Triangles, parallels, Euclid and Riemann

[lbiarge]

In my note I like to say near to:

 

An area is enclosed by 4 lines (a rectangle), 1 of that lines is fix, other use the common point and the other 2 are perpendicular to the 2 parallels,

According to that can to be many parallels in the same point, all this parallels need to be equidistant to the first line and by that:

 

- if they are equidistant and by that 3 lines are the same (the first) and the 2 perpendiculars, but in the same point are many parallels: How is possible that the area remain the same with other line that is not the same parallel? Or the area may be different with another parallel equidistant to the first line and by the same point?

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called 2-dimensional Euclidean space or bi-dimensional Euclidean

-Wikipedia

 

I'm confused. Are we talking about 2-d or 3-d

If we are talking 3-d then I agree. 2-d not flat is extrinsic curvature. If you are 3-d or more the curvature is obvious, not so much if you are 2-d.

 

According to this in non euclidian geometry not exist the parallels and not exist the 2d figures like triangles. Triangles, square, .. and parallels are 2d

"Triangles are assumed to be two-dimensional plane figures," - http://en.wikipedia.org/wiki/Triangle

Really parallels can to be not 2d, but parallels are equidistant and by that form a 2d flat plan.

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called 2-dimensional Euclidean space or bi-dimensional Euclidean

-Wikipedia

 

I'm confused. Are we talking about 2-d or 3-d

If we are talking 3-d then I agree. 2-d not flat is extrinsic curvature. If you are 3-d or more the curvature is obvious, not so much if you are 2-d.

 

I believe here is error, 3d also is euclidian, euclidian more clear is to say where the plane is not curved.

Really the Euclidian geometry use 2d, 3d and 4d (with time), the non euclidian geometry use curved (hyperbolic, ...) but also is 2d in the form that consider their 2d (from an Euclidian geometry a not euclidian flat seem 3d and viceversa).

 

Another time remembering that I say that we cannot proof that we live in an Euclidian universe but in same time in our universe Euclidian or non Euclidian exist the 2d flat and our triangles add 180º in their angles.

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called 2-dimensional Euclidean space or bi-dimensional Euclidean

-Wikipedia

 

I'm confused. Are we talking about 2-d or 3-d

If we are talking 3-d then I agree. 2-d not flat is extrinsic curvature. If you are 3-d or more the curvature is obvious, not so much if you are 2-d.

 

This is bad, bad, bad, in Euclidian exist 1d,2d,3d and 4d (with time), and in non-Euclidian also 1d,2d,3d and 4d

 

This is near to say that until near 1 century ("The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Around 1818," - http://en.wikipedia.org/wiki/Non-Euclidean_geometry) maths not know the sphere.

 

So all that phrases of wikipedia are error: "When n = 2, the set of all such locations is called 2-dimensional Euclidean space or bi-dimensional Euclidean" is error and would to be "When n = 2, the set of all such locations is called 2-dimensional space or bi-dimensional"

 

This is a very big error to say that Euclidian is 2d, is like to say that the 3d were unknow until 1818

Also this seem to say that 2d is Euclidian and 3d is non Euclidian, but this affirmation is false.

[moth]

I"Triangles are assumed to be two-dimensional plane figures," - http://en.wikipedia.org/wiki/Triangle

Good edit:

Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below)

 

as for the rest of it you should take it to Wikipedia.

[lbiarge]

This formula of an sphere was first derived by Archimedes, (http://en.wikipedia.org/wiki/Sphere) and Archimedes live in 287 BC – c. 212 BC - http://www.google.com/search?q=formula+esfera

 

Archimedes was Euclidian, because between other option the non Euclidian begin in 1818

 

By this you can read that in Euclidian geometry exist the 3d.

Good edit:
Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below)

as for the rest of it you should take it to Wikipedia.

 

A triangle always is 2d, because with 3 point you can make a flat plane.

 

The other over non flat 2d like spherical triangles, I write over that in my first note. A semi-sphere is not a triangle,or another time draw me a triangle that their angles not add 180º

2 point define a 1d, 3 point define a 3d. A triangle has 3 point by that all tirangle can define a 2d flat in any geometry.

 

A cone and a semi-sphere is not a triangle, but you can obtain a triangle joining their 3 points.

[lbiarge]

Sorry. 3 point define a 2d (not 3d)