How can a 1-dimensional being prove they live on a circle?

  • Context: Graduate 
  • Thread starter Thread starter ugalpha
  • Start date Start date
  • Tags Tags
    Circle Universe
Click For Summary

Discussion Overview

The discussion revolves around the question of how a one-dimensional being could mathematically prove that it lives on a circle. Participants explore the implications of dimensionality, topology, and the nature of mathematical proof in a one-dimensional universe.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that a one-dimensional being could determine its existence on a circle by circumnavigating its world and returning to a starting point.
  • Others argue that a circle is inherently a two-dimensional object, and a one-dimensional universe would not possess a shape, leading to confusion about the nature of dimensionality.
  • A participant mentions that a one-dimensional line can loop back on itself by identifying two endpoints, creating a non-trivial topology without requiring an additional dimension.
  • Some express skepticism about how a one-dimensional universe could have topology or a discernible shape, questioning the validity of the original premise.
  • References to differential geometry and topology are made, with some participants noting that these fields allow for the discussion of intrinsic properties without embedding in higher dimensions.
  • There is a discussion about the definitions of straight and curved lines in the context of one-dimensional spaces, with varying interpretations of what constitutes a one-dimensional shape.

Areas of Agreement / Disagreement

Participants do not reach consensus on whether a one-dimensional being can prove it lives on a circle. There are competing views regarding the nature of one-dimensional spaces and their ability to exhibit topological properties.

Contextual Notes

Some participants highlight the limitations of understanding one-dimensional spaces, particularly regarding the assumptions about shape and dimensionality. The discussion also touches on the historical development of concepts in topology and differential geometry, which may not be universally understood among participants.

  • #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?
 
Space news on Phys.org
  • #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
 
  • #36
minio said:
I just want to ask - this is true only for cylindrical geometry, not for sphere, right?

Yes. In a sphere you will know your space is curved *intrinsically*, so again you don't have to talk about any space outside of the sphere, that's the point of intrinsic geometry.
 

Similar threads

Graduate How do we say that the universe we live is 3 dimensional?
  • · Replies 4 ·
Replies
4
Views
2K
Find Values of Osculating Circle Tangent to a Parabola
  • · Replies 4 ·
Replies
4
Views
2K
MHB How to Prove a Complex Number Equation and Its Trajectory Forms a Circle?
  • · Replies 2 ·
Replies
2
Views
3K
High School What would the universe be like with less mass and energy?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Problem understanding how universe is infinitely flat
  • · Replies 10 ·
Replies
10
Views
9K
Undergrad Different norms and how L1 leads to the sparsest solution
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Proton decay at the end of the Universe
  • · Replies 13 ·
Replies
13
Views
2K
High School What is the important scale of things in the Universe?
  • · Replies 11 ·
Replies
11
Views
10K
Graduate Nature and The Universe, How can Nature being conscious, know what it is doing ?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Big Bang, big rip, lQc, has anyone ever run a computer simulation?
  • · Replies 5 ·
Replies
5
Views
2K