]

Mathematics (for example euclidean geometry) does not claim to explain something real. Physics claims this and uses mathematics to approximate reality in their field of interest.

[wendy]

I was doing my Geo. homework when something occured to me: isn't all of spacetime curved? For example, take a table sitting in front of you. You may place a ruler on it and find it perfectly straight, but lo and behold: all fout legs are touching the Earth, and the Earth is curved. Therefore, the table must be curved.

All space is curved. It is not curved in anything. See for example, my website for how curvature works.

Due to the curvature of spacetime, all lines in space, no matter how big or small are curved towards an object with large mass. So, doesn't that make Euclidean Geometry, which relies on the fact that lines and planes are straight, worthless?

Euclidean geometry is the first approximation, and remains valid until gravity becomes significantly intense.

There are 5 postulates that require Euclidean Geometry to be true; Here are the ones that are proved false by the curvature of spacetime:

1) For any two points, there is exactly one line that contains them.
If lines are curved in spacetime, then two points have an infinite number of curved lines containing them.

The same proposition is true in spheric and hyperbolic geometries.

2)For any three non-collinear points, there is exactly one plane that contains them Same as above, except replace two with three and lines with planes.

ditto.

3)If two points are in a plane, then the line containing them is in the plane This can be true, but only if the line and the plane are equally curved.

Since a point is the intersection of three planes, then one might contain both A and B. so if a line contains both A and B, then any plane that contains one contains the other. This is not specifically euclidean.

4)If two planes intersect, then they intersect at exactly one line If the two planes were curved, then they could intersect at two lines.

A set of freely curved planes can intersect an infinite number of times.

Tell me what you think .

P.S. The fifth postulate is "A line, a plane, and space each contain an infinite number of points. Some points in a plane are noncolliniear. Some points in space are noncoplanar".

noncollinear = not on the same line
noncoplanar = not on the same plane.

This is evidently wrong: a plane is an equal sign, a line is two equal signs, and a point is three equal signs. A plane is not comprised of points, becasue it is more fundemental than the point.

The notion that a line is made of points (rather than, say, one can make points in the line), leads to all sorts of self-evident falacies.

W

[Nick]

This is evidently wrong: a plane is an equal sign, a line is two equal signs, and a point is three equal signs. A plane is not comprised of points, becasue it is more fundemental than the point.

The notion that a line is made of points (rather than, say, one can make points in the line), leads to all sorts of self-evident falacies.

That's weird.. I never noticed that. The postulate is straght from my textbook, too!

[jinydu]

Your textbook may have oversimplified things somewhat. Try reading the book "Journey Through Genius". I think the author was named "William Durham", or something like that. It explains all 5 postulates of Euclidean geometry exactly as Euclid stated them.

[houserichichi]

Or pick up a copy of Euclid's Elements, the second most read book in the history of mankind.

moonlord: removed off-topic posts about the top selling books in history...