Triangles, parallels, Euclid and Riemann

(work against the mainstream) This is an IMAGINE

Author : Luis Biarge Baldellou – Email:


In classical maths from Euclid believed that a triangle is 180º and that in 1 point only can travel a parallel to other. –

So determine that “At most one line can be drawn through any point not on a given line parallel to the given line in a plane”

Lately Non-Euclidean geometry says that “Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line” –

According to this new geometry (has more that a century), by a point can exist more that one line, so according to parallel definition this more that 1 line would be parallels to the first line but not parallels between theirs (they have 1 comm0n point).

In same form would occurs the same in the infinites points that use these multiple parallels, all that billions of lines would be parallel to the first line and not parallels between theirs.

By reciprocity in all or many of the points of the first line would exist many parallel lines parallel to any point of the millions parallels lines to the first, but not parallels between theirs.

According to this we cannot say that a parallel from a parallel are also both parallels. By that probably the geometry would need to go out of the maths.

But there are more:

According to hyperbolic geometry (the acute case) and elliptic geometry (the obtuse case) the sum of angles of a triangle can to be more or less of 180º : “The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180° if the geometry is elliptic” according to

By that they give an example of a triangle formed by the equator of the Earth and meridians that with 2 angles of 90º have another angle in the pole and so add more of 180º.

According to this:

1 – Know parallel lines can not be parallels, because according to this 2 lines with 90º angle (parallels) make a triangle. So to the before note of that in a point can to be more or a parallel also can affirm that a parallel also can not to be a parallel.

2 – A semi-sphere is not a triangle, in the example the line of the equator is equidistant to the pole angle and by that is a line but is not straight.

Really in semi-spheres, cones and semi-cones there is near a triangle (it has 2 angles of 90º and 1 more), but semi-spheres and semi-cones are not triangles and the line between the 2 angles of 90º is equidistant to the other angle (really is the perimeter of the cone without the base like a cornet)

If I could make a triangle with lines not straight I could make triangles from 0º to 360×3º (less the minimum angle we consider x 3), but a triangle has 3 straight lines and 3 angles.

Really in hyperbolic and elliptic geometries the triangles really are also of 180º. How? easy: make a triangle in a sheet of paper and now you can curve the paper in hyperbolic to see the result of a hyperbolic triangle, … also you can bend the paper in the form you like to understand how would be a triangle in any other geometry environment.

Also for parallels in a hyperbolic geometry would seem a parallel different from a elliptic geometry but an space cannot to be at same time hyperbolic and elliptic and like in the case of triangle the result is to make 1 or more parallel in a sheet of paper and bend the paper if you bend the paper the position of the parallel and point change thinking in a 3d space because the space has changed. By that really is 1 only point or parallel but like in the case of sheet of paper bended the parallel and point has changed their position. You cannot thing in a 3d space in that change the parallel because if you change the geometry also the point changes of position. Proof with the sheet of paper to see that positions of the points are different but by the same point only is 1 parallel. To say that are many parallels in a point would be like to say that New York is in infinite possitions because the Earth turn and has translatation without consider that time is a dimension.

Also in sphere and cones,… you can use the triangle of the sheet of paper to see the form of a triangle in this form.

If this they say a train that travels by parallel railroad could not travel because the railroad would change the separation.


According to relativity theory a man that travel in a rocket a hight speed the time go slowly (person 1).

So according to this consider initial time x for the rocket and another person in Earth (personn 2), 1 year later in Earth (consider time present) occurs 2 things.

1 – present time is for person 1 and 2

2 – if we consider time x + 1 year the time is different by the 2 persons.

In same form the parallel or the triangle change visualy from Euclidian to Hyperbolic but really not change and also the xyz not change because the visual 3d xyz is like consider in x + 1 year that the time is same for both persons. So really is 1 parallel in any point.


Do you can affirm or deny that we live in a Euclidian universe?, you can affirm or deny that we cannot to live in a hyperbolic universe? Can you give any proof to confirm by positive or negative the difference over an universe Euclidian, Hyperbolic, …?

Do you believe that in a hyperbolic universe the triangle would not be flat and the sum of their angles is not 180º?. Do you believe that in an hyperbolic universe the parallels are not equidistants?

Remember that a triangle can to be made in 2d and at least in 3d because in 3d 3 points make a plane.

I don’t understand very well how mathematicians have admitted this impossible near of 100 years because is worse that a bad novel fiction. The classical scientists create in Earth parallels and meridians for localization in Earth because the meridians are not parallels, they understood well the maths principles but actual scientists have mistake all this information.

If this Non-Euclidean geometry would true:

1 – would not exist the parallels because many parallels in 1 point is same that not exist parallels because have 1 common point, also the triangle with parallel lines that have a common angle.

2 – Geometry would to go out of maths because to say that a triangle can to have more and less of 180º is so math that to say that 2+2 can to be 4 and more and less of 4.

3 – Trains could not work because would not exist parallel lines.

With same example of parallels we would say that 10 is many result like 1+1=10, 1+2=10,1+3=10, … 5+5=10, 8+8=10 because 1+1 is 10 in binary, 5+5=10 in decimal and 8+8=10 in hexadecimal and also with same example of triangles we would with 3 point in 3d you can draw infinites triangles because can to be in hyperbolic, elliptic and euclidean geometry without have importance that the points change of position like in the example of the sheet of paper when you bend it and by all this kill the maths because are not exact  (in this examples arithmetic and geometry). In same form a point that  change from Euclidian to hyperbolic also change the position xyz because consider the same xyz and by that multiple parallels in a point is like consider 10 absolut and by that that 2=…=16=….

Parallels in mathematical formula: according to Euclidian in a point only is a parallel to a line, in mathematical form, if we take the line, their parallel in that point and a section of theirs and the perpendiculars we obtain a square or a rectangle, according to area formula this acotation has an area that is the result of both distances, the distance of the section of the line and the perpendicular distance from line to the point of the other parallel. So for example to be easy the xy area is a. According to that is posible many parallels they will give an area with any parallel and by that will have the areas xy=a xy=b xy=c xy=d where x is the section of the line, y the distance from the line to the point and a b c and d the result of the rectangular area that is enclosed by x and 1 of the parallels that is in the point (you say that can to be more that 1 parallel in a point of a line). So if a = b = c = d same value area can give with differents lines?

According to the definition of parallels in math : “Parallel lines remain the same distance apart over their entire length. No matter how far you extend them, they will never meet.”

So if by a point can to be more that 1 parallel also need to say that there are many point in same distance and that can draw many different lines equidistant to the line and that with a common point.

Also an area is the space between a segment of 2 parallels and theirs perpendiculars, so in that form its possible to change 3 of the 4 lines with the same result in area, and all theirs other areas have 1 common point outside of the line that remain the same in all the examples.

In same math dictionary: “Area is a measure of the size of a 2-dimensional surface. For example in the rectangular shape on the right is 8 meters wide by 2 meters high. As you can see it can hold 8 square meters. So we say it has an area of 8 square meters. This is written sometimes a”

By that the area need to be in the same 2-dimensional surface for use the same point and line or maybe other 2d with the same point and line?

Burden of proof

You can proof what I say is false (remember that who affirm anything is who need to give the burden of proof):

1- Proof with real example that in a point you can put more of 1 parallel to a line.

2- Draw a triangle with 2 angles of 90º.

Against this, I can proof both:

1- I can proof that only exist 1 parallel to a line (same not straight like railroad) in 1 point, in hyperbolic, euclidian and elliptic geometry. That what you say is the same point is not true, like I say with a sheet of paper the point change their position respect a 3d euclidian coordinates (that we thing is 3d space), and by that what you say is the same point is really other point like occurs when you bend a sheet of paper or a object 3d.

2 – I can proof that any triangle in hyperbolic, euclidian and elliptic geometry the addition of all angles sum 180º (I can proof that later you translate to straight lines like the sheet of paper of the example all are exactly of 180º only). Cones, semi-cones and semi-spheres have 3 angles but they are not triangles).


Author: Luis Biarge Baldellou


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