Explore the Two-Torus T²: A Visual Guide and Explanation

  • Context: Graduate 
  • Thread starter Thread starter waterfall
  • Start date Start date
  • Tags Tags
    Picture
Click For Summary

Discussion Overview

The discussion centers around the visualization and understanding of the two-torus T², including its representation in different dimensions and its topological properties. Participants seek visual aids and explore related concepts in higher-dimensional topology.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant expresses difficulty in finding images of a two-torus T² and requests assistance.
  • Another participant suggests that a two-torus is simply a doughnut shape and provides a link to a Wikipedia page for reference.
  • A third participant shares a specific image of the standard embedding of T² in R³.
  • One participant reflects on the simplicity of the torus and mentions interest in higher-dimensional topologies, specifically asking for resources on 6-dimensional Calabi-Yau manifolds and their properties.
  • A later reply notes that the standard embedding of T² in R³ is not flat and suggests considering it as a product of circles in R⁴ for a different perspective.

Areas of Agreement / Disagreement

Participants express differing views on the dimensionality and properties of the two-torus, with some focusing on its standard representation while others are interested in more complex topological structures. The discussion remains unresolved regarding the best way to visualize and understand these concepts.

Contextual Notes

There is a lack of consensus on the dimensionality and flatness of the two-torus, as well as the appropriate visual representations for higher-dimensional topologies. Some assumptions about the nature of embeddings and dimensionality are not fully explored.

waterfall
Messages
380
Reaction score
1
Hi, I can't find any picture of a two-torus T² in the net after much searching. Does anyone have a link or can draw what it looks like? Thanks.
 
Physics news on Phys.org
oh. so it's just a donut.. I was thinking of many dimensional donuts...

if anyone has links to videos which show clearly how complex topologies like 6 dimensional Calabi-Yau can be ricci-flat, pls share it so it can be appreciated visually... thanks.
 
The standard embedding of T^2 in R^3 is of course not flat. You need to consider it as a product of circles in R^4 for that.
 

Similar threads

Graduate Can I find a smooth vector field on the patches of a torus?
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Is This the Correct Way to Attach Möbius Strips to a Torus?
  • · Replies 12 ·
Replies
12
Views
5K
Graduate What is the connection between the mapping class group of a torus and Gl(2,Z)?
  • · Replies 2 ·
Replies
2
Views
4K
Guiding Magnetic Fields in Solenoid Coils: Design and Material Considerations
  • · Replies 22 ·
Replies
22
Views
4K
Question on Android phone system: Moving files & the Gallery
  • · Replies 16 ·
Replies
16
Views
6K
Undergrad Quantum Circuit Confusion On Time Evolution
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Visualization of complex vectors and dot product
  • · Replies 14 ·
Replies
14
Views
4K
Finding Planar Representation of Torus with n Holes
  • · Replies 5 ·
Replies
5
Views
2K
Requesting Resources and Animations to understand Solid State Physics
  • · Replies 4 ·
Replies
4
Views
2K
Experiment - how temperature changes along a tube, using heaters
  • · Replies 14 ·
Replies
14
Views
2K