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

ugalpha

How could i mathematically proove that im living on a circle. Almost got it last night , just need an insight to figure out an equation.

Ty.

supermarina

Circle is object in two dimensions!

ugalpha

the circle being 1,2,3,4,5 or 6 dimensions is not important. I need the maths to get me there.

marcus

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.

shifty88

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.

yenchin

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.

shifty88

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?

yenchin

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.

shifty88

But how does this apply to the Op's question and a 1 dimensional existence.
How can there be topology in the 1D

yenchin

See Fig 1. of this article.

phinds

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.

marcus

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.

Thanks yenchin! also for your reply in this other thread, where Chalnoth also joined in with some helpful amplification.
http://physicsforums.com/showthread....07#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.

shifty88

Thanks for clearing that up for me.

marcus

Wow! Matt Strassler does a great job of explaining here!
http://profmattstrassler.com/article...ic-confusions/
I spelled your Figure 1 link out simply to amplify: his post is "Big Bang Classic Confusions" and we could make regular use of it here, it deals with several issues besides the dimensional and topological one.

daveb

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.

alt

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' ?

phinds

Why are you imposing "straight" on the concept of one-dimensional? The OP correctly did not.