IF i was a 1 dimensional being living in a 1 dimensional Universe being a circle.

In summary, the 1D universe could be a circle if a 1D being did determine the shape of his universe by marking a position and seeing if he came back upon himself. However, this is not necessary and a 1D universe could exist without shape.
  • #1
ugalpha
22
0
How could i mathematically proove that I am living on a circle. Almost got it last night , just need an insight to figure out an equation.

Ty.
 
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  • #2
Circle is object in two dimensions!
 
  • #3
the circle being 1,2,3,4,5 or 6 dimensions is not important. I need the maths to get me there.
 
  • #4
ugalpha said:
How could i mathematically proove that I am living on a circle...

I like the question! In differential geometry you do not need an n-dimensional space to be embedded in a space of higher dimension. Probably I disagree with Marina. You can define what the 2-sphere is without it having to live in a 3D environment.

So a ring can be defined and studied without assuming the existence of any surrounding 2D space.

However it is trivial. there is no intrinsic way to define a curvature that could be non-zero. There is (at least in my opinion) no way that the 1-dim creature can discover the topology of his world unless he makes a voyage of exploration. He must circumnavigate his world.

He must tell his 1D friends to stay at one place, to mark that place, and then he must travel as far as necessary in one direction. when he sees his friends again he will know that it is a ring.
 
  • #5
surely a ring has more than on dimension, two at the least

A 1D universe would just be a line if a 1D being did try to determine the shape of its universe by marking a position and seeing if he came back upon himself suggests that the universe is 2D or greater. A 1D universe would not have a shape to it. Or am I mistaken? show me a shape that has 1 dimension

As for the Op, I have no idea how a 1D being would be able to prove he is on a 1D universe using mathematics

EDIT - It has come to my attention that even a line is a 2D object. I cannot see or imagine how a 1D universe would work, it would have no shape in any sense of the word, making your question impractical.
 
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  • #6
shifty88 said:
surely a ring has more than on dimension, two at the least

A 1D universe would just be a line if a 1D being did try to determine the shape of its universe by marking a position and seeing if he came back upon himself suggests that the universe is 2D or greater. A 1D universe would not have a shape to it. Or am I mistaken? show me a shape that has 1 dimension

As for the Op, I have no idea how a 1D being would be able to prove he is on a 1D universe using mathematics

EDIT - It has come to my attention that even a line is a 2D object. I cannot see or imagine how a 1D universe would work, it would have no shape in any sense of the word, making your question impractical.

You can have a one-dimensional line that is effectively looping back on itself by just identifying two endpoints of the line segment and thus giving it non-trivial topology. This does not require another dimension.
 
  • #7
You can have a one-dimensional line that is effectively looping back on itself by just identifying two endpoints of the line segment and thus giving it non-trivial topology. This does not require another dimension.

Does anyone have any links that can support this sentence?
I do not see how a 1 dimensional universe can loop back on itself without the presence of an extra dimension

I imagine a 1D universe to consist of just length, without width or depth, without these dimensions their would be no discernible shape to it.
Have I misunderstood something here?
 
  • #8
shifty88 said:
Does anyone have any links that can support this sentence?
I do not see how a 1 dimensional universe can loop back on itself without the presence of an extra dimension

I imagine a 1D universe to consist of just length, without width or depth, without these dimensions their would be no discernible shape to it.
Have I misunderstood something here?

Yes, you are missing out topology and a whole field of mathematics known as differential geometry. Modern differential geometry basically started with Riemann's realization in the 19th century that we can talk about geometry *intrinsic* to a surface without embedding it in a higher dimensional space. Perhaps this will help.
 
  • #9
But how does this apply to the Op's question and a 1 dimensional existence.
How can there be topology in the 1D
 
  • #10
  • #11
yenchin said:
See Fig 1. of this article.

Thanks for that. Helped me see the possibilities in a way I wasn't thinking of them. I was stuck on the concept that the 1D universe had to be an open line segment, somehow didn't wedge it into my brain that a circle is a valid version of a 1D universe.
 
  • #12
shifty88 said:
But how does this apply to the Op's question and a 1 dimensional existence.
How can there be topology in the 1D

For a math major in the 1960s one's first exposure to topology was the topology of the line (the real numbers) which of course is 1 dimensional. Many of the definitions can be illustrated. Topology is a rich field.

yenchin said:
Yes, you are missing out topology and a whole field of mathematics known as differential geometry. Modern differential geometry basically started with Riemann's realization in the 19th century that we can talk about geometry *intrinsic* to a surface without embedding it in a higher dimensional space. Perhaps this will help.

Thanks yenchin! also for your reply in this other thread, where Chalnoth also joined in with some helpful amplification.
https://www.physicsforums.com/showthread.php?p=3724707#post3724707

As I see it, some of these elementary math questions might be better pursued in the math forum. Or in Relativity forum. After all, cosmology (the topic here) is based on GR which in turn is based on differential geometry as developed in the 19th century by Riemann.
If you want to discuss the basics of the math foundations---the language used to formulate cosmic models---differential geometry---maybe this is not the place.
 
  • #13
yenchin said:
See Fig 1. of this article.

Thanks for clearing that up for me.
 
  • #15
shifty88 said:
I imagine a 1D universe to consist of just length, without width or depth, without these dimensions their would be no discernible shape to it.
Have I misunderstood something here?

Remember, a line does have only length, even if it happens to be a line that loops upon itself (think the circumference of a circle), a 2 dimensional object has area (think the surface of a sphere). A 3 dimensional object has volume, etc. The only reason some might think that a circle (or any line) is 2 dimensions is that we are viewing it from the perspective of 3 dimensions.
 
  • #16
How can a straight (one dimensional) line loop around anything ? Doesn't make sense to me. Or has 'straight' been redefined to mean 'slightly curved' ?
 
  • #17
alt said:
How can a straight (one dimensional) line loop around anything ? Doesn't make sense to me. Or has 'straight' been redefined to mean 'slightly curved' ?

Why are you imposing "straight" on the concept of one-dimensional? The OP correctly did not.
 
  • #18
alt said:
How can a straight (one dimensional) line loop around anything ? Doesn't make sense to me. Or has 'straight' been redefined to mean 'slightly curved' ?

Don't even think of a shape at all.
Takes the numbers 1 to 10. That is a straight, 1 dimensional line.
Now when you count to 10, what's next after 10? Stop or loop back to 1?

I'm not sure what a 'not straight' 1d line would be like...
Just a guess, but I would suppose it would be a non-linear line. As in not all points are evenly distributed. Count from 1 to 10 but skip 3,4 and 7.
 
  • #19
alt said:
How can a straight (one dimensional) line loop around anything ? Doesn't make sense to me. Or has 'straight' been redefined to mean 'slightly curved' ?

Think of a straight line in cartesion coordinates, say, y = 2. The equation of this is a constant function, with no value that varies (for every x value, y is the same). Now, take a circle in polar coordinates, defined by the equation r = 2. This is a constant function, also with no value that varies (for every theta, r is the same).

I think the problem is that there are two distinct ideas of dimension that are being used here. One is mathematical, and the other is visual. People probably tend to think in terms of cartesian coordinates, so anything they think of is visually embedded in the 3-d certsian system, and a circle "visually" to us is in 2 dimensions (say, x and y). However, in mathematics,, and specifically differential geometry, as has been pointed out, it is unnecessary to embed any curve in a higher dimensional space, so that a "line" in some arbitrary space is defined by the curvature of that space.
 
  • #20
daveb said:
I think the problem is that there are two distinct ideas of dimension that are being used here. One is mathematical, and the other is visual. People probably tend to think in terms of cartesian coordinates, so anything they think of is visually embedded in the 3-d certsian system, and a circle "visually" to us is in 2 dimensions (say, x and y). However, in mathematics,, and specifically differential geometry, as has been pointed out, it is unnecessary to embed any curve in a higher dimensional space, so that a "line" in some arbitrary space is defined by the curvature of that space.

I was of the opinion that we were talking of visual dimensions, but that's been cleared up for e since, i was unaware of topology and all that jazz, but now i am I have plenty of studying to do in my spare time now. Thanks to this thread. If anyone can provide more reading material on this subject i would really appreciate it.
 
  • #21
shifty88 said:
I was of the opinion that we were talking of visual dimensions, but that's been cleared up for e since, i was unaware of topology and all that jazz, but now i am I have plenty of studying to do in my spare time now. Thanks to this thread. If anyone can provide more reading material on this subject i would really appreciate it.

This is a great book.

You may also enjoy playing this.
 
  • #22
phinds said:
Why are you imposing "straight" on the concept of one-dimensional? The OP correctly did not.

I was thinking one dimension, one direction only, therefore straight. Surely if it deviates from that one dimension / direction, it is no longer straight ? A non straight line would have two dimensions wouldn't it ? No matter how imperceptable to that 1D being ?
 
  • #23
Fuzzy Logic said:
Don't even think of a shape at all.
Takes the numbers 1 to 10. That is a straight, 1 dimensional line.
Now when you count to 10, what's next after 10? Stop or loop back to 1?

I'm not sure what a 'not straight' 1d line would be like...
Just a guess, but I would suppose it would be a non-linear line. As in not all points are evenly distributed. Count from 1 to 10 but skip 3,4 and 7.

But I have to think of shape. I've seen it stated many times here and elswhere, that mathematics is but a language. A language with which to interpret the real world - in this case, shape.

Standing alone, and without any reference to the real world, the numbers 1-10 mean nothing (to me).
 
  • #24
A 1 dimensional line can curve in higher dimensions, just like a 2 dimensional sheet of paper can be curved in a 3rd dimension.

A sheet of paper, curved, folded or spindled still has a 2 dimensional surface.

Why?

To uniquely describe any point on the sheet of paper requires two and only two coordinates: x and y.

Same with a line. When curved, it is still one dimensional. To uniquely describe any point on the line requires one and only one coordinate.
 
  • #25
I can't see how a line, once curved, doesn't trace a two dimensional path.

Nor how a sheet of paper, when curved or folded or spindled, doesn't enter a 3D space.
 
  • #26
alt said:
I can't see how a line, once curved, doesn't trace a two dimensional path.

Nor how a sheet of paper, when curved or folded or spindled, doesn't enter a 3D space.

They are curved in a higher space but the space itself is still only as dimensional as the number of coordinates it takes to define a unique point.

I can't say it more clearly than that. Regardless of how you bend fold or spindle a sheet of paper, you still only need two coordinates to uniquely define a point in it. You do not need a third coordinate.
 
  • #27
DaveC426913 said:
They are curved in a higher space but the space itself is still only as dimensional as the number of coordinates it takes to define a unique point.

I can't say it more clearly than that. Regardless of how you bend fold or spindle a sheet of paper, you still only need two coordinates to uniquely define a point in it. You do not need a third coordinate.

I think the issue was that, earlier in the thread, there were claims that a line could loop back on itself without curving in a higher dimension.
 
  • #28
DaveC426913 said:
A 1 dimensional line can curve in higher dimensions, just like a 2 dimensional sheet of paper can be curved in a 3rd dimension.

A sheet of paper, curved, folded or spindled still has a 2 dimensional surface.

Why?

To uniquely describe any point on the sheet of paper requires two and only two coordinates: x and y.

Same with a line. When curved, it is still one dimensional. To uniquely describe any point on the line requires one and only one coordinate.

Thanks for this. Totally cleared that all up for me.

I'd say the confusion about the 1D line curving into a circle is that the people are looking at the circle as an area and not at the circumference which is what the focus is really about. The circumference is still only a line. The line has only a length and picking an arbitray point to start you can relate how far you have traveled with only a single coordinate: the distance from where you started. Regardless of whether there is a 2nd dimension in which the circle exists it isn't necessary to describe it in order to describe your location along the circumference.
 
  • #29
e^(i Pi)+1=0 said:
I think the issue was that, earlier in the thread, there were claims that a line could loop back on itself without curving in a higher dimension.

The point is: while a line could not loop back on itself to form a circle without curving in a higher dimension, we could have a line modulo an equivalence relation so that it is topologically effectively a circle, and thus no higher dimension is required in this case. Since we are talking about cosmology, a resident *in* a universe only has access to its intrinsic geometry, and thus won't be able to tell the difference between a line curving in higher dimension [which is *outside* the universe] and a topologically nontrivial line.

Similarly in a two dimensional infinite cylindrical universe, the inhabitants measure each triangle to sum up to 180 degrees, intrinsically the universe has *flat* geometry, with one compact direction [so that if they travel far enough along that direction they return to where they started]. There is no difference, according to the inhabitants, whether their world is a cylinder curving in higher dimensional space [the curvature here is the mean curvature, which is *extrinsic*, not the Gaussian curvature which is *intrinsic*], or simply a flat strip of plane [itex][a,b] \times \Bbb{R}[/itex] with opposite sides topologically identified.
 
  • #30
e^(i Pi)+1=0 said:
I think the issue was that, earlier in the thread, there were claims that a line could loop back on itself without curving in a higher dimension.

If I understand correctly, it can. But I don't have the math to demonstrate it. I'd have to defer to someone with more chops mathwise than I.
 
  • #31
yenchin said:
Similarly in a two dimensional infinite cylindrical universe, the inhabitants measure each triangle to sum up to 180 degrees, intrinsically the universe has *flat* geometry.
I just want to ask - this is true only for cylindrical geometry, not for sphere, right?
 
  • #32
phinds said:
Why are you imposing "straight" on the concept of one-dimensional? The OP correctly did not.

I'm sorry for taking this argument 2 days back, but I think I can see the problem OP is having concerning curved 1-dimensional lines. Being curved implies being oriented in a different direction, and thus, along a different axis. So to give a description of a point along a line in an ℝ^n space would require n different coordinates. But, correct me if I'm wrong, even though the line is described in an ℝ*^n space, the line itself has only one dimension, it's length.
 
  • #33
rustynail said:
I'm sorry for taking this argument 2 days back, but I think I can see the problem OP is having concerning curved 1-dimensional lines. Being curved implies being oriented in a different direction, and thus, along a different axis. So to give a description of a point along a line in an ℝ^n space would require n different coordinates. But, correct me if I'm wrong, even though the line is described in an ℝ*^n space, the line itself has only one dimension, it's length.

Correct! Very astute observation!
 
  • #34
rustynail said:
I'm sorry for taking this argument 2 days back, but I think I can see the problem OP is having concerning curved 1-dimensional lines. Being curved implies being oriented in a different direction, and thus, along a different axis. So to give a description of a point along a line in an ℝ^n space would require n different coordinates. But, correct me if I'm wrong, even though the line is described in an ℝ*^n space, the line itself has only one dimension, it's length.
I thought that's what I said in post 24:
https://www.physicsforums.com/showpost.php?p=3731892&postcount=24
and post 26:
https://www.physicsforums.com/showpost.php?p=3733530&postcount=26
 
<h2>1. What is a 1 dimensional being?</h2><p>A 1 dimensional being is a hypothetical creature that exists in a world with only one spatial dimension. This means that it can only move in one direction, either forward or backward.</p><h2>2. What is a 1 dimensional Universe?</h2><p>A 1 dimensional Universe is a theoretical universe that only has one spatial dimension. This means that it has no width or depth, and all objects in this universe would only exist as a line or a point.</p><h2>3. What does it mean to be a circle in a 1 dimensional Universe?</h2><p>In a 1 dimensional Universe, a circle would appear as a single point. This is because a circle is a 2 dimensional shape, and in a 1 dimensional world, it would only have one dimension to exist in.</p><h2>4. How would a 1 dimensional being perceive its surroundings?</h2><p>A 1 dimensional being would only be able to perceive objects that exist in its one-dimensional world. It would not be able to see or understand objects that have width or depth, as these dimensions do not exist in its universe.</p><h2>5. Is it possible for a 1 dimensional being to exist?</h2><p>It is currently not possible for a 1 dimensional being to exist in our universe. Our universe has three spatial dimensions (length, width, and depth), and it is impossible for a being to exist with only one of these dimensions. However, in a theoretical 1 dimensional universe, it is possible for a 1 dimensional being to exist.</p>

1. What is a 1 dimensional being?

A 1 dimensional being is a hypothetical creature that exists in a world with only one spatial dimension. This means that it can only move in one direction, either forward or backward.

2. What is a 1 dimensional Universe?

A 1 dimensional Universe is a theoretical universe that only has one spatial dimension. This means that it has no width or depth, and all objects in this universe would only exist as a line or a point.

3. What does it mean to be a circle in a 1 dimensional Universe?

In a 1 dimensional Universe, a circle would appear as a single point. This is because a circle is a 2 dimensional shape, and in a 1 dimensional world, it would only have one dimension to exist in.

4. How would a 1 dimensional being perceive its surroundings?

A 1 dimensional being would only be able to perceive objects that exist in its one-dimensional world. It would not be able to see or understand objects that have width or depth, as these dimensions do not exist in its universe.

5. Is it possible for a 1 dimensional being to exist?

It is currently not possible for a 1 dimensional being to exist in our universe. Our universe has three spatial dimensions (length, width, and depth), and it is impossible for a being to exist with only one of these dimensions. However, in a theoretical 1 dimensional universe, it is possible for a 1 dimensional being to exist.

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