Concentric circles are parallel?

[phya]

I agree with HallsofIvy. If you have two straight lines that are parallel (no intersection points) and you translate one of them so that it coincides with the other line, then the two lines intersect at every point.

No doubt it's a problem with your command of English. "To intersect" does not necessarily mean that the lines have to cross at some nonzero angle.

The superposition straight line, their included angle is zero mutually, but intersects the straight line, their included angle is not zero. Therefore is different.

 

[phya]

I agree with HallsofIvy. If you have two straight lines that are parallel (no intersection points) and you translate one of them so that it coincides with the other line, then the two lines intersect at every point.

No doubt it's a problem with your command of English. "To intersect" does not necessarily mean that the lines have to cross at some nonzero angle.

In the attached figure is an animation, explained that anything is the superposition, any intersection, they are different.

 

[Mark44]

The animation clearly shows that when one line is superimposed on another, there are an infinite number of intersection points.

 

[phya]

The animation clearly shows that when one line is superimposed on another, there are an infinite number of intersection points.

there are an infinite number of intersection points. ？？？

 

[Mark44]

So you now agree with me and HallsofIvy?

 

[G037H3]

there are an infinite number of intersection points. ？？？

a line is a set of infinite points sharing a value, a two dimensional continuum of those points

if a line has an infinite number of points, and another line merges with the first, then both have an infinite number of points, and an infinite number of intersection points

if you have two lengths of rope that are infinitely long, and place one on top of the other, then there are an infinite number of places you could stick a nail through both ropes

 

[Mentallic]

Let's not use any analogies from real life, or we'll never hear the end of how the ropes can't be truly merged, only touching each other.

 

[G037H3]

Let's not use any analogies from real life, or we'll never hear the end of how the ropes can't be truly merged, only touching each other.

I thought that he would be able to reason that, since we're talking about plane geometry, a distinction in 3 dimensions can't be made, since it is a top-down view...

 

[phya]

So you now agree with me and HallsofIvy?

No, I did not agree, I thought in the animation shows the superposition is the parallel one kind, but in animation intersection, is not is parallel.

 

[phya]

Let's not use any analogies from real life, or we'll never hear the end of how the ropes can't be truly merged, only touching each other.

Actually, after the superposition straight line is a straight line, is two straight lines becomes a straight line, therefore I asked that the straight line is own and oneself is parallel.

 

[phya]

The parallel essence is the distance maintains invariable, but is not the intersection does not intersect. .

 

[Mentallic]

Why? Because you say so?

Parallel is defined as two or more lines that maintain a constant distance between each other and as such, they don't intersect.

Notice: two or more lines.

 

[phya]

Why? Because you say so?

Parallel is defined as two or more lines that maintain a constant distance between each other and as such, they don't intersect.

Notice: two or more lines.

Actually, parallel was aims at two lines, the parallel essence was two line distances maintains invariable.

 

[phya]

Why? Because you say so?

Parallel is defined as two or more lines that maintain a constant distance between each other and as such, they don't intersect.

Notice: two or more lines.

Parallel aims, a pair, a pair of line.

 

[Mentallic]

the parallel essence was two line distances maintains invariable.

Yes, two distinct lines.

 

[phya]

Yes, two distinct lines.

But the superposition line is two line special situations,

 

[phya]

Two congruent triangles may superpose a triangle. Two straight lines are congruent, two straight lines may also superpose a straight line.

 

[Mentallic]

Do you realize you've nearly hit 150 posts and most - if not all of it - has consisted of crackpottery?

 

[G037H3]

phya, geometry is one of those things where you should accept the ideas put forth by the people who created and developed it...the definitions have been argued about for thousands of years and conclusions have been reached based on logical arguments

if you really want to learn geometry, go pick up all three volumes of https://www.amazon.com/dp/0486600882/?tag=pfamazon01-20 and work through it; it explains all of the definitions and ideas in great detail

 

[phya]

phya, geometry is one of those things where you should accept the ideas put forth by the people who created and developed it...the definitions have been argued about for thousands of years and conclusions have been reached based on logical arguments

if you really want to learn geometry, go pick up all three volumes of https://www.amazon.com/dp/0486600882/?tag=pfamazon01-20 and work through it; it explains all of the definitions and ideas in great detail

Yes, the Euclidean geometry already had over a thousand year history, but afterward presented the non-Euclidean geometry, this explained that the geometry is in the development.

 

[G037H3]

Yes, the Euclidean geometry already had over a thousand year history,

2500 years, and Greek geometry is the first use of rigorous proof in science.

but afterward presented the non-European geometry,

Non-Euclidean or Non-European? Regardless, if you want to learn plane geometry, take my suggestion, a few hours of study will blow your mind.

this explained that the geometry is in the development.

Plane geometry is well understood lol.

 

[phya]

You must acknowledge that parallel is the distance maintains invariable, because the distance has changed, therefore only will then intersect. The concentric circle and straight line parallel is similar, they are the distance maintain invariable. Whether you do acknowledge this point?

 

[phya]

In Europe, the supposition we give human's definition are “the white skin biology”, afterward Columbus's ship to the Americas, the crews has discovered some living thing, is similar with the human, but their skin is the black, therefore the crews had the argument, the most people had thought that these living thing were not the human, because they did not conform to human's definition, most only might call them the kind of human biology, but the small number of people believed that these living thing were also the human, was only their skin's color is different.
Are we about the parallel line question argument are also so?

 

[G037H3]

In Europe, the supposition we give human's definition are “the white skin biology”, afterward Columbus's ship to the Americas, the crews has discovered some living thing, is similar with the human, but their skin is the black, therefore the crews had the argument, the most people had thought that these living thing were not the human, because they did not conform to human's definition, most only might call them the kind of human biology, but the small number of people believed that these living thing were also the human, was only their skin's color is different.
Are we about the parallel line question argument are also so?

That's 100% wrong. Europeans have known of the swarthy races (subspecies) for a very long time. Aryan invasion of India? was at least 3,500 years ago.

If you don't want to actually study the nature of the things you're talking about, fine. But don't try to change standard definitions to suit your opinion when there is material available for you to study so you can understand why things are labeled as they are.

 

[Mentallic]

So all you're hoping in doing is that you'll make a contribution to mathematics somehow? You need a reputation first, and that is to know that specific topic inside out.

There is a reason we coined the term crackpot to describe those that suggest new crazy far out theories in science with little to no math ability or even a respectable knowledge in the topic at hand. If you just looked at all the crackpot theories in relativity...

And judging by some of your ideas, mainly that the absolute of a number is an "unsigned" number and not positive because that is being prejudice, then I can only suggest that you put your theories away in the basement, study the maths for many years to come - particularly geometry. It will give you the time to fully appreciate what the collective thinking of millions of mathematics over thousands of years have been able to produce - and once you're grown ripe in age and have a firm position in the understanding of modern geometry, take a look at those dusty old tomes again that you threw into the basement. See if those theories are still sound, and if they are, pursue them further with your new status of being a professional in that topic.

 

[phya]

That's 100% wrong. Europeans have known of the swarthy races (subspecies) for a very long time. Aryan invasion of India? was at least 3,500 years ago.

If you don't want to actually study the nature of the things you're talking about, fine. But don't try to change standard definitions to suit your opinion when there is material available for you to study so you can understand why things are labeled as they are.

I was only said that if, but not really thought any discovery black skin's person.

 

[phya]

Will discuss the issue in here not to know the parallel definition.

 

[phya]

equivalent statements to the parallel postulate :
There exists a pair of straight lines that are at constant distance from each other.

Therefore a straight line own and oneself is also parallel, because in this case, the straight line and the straight line distance is zero.

 

[phya]

At most one curve can be drawn through any point not on a given curve parallel to the given curve in a plane.

 

[phya]

L1 and L2 whether still parallel?

In the analytic geometry,
Supposition
the straight line L1 equation is y=kx,
the straight line L2 equation is y=kx+c,

then L1∥L2 is parallel,
if reduces c, then still L1∥L2.
When c=0,
L1 and L2 are coincident lines,
L1 and L2 whether still parallel?

 

[phya]

In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. http://mathworld.wolfram.com/ParallelLines.html" , Therefore, The parallel essence is not does not intersect, but is away from constantly invariable.

 

[CRGreathouse]

In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. http://mathworld.wolfram.com/ParallelLines.html" , Therefore, The parallel essence is not does not intersect, but is away from constantly invariable.

So would the helix x = sin t, y = cos t, z = t be phya-parallel to the line x = 0, y = 0, z = t?

 

[phya]

Curve parallel axiom is:

If the plane B parallel to the plane D, the distance between them is E, A straight line is a straight line on the plane B, obviously, A parallel to B, A to B on the distance of any point is E.

B in the plane, we bend A, so A a circle C, so, C to B at any point to the distance is still E, then, C is parallel to the B it? Obviously, the answer is yes.

Because, B to D on the distance of any point is E, therefore, B is parallel to D, or, at any point, B parallel to the D, so, B parallel to the D. The arbitrary point A parallel to B, so C is also parallel to the B.

Therefore, the curve can be parallel to the plane.

Therefore, the curve and the curve can be parallel to each other, as long as the distance between them remained unchanged.

Curve parallel axiom is:

At most one curve can be drawn through any point not on a given curve parallel to the given curve in a plane.

Attached is an animated map shows the curve parallel to the truth.

 

[phya]

So would the helix x = sin t, y = cos t, z = t be phya-parallel to the line x = 0, y = 0, z = t?

I may tell you in the appendix animation the spiral line parallel truth. In this animation, red line and blue color line parallel, if their distance is invariable, on the column surface, they becomes two spiral lines, that they were still parallel.

 

[phya]

So would the helix x = sin t, y = cos t, z = t be phya-parallel to the line x = 0, y = 0, z = t?

The circle is also a parallel line actually. The circumference and the center of circle are parallel.

 

[Mentallic]

A point is phya-parallel too?

 

[sachinism]

A point is phya-parallel too?

whats meant by phya-parallel ?

 

[Mentallic]

I'm not exactly sure. It's Phya's definition of parallel so you should ask him

 

[HallsofIvy]

Phya has been asked repeatedly, through 189 posts on this thread and one or two other threads on basically the same thing, to explicitely give his definition of "parallel". He has not yet done so. I rather suspect that he has no idea what a mathematical definition is.

 

[phya]

whats meant by phya-parallel ?

Actually I have said many times, parallel is the constant distance. But some people just don't listen.

equivalent statements to the parallel postulate :

There exists a pair of straight lines that are at constant distance from each other.

 

[HallsofIvy]

Actually I have said many times, parallel is the constant distance. But some people just don't listen.

equivalent statements to the parallel postulate :

There exists a pair of straight lines that are at constant distance from each other.

That is true in Euclidean geometry but you have also said that you are not talking about "parallel lines".

 

[HallsofIvy]

In the analytic geometry,
Supposition
the straight line L1 equation is y=kx,
the straight line L2 equation is y=kx+c,
then L1∥L2 ,
if c→0, then still L1∥L2.
When c=0,
L1 and L2 superposition,
if L1 and L2 not parallel, then L1 and L2 will not superpose, will intersect.
L1 and L2 superpose, not intersect.
Therefore still L1∥L2,
Therefore the straight line own and oneself is parallel, otherwise the straight line will not be a straight line, will intersect.
The curve is also so, the curve is also own and own parallel, therefore the curve is also may mutually parallel.
Does my this logic have what question?

Actually, you have given a number of different definitions of "parallel" which are equivalent in Euclidean geometry.

But you titled this thread "Concentric circles are parallel?" which implies that you are NOT using Euclid's definition of "parallel" which requires lines, not curves. The statement that "two curves are parallel to each other if they are always the same distance apart" is NOT equivalent to Euclid's definition.

You have also asserted over and over again that "a curve is parallel to itself" despite repeated attempts to tell you that that violates the definition given, even for parallel curves, in textbooks. You seem to be asserting that all textbooks are wrong just because you do not agree with them.

You also are skipping over the question of how you measure the 'distance between lines'. The standard definition of the distance between a point on one curve and a second curve is the distance measured along a line perpendicular to the second line. But are you aware that two curves with a constant "distance" between them, in that sense, may intersect?

 

[phya]

That is true in Euclidean geometry but you have also said that you are not talking about "parallel lines".

The attention, my definition contains the concentric circle parallel.

 

[CRGreathouse]

I take from posts #183 and #184 that curves A and B are phya-parallel if they can be parameterized as functions a(t) and b(t) such that for all t, the distance from a(t) to b(t) is constant.

My guess is that "if" above can be replaced by "iff": that is, if you cannot parameterize the curves A and B with equidistant functions, then A and B are not phya-parallel.

phya, is this right?

 

[phya]

A point is phya-parallel too?

Circumference is parallel to the center of the circle, so the circle became a circle, because the ellipse is not parallel, so the ellipse as the ellipse.

 

[phya]

Actually, you have given a number of different definitions of "parallel" which are equivalent in Euclidean geometry.

But you titled this thread "Concentric circles are parallel?" which implies that you are NOT using Euclid's definition of "parallel" which requires lines, not curves. The statement that "two curves are parallel to each other if they are always the same distance apart" is NOT equivalent to Euclid's definition.

You have also asserted over and over again that "a curve is parallel to itself" despite repeated attempts to tell you that that violates the definition given, even for parallel curves, in textbooks. You seem to be asserting that all textbooks are wrong just because you do not agree with them.

You also are skipping over the question of how you measure the 'distance between lines'. The standard definition of the distance between a point on one curve and a second curve is the distance measured along a line perpendicular to the second line. But are you aware that two curves with a constant "distance" between them, in that sense, may intersect?

If our original definition crow is the black, afterward we discovered that the crow also has the white, at this time, whether we should revise the original definition?

Newton's time's definition is different with Einstein's time's definition, after the theory of relativity appears, we should revise Newton's definition?

 

[phya]

I take from posts #183 and #184 that curves A and B are phya-parallel if they can be parameterized as functions a(t) and b(t) such that for all t, the distance from a(t) to b(t) is constant.

My guess is that "if" above can be replaced by "iff": that is, if you cannot parameterize the curves A and B with equidistant functions, then A and B are not phya-parallel.

phya, is this right?

If in the attached figure animation shows: If B plane and D plane parallel, the distance is E. A is a straight line, and A on B, therefore, A to the B distance is also E, when A becomes circumference C, C to the B distance is also E obviously, therefore, the distance is constant, therefore, C is also parallel to B.

 

[phya]

If a line (straight line either curve) to another line (straight line or curve) the distance maintains invariable, then these two lines (straight line or curve) are parallel.

 

[Dr Lots-o'watts]

Straight lines have the distinction that if I measure the distance at say 45 deg, the distance separating them is indeed invariable.

Not so for any other other type of (curved) line. Plus at some point, you have to decide if phya-parallel includes intersections, touching, phase shifts, is it limited to co-planar curved lines or not? Is a circle phya-parallel to the dot at its center? If so, then can I say the center-dot is parallel to the circumference? Can a dot then also be parallel to an ellipse? If so, then will a circle be parallel to an ellipse? If the circumferences are dotted, then two concentric (such a nice word - concentric! such a shame it would be to delete it from our vocabulary) circles don't have the same number of dots so there is no correspondence, are they still phya-parallel? If two concentric circles are phya-parallel, then are there other lines, that are of different lengths that can be parallel? What about two spirals, occupying 3 dimensions? Does phya-parallel include spirals of different wavelengths, of different radius? Are two squares, one within the other, phya-parallel? Then why not two triangles? Why not two houses? Are Russian dolls phya-parallel to each other? Where do I end?

Ah the simplicity of Euclid's definition!

(I do hope this debate is not an attack on Euclid himself because he's Greek or something, because I've always seen math as a nice place where everybody was civilized and able to get along and not waste time on such issues. Yay for Euclid and al-Khwārizmī! The ultimate tag team! http://www.fansonline.net/images/wrestling/TheMegapowers2.jpg)

 

[CRGreathouse]

If in the attached figure animation shows: If B plane and D plane parallel, the distance is E. A is a straight line, and A on B, therefore, A to the B distance is also E, when A becomes circumference C, C to the B distance is also E obviously, therefore, the distance is constant, therefore, C is also parallel to B.

That didn't answer the question. Even this many pages into the thread, you still haven't told us what you mean by parallel (what the rest of us call phya-parallel).

I'm done with this thread. If someone is able to formalize a definition for phya-parallel, feel free to message me and I'll check back to see what it is. Until then, have fun!