Triangles, parallels, Euclid and Riemann

[lbiarge]

In classical maths from Euclid believed that a triangle is 180º and that in 1 point only can travel a parallel to other. – http://en.wikipedia.org/wiki/Parallel_postulate

 

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” – http://en.wikipedia.org/wiki/Non-Euclidean_geometry#History

 

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 common 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.

 

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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 http://en.wikipedia.org/wiki/Non-Euclidean_geometry#History

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.

 

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.

 

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.

 

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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).

 

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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.

 

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Luis Biarge Baldellou

This note is posted in public domain and copy in this page: http://imagineonscience.wordpress.com/triangles-and-parallels/

Thanks.

[Bignose]

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

I really don't understand what most of the above is on about. But I can answer this:

 

Spherical triangles

 

 

C on the north pole,

A on the equator on the prime meridian

B on the equator on the 90* meridian --- all 3 angles will be 90* then.

Edited February 8, 2013 by Bignose

[moth]

I think the other part of your question is about Euclid's paralell postulate.

This TED talk is an intuitive description of hyperbolic geometry:

http://embed.ted.com/talks/margaret_wertheim_crochets_the_coral_reef.html

Edited February 8, 2013 by moth

[Amaton]

 

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

 

Parallelism is (ordinarily) a Euclidean property, and in this context seems impossible for obvious reason.

 

However, we could consider the projective plane. So intuitively, a given pair of parallel lines would intersect at a "point at infinity" (though really, the Wiki states that there are no parallel lines in the projective plane as per the preceding condition).

 

 

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

 

In any ordinary geometry, is a tri-angle not always defined to have 3-angles?

[lbiarge]

I really don't understand what most of the above is on about. But I can answer this:

 

Spherical triangles

 

 

C on the north pole,

A on the equator on the prime meridian

B on the equator on the 90* meridian --- all 3 angles will be 90* then.

 

AC and AB are meridians not parallels

 

the line from A to B is equidistant to C, this is a semi-sphere not a triangle.

 

A cone and a semi-cone also have 2 angles of 90º but the line of equator is equidistant to the pole.

 

A cone and a semi-sphere are not triangles.

 

Also 3 angles unit by not straight lines can to sum from 0º to 360x3º

 

A triangle need to be 3 straight lines, not only 3 lines.

 

I think the other part of your question is about Euclid's paralell postulate.

This TED talk is an intuitive description of hyperbolic geometry:

http://embed.ted.com/talks/margaret_wertheim_crochets_the_coral_reef.html

 

In that hyperbolic geometry:that show like triangle are 3 angles unit by not straight lines.

 

A triangle is formed by 3 angles and straight lines union, if you take that supposse triangle and put in a plane you can see they have not straight lines.

 

Draw me a triangle with 2 angles of 90º. Please.

 

Thanks.

 

Parallelism is (ordinarily) a Euclidean property, and in this context seems impossible for obvious reason.

 

However, we could consider the projective plane. So intuitively, a given pair of parallel lines would intersect at a "point at infinity" (though really, the Wiki states that there are no parallel lines in the projective plane as per the preceding condition).

 

 

In any ordinary geometry, is a tri-angle not always defined to have 3-angles?

 

Do you speak of maths of doy you speak over fiction?

 

Do you know any train that travel by 2 ways that join in any point of the world?

 

Do you really believe that 2 parallels join in any point of the universe?

 

Do you know any triangle with 2 or 4 angles?

 

Please speak over maths not over fiction.

 

Thanks.

 

Parallelism is (ordinarily) a Euclidean property, and in this context seems impossible for obvious reason.

 

However, we could consider the projective plane. So intuitively, a given pair of parallel lines would intersect at a "point at infinity" (though really, the Wiki states that there are no parallel lines in the projective plane as per the preceding condition).

 

 

In any ordinary geometry, is a tri-angle not always defined to have 3-angles?

 

To say that "So intuitively, a given pair of parallel lines would intersect at a "point at infinity"" is near same that to say that x and x+1 in near infinite has a different distance less or more that 1 and by that that maths is not science.

[moth]

Your understanding of the words "straight line" seems too limited for this topic. Are you familiar with geodesics?

The TED video demonstrates how, given a line and a point not on the line, how to draw many different lines through the point that do not intersect the given line. Did you watch it?

[lbiarge]

 

Your understanding of the words "straight line" seems too limited for this topic. Are you familiar with geodesics?

The TED video demonstrates how, given a line and a point not on the line, how to draw many different lines through the point that do not intersect the given line. Did you watch it?

 

1 - first TED video is more easy to see here (add h por http):

 

2 - Curious like a person can say many words and say nothing. (in the video).

 

3 - I believe I understand geodescis and also like gravity curve the geodesic where travel the photons, ...

 

According to geodesic you consider that a point 3d xyz after a conversion o geodesical change from Euclidian to hyperbolic, ... the point xyz remain at same point 3d because you represent that point in a 3d coordenates without consider that hte geometry has changed.

 

An example: 10 is a mathematical value, but according by the base can to means 2 (binary), 10 (decimal), 16 (hexadecimal), ... but according you say only count that it's 10, by that you consider that 10 binary = ... = 10 hexadecimal and by that that 2 = .... = 16

 

Another example, light in universe travel by the geodesic, so near a mass the "straght line" without consider the mass is curved, really is straight but according to your geometry it would be curved and by that in a point would to be many parallels but really if you consider the so curved line light a "straght line" that change the geometry from euclidian to hyperbolic or another really only can to be 1 parallel in that point.

 

So wiithout consider that the geodesic is curved by gravity seem that the light is curved but knowing that mass curve the geodesic understand that really travel a "straght line".

 

4 - The TED video show a piece with more that one parallel that supposed parallels cannot to be parallels between theirs (1 common point) or only 1 parallel at any time and changing the geometry and by that the point xyz change.

 

Really the geometry in parallels and triangle is like a sheet of paper, draw a triangle or a parallel, know you can roll the paper in hyperbolic form and the parallel is only one, the point xyz change and the triangle adopt the adecuate form.

 

Please draw me more that 1 parallel in 1 point (withot the TED absurd form) or draw me a triangle with 2 angles of 90º, not affirm, draw me,

 

When I speak over cones I really speak over perimeter of cone without the base, this has 2 angles of 90º, and 1 more angle but the line that joint the 2 angles of 90º is equidistant to the other angle and by that is not straight and by that is not a triangle.

 

The example of the meridians in same form not is usefull because the line from equator is equidistant to the pole.

 

Thanks.

 

Another example:

 

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

 

So according to this in time x the rocket and another person in Earth (personn 2) consider time x, 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 a triangle or parallel in Euclidian geometry can to be represented in hyperbolic, ... but the position xyz change, not remains same in same form that in 2 the x + 1 year is not the same for both persons of the example or that 10 is not the same in binary that in decimal.

 

In the example of corals, ... the 2 sides of the coral or the worms only are paralels in a geometry according to that and the point is the same that in a flat Euclidian geometry when are parallels and the xyz point change and by that are not multiples parallels in a point because the point xyz always is the same, only change the geometry according to the movement of the worm.

Edited February 8, 2013 by lbiarge

[moth]

If you could give your definition of "straight line" it might make it easier to understand where the confusion begins.

It's no more possible to draw the picture of a triangle you asked for than it is to fit a piece of paper to a sphere or saddle-shape, and for the same reason:although the (surface of) sphere,saddle, or plane are all 2-d they don't share the same geometry.

[lbiarge]

If you could give your definition of "straight line" it might make it easier to understand where the confusion begins.

It's no more possible to draw the picture of a triangle you asked for than it is to fit a piece of paper to a sphere or saddle-shape, and for the same reason:although the (surface of) sphere,saddle, or plane are all 2-d they don't share the same geometry.

 

Please, better give me you your definition of triangle and parallel.

 

You can draw a triangle in the sphere and the put in 2d.

 

Or a triangle is not 2d?

 

If you could give your definition of "straight line" it might make it easier to understand where the confusion begins.

It's no more possible to draw the picture of a triangle you asked for than it is to fit a piece of paper to a sphere or saddle-shape, and for the same reason:although the (surface of) sphere,saddle, or plane are all 2-d they don't share the same geometry.

 

Sorry the answer.

 

Maybe that I don't understand well what is triangle, parallel, 2d and 3d.

 

But I believe that a triangle is a 2d figure, by that a draw of a spherical triangle need to be also a 2d triangle.

 

A draw of Earth meridians and equator seem a triangle but it's not a triangle, in a triangle a line cannot to be equidistant to one angle.

 

Thanks and sorry for my answer.

 

Another question if in a point can to be more of 1 parallels to a parallel: How is possible that all this parallels in that point are parallels to the first line and not parallels between theirs?

 

Can to be 1+a = 1+b = 1 + c and a, b y c to be differents?

 

Can to be a = b = c mathematic and the parallels not parallels between theirs and also mathematics? Geometry is part of the mathematics?

 

10 = 10 = 10 , .. but 10 binary = 10 hex, that means that 2 = 16 and mathematics exact? If we change the base and consider is equal is the same that change the geometry and consider is equal.

[moth]

Please, better give me you your definition of triangle and parallel.

 

You can draw a triangle in the sphere and the put in 2d.

 

Or a triangle is not 2d?

I'm not a math guy but I believe the definitions are something like:

a line is the shortest distance between two points.

a triangle is three lines arranged so each line intersects with the other two.

parallel lines do not have an intersection.

a triangle is 2-d but it doesn't have to be "flat".

[lbiarge]

I'm not a math guy but I believe the definitions are something like:

a line is the shortest distance between two points.

a triangle is three lines arranged so each line intersects with the other two.

parallel lines do not have an intersection.

a triangle is 2-d but it doesn't have to be "flat".

 

Sorry, any puntctuations:

 

- "a line is the shortest distance between two points." - this is a "straight line", a line joint 2 points in any form. A line can join a distance of 1 meter with a 1000 km.

 

- "parallel lines do not have an intersection" and also are equidistant in all the points, many times 2 lines without intersection in 3d are not parallels.

 

- "a triangle is 2-d but it doesn't have to be "flat"" - Do you know a plane not flat?, flat is by definition 2d, a 2d not flat need to be 3d and by definition 2d in not 3d. If a 2d has z dimension is 3d.

 

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More:

 

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, ...?

 

More:

 

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?

 

I affirm: in a hyperbolic universe exist triangles in their flat 2d with addition of angles of exactly 180º.

 

More:

 

I use a sheet of paper, here I make some parallels, then I put 1 of their in a static positions and I put in a nail a point of other parallel to their, then according to Euclidian in a point there is only a parallel, now I roll the paper and put in the nail another point of another parallel, according a 3d coordinates in that nail are many parallels but really are differents point with differents parallels that I have draw in the same paper, but in any proof the geometry of the paper change.

 

Another example: I put a world globe and I point New York, now I turn the globe and in the same point is Sidney, the I can say that New York and Sidney are in the same point, and so with all the world or same all the universe (another example is the center of a gps screen).

 

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Another example: 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 where are differents that give the same value area with differents lines?

 

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In hectare the space is considered 2d but really the land is not 2d, so there are hectares with 3d that really not give the same real area that another (1 really flat o leveled) and another not leveled, by result if the not leveled hectare is really well measure give more area. In same form if you change the geometry seem that in same point can to be more that 1 parallel but really the area is different and the geometry change.

 

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In the TED video all is false, this woman say in time 8.12 that there is not parallel in the point with the equatorial line, that's false, in our globe (the world) only is a parallel in the equator but in other point of the globe is a parallel but not point the equator because parallels are equidistant, in time 9.58 show a jersey with 3 lines with a common point, this 3 lines cannot to be parallels to a side, because a parallel is equidistant. For example she show a worm, for example the eye of the worm make a parallel line to any of the sides of the worm, the worm can changing geometry change the point at the distance of was the eye but really that is not the point and another parallel, the point continue in the eye and if the worm short the thickness change the geometry and the point continue in the eye and the same parallel line, in the point where was the eye now is another point.

 

Maybe I create a new topic asking who can affirm or deny we live in a Euclidian or hyperbolic universe and how can proof that.

[moth]

First thing, I'd like to apologize for causing you to feel an apology was needed. We're just talking here, no apologies necessary so far.

One of the reasons you cannot draw parallel lines on a sphere is that a "straight line" on a sphere is a Great Circle.

 

In a way, Geometry is about definitions. If you don't use the standard definition of line then any argument you make about parallel lines will be invalid.

Same for triangles if you don't know what Extrinsic Curvature is you can't argue about the sum of the internal angles of a triangle.

 

I don't think it's known if we live in Euclidean space or some other Geometry is a better fit to the universe.

[lbiarge]

First thing, I'd like to apologize for causing you to feel an apology was needed. We're just talking here, no apologies necessary so far.

One of the reasons you cannot draw parallel lines on a sphere is that a "straight line" on a sphere is a Great Circle.

 

In a way, Geometry is about definitions. If you don't use the standard definition of line then any argument you make about parallel lines will be invalid.

Same for triangles if you don't know what Extrinsic Curvature is you can't argue about the sum of the internal angles of a triangle.

 

I don't think it's known if we live in Euclidean space or some other Geometry is a better fit to the universe.

 

 

”In a way, Geometry is about definitions. If you don't use the standard definition of line then any argument you make about parallel lines will be invalid.

Same for triangles if you don't know what Extrinsic Curvature is you can't argue about the sum of the internal angles of a triangle.”

 

 

 

I consider this answer a low blow and consider I don’t like to use this system. I prefer before to use these techniques to end this conversation.

 

I know really what is parallel lines, and also triangle, maybe you doubt in my known but this not negate this.

 

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The maths have a characteristic and this is that are exact. A math not exact is not math. The geometry is part of maths by that it need to be exact and by that a geometry not exact would to take out of maths. In maths 2 + 2 are 4 in Euclidian and not Euclidian universes, by that if geometry change from one universe of the other is not maths.

 

 

 

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If we negate the existence of flat triangles (2d) in a not Euclidian geometry we are negating that in that geometry exist 1d and 2d, because if we admit 1d not flat would be 2b that is negated.

 

 

 

Maybe also that we live in a not Euclidian universe, by that we negate the existence of triangle flat (2d) in the Euclidian because for a people that live in a not Euclidian universe they seem their universe like flat and the real Euclidian universe like not Euclidian.

 

 

 

Another example: a sail of a ship triangular, this is triangular and their angles add 180º, you and wind can bend in any direction but continue in triangle, you can bend to spherical geometry and other not Euclidian geometries but at last if you put in a flat is a triangle.

 

 

 

Like we cannot to know if we live in a Euclidian or not universe and it’s very possible that our universe is not Euclidian this represent that in our universe the triangles will not add 180º and not to be flat (like you say) and lines have more of 1 line in 1 point.

 

 

 

The probability to live us in a Euclidian universe probably is less of 1/1000000 because the only Euclidian probably is without any curve, all the other probabilities are not Euclidian.

 

"One of the reasons you cannot draw parallel lines on a sphere is that a "straight line" on a sphere is a Great Circle."

 

Math is exactly and a big thing is same that a little thing so if you can "draw parallel lines on a sphere is that a "straight line" on a sphere is a Great Circle."" also same in little circle.

 

Repeating, if you can so that only in great circle and not in little circle is not maths.

 

 

 

First thing, I'd like to apologize for causing you to feel an apology was needed. We're just talking here, no apologies necessary so far.

One of the reasons you cannot draw parallel lines on a sphere is that a "straight line" on a sphere is a Great Circle.

 

In a way, Geometry is about definitions. If you don't use the standard definition of line then any argument you make about parallel lines will be invalid.

Same for triangles if you don't know what Extrinsic Curvature is you can't argue about the sum of the internal angles of a triangle.

 

I don't think it's known if we live in Euclidean space or some other Geometry is a better fit to the universe.

 

4 + 1 = 5 and 34646494446461 + 1 = 34646494446462

if a big circle is different to little circle is not math like if x+1 in any number is not the next integer number the maths would to need dissapear.

 

in this x and x + 1 are intengers.

Edited February 10, 2013 by lbiarge

[moth]

select two points on a sphere. now imagine all the circles on the sphere that intersect both points.

measure the distance along the circles circumference between the two points,which circle minimizes the distance between the two points?

You haven't said what your definition of a line is, but by the definition: a line is the shortest distance between two points, a Great Circle is a line. Other circles are not.

[lbiarge]

select two points on a sphere. now imagine all the circles on the sphere that intersect both points.

measure the distance along the circles circumference between the two points,which circle minimizes the distance between the two points?

You haven't said what your definition of a line is, but by the definition: a line is the shortest distance between two points, a Great Circle is a line. Other circles are not.

 

line definition by http://www.mathopenref.com/line.html for example and about google the page is "Line - math word definition - Math Open Reference"

 

1 - "A geometrical object that is straight, infinitely long and infinitely thin."

 

2 - shorter : "A straight line is the shortest distance between any two points on a plane"

 

by this: A sphere is a plane?, if you work with that geometry there is not 2d in that not Euclidian geometry?

 

I don't believe the shorter line between New York and Sidney is the line over the sphere, the shorter is by 2d by the center of the Earth. If you put the World map in 2d (flat) the the shorter line is by the surface but only if you consider 2d in the sphere like atlas or maps.

 

"which circle minimizes the distance between the two points?" I only can considerer the circle that minimize the distance only considering they like 2d not 3d, like 3d always there is a line shorter.

 

 

 

More: according to this line definition a circle cannot to be considerer a line because not go until infinite. Look : 1 - "A geometrical object that is straight, infinitely long and infinitely thin." and a circle is closed.

 

Is very probably that our universe is not euclidian Universe, according to this and that you deny that in not euclidian geometry exist 2d, then we need to affirm that 2d and 1d not exist and our triangles have not 180º.

 

The probability of live in a Euclidian universe is less that 1/100000000000, only a flat universe is Euclidian, all the other probabilities like the draws of TED video are not.

[lbiarge]

Being more exact you cannot speak over circles, circles are 2d, like triangles and parallels and you speak over a non Eucidian without existence of 2d.

 

Really seem that you admit 2d if it's in the direction you admit and not in the spherical dimension, but a circle is 2d and without admit 2d flat existence circle also cannot exist.

If circle exist also exist triangle and parallels flat = 2d, if you not admit triangles flat and not exactly 180º in angles then you also cannot admit cicles existence.

Edited February 11, 2013 by lbiarge

[moth]

It seems you believe 2-d means "flat" but it simply means 2 coordinates are enough to locate any point on the object.

[lbiarge]

It seems you believe 2-d means "flat" but it simply means 2 coordinates are enough to locate any point on the object.

 

No.

 

If is sufficent is because is 2d.

 

For example : city, street, number, date, this is a 4d, if you not give date is 3d and by that without temp, but not for sufficent.

 

In this example: If you give only 3 coordinates of a 4d at what time refer you? at all or at none, you can say tomorrow in your house if only give your house, or imagine you give time but your direccion only in 2d without the floor, ...

 

In same form if you only give 1 dim of a square is insufficient, and so all.

 

A non flat coordinate cannot to given with only 2 dimensions, for exmaple a map is converted to 2d from 3d but all people know this, but against this never can to use floors and undergrounds.

 

Sorry, but any mathematical need to know that given 2d of a information 3d is insufficient and not like you say.

 

Really is imposible to name a x dimension with x-1 coordinates. (graphs, directions, time, ...)

[moth]

I can locate any point on the surface of the Earth using latitude and longitude, the surface of the Earth is 2-d.

I'm not sure I understand what you are saying, are you saying non-Euclidean geometry is the same as Euclidean or non-Euclidean is impossible?

[lbiarge]

I can locate any point on the surface of the Earth using latitude and longitude, the surface of the Earth is 2-d.

I'm not sure I understand what you are saying, are you saying non-Euclidean geometry is the same as Euclidean or non-Euclidean is impossible?

 

A map is a conversion of the word in 2d without use the z coordinate, but next time if you live or send a post to anything in not the floor not put the floor in your adress.

 

At same time you can say you are in New York because a time you were there, because for you the time (4d) is not important and by that you can speak over a 4d system only with 3 coordinates and for you has not importance time.

 

In general we can speak over any x dimension only giving 1 only coordinate: If you can speak over 4d with 3 coordinates also with only 2 and only with only 1.

 

In Euclidian and not Euclidian geometry need to exist 2d and by that in all cases a triangle is 2d and by that in any geometry the sum of angles allways is 180º in a triangle.

 

A x dimensional geometry or system can not be described with x-1 coordinates.

 

Against this if a x system could to be described with x-1 coordinates in same relation any x dimensional system would to be described with only 1 coordinate.

[moth]

A map is a conversion of the word in 2d without use the z coordinate,

 

If you try and glue a map to a globe you will find the land and oceans are a different shape on the globe than they appear on the map. One is curved, the other is flat.

 

in all cases a triangle is 2d and by that in any geometry the sum of angles allways is 180º in a triangle.

 

I disagree.

[lbiarge]

According 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." from http://www.mathopenref.com/parallel.html

 

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 differents 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 lins with the same result in area, and all theirs other areas have 1 common point outside of the line that remeain the same in all the examples.

 

In same 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" - http://www.mathopenref.com/area.html

 

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

[moth]

If you replace Euclid's idea about no intersection with the idea of equal length perpendiculars everywhere between the parallel lines, then at most there is one parallel to a line.

I think on a hyperbolic surface, a "line" equidistant from a straight line would not be the shortest path between two points and that makes it a curve, and on a sphere... well, you know.

I'm not sure I understand the rest of your post, sorry.

Could you try and say the same thing using different words ?

[lbiarge]

 

If you replace Euclid's idea about no intersection with the idea of equal length perpendiculars everywhere between the parallel lines, then at most there is one parallel to a line.

I think on a hyperbolic surface, a "line" equidistant from a straight line would not be the shortest path between two points and that makes it a curve, and on a sphere... well, you know.

I'm not sure I understand the rest of your post, sorry.

Could you try and say the same thing using different words ?

 

You continue thinking that a not euclidian geometry is not flat, by that begin in 3d, because a 2d not flat is 3d.

 

By that you consider that the shortest distance would to be a curve. Like a geometry without 2d

[moth]

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.