# How is this a representation of a 3 dimensional torus?

• I
• docnet
In summary, a torus in differential geometry can be defined by a pair of equations representing two orthogonal 2-dimensional Euclidean spaces. This can be visualized as a circle rotating around another circle. In 4-dimensional space, the torus meets two orthogonal 2-dimensional spaces in circles. Alternatively, the torus can be represented as the product of two orthogonal Euclidean planes, with the equations defining the intersection of two cylinders. Thank you for the clarification.
docnet
Gold Member
TL;DR Summary
How is this expression of a torus?
In a differential geometry text, a torus is defined by the pair of equations:

I initially thought this was somehow a torus embedded in 4 dimensions, but I do not see how we can visualize two orthogonal 2-dimensional Euclidian spaces. How is this a representation of a 2 dimensional torus embedded in 3 dimensions, with 4 presumably orthogonal coordinates?

Last edited:
This is a two dimensional torus.

The idea is you can imagine a torus as a circle rotated around in a circle. The first equation defines your location on the first circle, and the second the second circle.

docnet
in 4 space with coords p,q,r,s, the subspaces p=q=0 and r=s=0 are two orthogonal euclidean 2 dimensional spaces. this torus meets each of them in a circle.

Alternatively, R^4 = R^2 x R^2 is the product of two orthogonal euclidean planes. your two equations define two (3 dimensional) "cylinders", one in each of these products, i.e. S^1 x R^2 and R^2 x S^1. Setting both equations equal to zero defines the intersection of these two cylinders, namely (S^1xR^2)intersect(R^2xS^1) = S^1xS^1.

docnet
thank you for your replies. It makes more sense now. It is a product of the sets {p^2+q^2=1} and {r^2+s^2=1}.

mathwonk

## 1. What is a 3 dimensional torus?

A 3 dimensional torus is a geometric shape that is formed by rotating a circle around an axis in three-dimensional space. It is a three-dimensional version of a donut or a tire, with a hole in the center and a circular cross-section.

## 2. How is a 3 dimensional torus represented?

A 3 dimensional torus can be represented mathematically using equations and coordinates in three-dimensional space. It can also be represented visually using computer graphics or physical models.

## 3. What is the significance of a 3 dimensional torus in science?

A 3 dimensional torus has many applications in science, including in physics, mathematics, and engineering. It is used to model various phenomena such as fluid flow, electromagnetic fields, and the structure of the universe.

## 4. How is this a representation of a 3 dimensional torus?

This representation of a 3 dimensional torus may be a visual or mathematical representation that demonstrates the key characteristics of a torus, such as its circular cross-section, hole in the center, and three-dimensional shape.

## 5. What are some real-world examples of a 3 dimensional torus?

Some examples of a 3 dimensional torus in the real world include the shape of a donut, the shape of a tire, and the shape of a smoke ring. It can also be found in various man-made structures such as pipes, tunnels, and roller coasters.

Replies
8
Views
820
Replies
3
Views
2K
Replies
28
Views
6K
Replies
1
Views
2K
Replies
8
Views
2K
Replies
12
Views
2K
Replies
6
Views
2K
Replies
8
Views
1K
Replies
18
Views
388
Replies
1
Views
2K