What is the symbol of this torus?

In summary, the conversation discusses the construction of a torus from a square or rectangle, and the notation and lattices involved in quotient groups. It also briefly mentions the E8 Lie group and its corresponding root lattice.
  • #1
Heidi
411
40
Homework Statement
tori
Relevant Equations
R2/Z2
Hi Pfs,
I can get a taurus from a square: I identify the oppsite sides. It has the symbol $R^2/ Z^2 $
Suppose now that i replace this square by a rectangle with the length L ans 2L. I identify the opposite sides in the same manner. The new taurus is also a quotient of $R^2$ but how to write it?
 
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  • #2
Heidi said:
Homework Statement:: tori
Relevant Equations:: R2/Z2

Hi Pfs,
I can get a taurus from a square: I identify the oppsite sides. It has the symbol $R^2/ Z^2 $
Suppose now that i replace this square by a rectangle with the length L ans 2L. I identify the opposite sides in the same manner. The new taurus is also a quotient of $R^2$ but how to write it?
"Taurus" is the Latin word for bull, and is also the name of a constellation of stars in the sky near Orion. The word you want is torus (pl. tori).

It appears to me that you're talking about quotient groups, but it's been so long since my year-long sequence in Modern Algebra that I don't remember how tori are related to quotient groups.

To actually make a torus from a square sheet of rubber, stretch the square in one direction to make a rectangle. Then connect the long edges of the rectangle to make a tube, and finally, connect the ends of the tube (the rolled short edges of the rectangle).
 
  • #3
Of course it was a torus and not a taurus :) even if we have bullfights in the south of France (i do not like).
 
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  • #4
Heidi said:
Homework Statement:: tori
Relevant Equations:: R2/Z2

Hi Pfs,
I can get a taurus from a square: I identify the oppsite sides. It has the symbol $R^2/ Z^2 $
Suppose now that i replace this square by a rectangle with the length L ans 2L. I identify the opposite sides in the same manner. The new taurus is also a quotient of $R^2$ but how to write it?

The quotient [itex]\mathbb{R}^2 / \mathcal{L}[/itex] where [tex]\mathcal{L} = \{ (na, mb) : (n,m) \in \mathbb{Z}^2 \}[/tex] is homeomorphic to [itex]\mathbb{R}^2 / \mathbb{Z}^2[/itex] under [itex](x,y) \mapsto (x/a,y/b)[/itex].

EDIT: [itex]\mathcal{L}[/itex] can also be written as [itex]a\mathbb{Z} \times b\mathbb{Z}[/itex] or [itex]a \mathbb{Z} \oplus b \mathbb{Z}[/itex].)
 
Last edited:
  • #5
I believe re your new quotient, that it's thee quotient is by ##(2\mathbb Z + \mathbb Z)##
But then again, I may be biased, as Gemini don't in general like those that are Taurus. ;).
 
  • #6
I saw the notation $R^{16} / E8 \oplus E8$.
Do you know what is the lattice in the quotient?
 
  • #7
Maybe the Lie group E8? Doesn't seem like it though.
 
Last edited:
  • #8
Why do i see not answered (a "-" in front of the number of answers)?
 
  • #9
Heidi said:
I saw the notation $R^{16} / E8 \oplus E8$.
Do you know what is the lattice in the quotient?

From https://en.wikipedia.org/wiki/E8_(mathematics):
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.

In this context, I suspect the root lattice is meant; the sum of two linearly independent copies of it ([itex]E_8 \oplus E_8[/itex]) would make a lattice in [itex]\mathbb{R}^{16}[/itex].
 

1. What is a torus?

A torus is a three-dimensional geometric shape that resembles a donut or an inner tube. It is created by rotating a circle around an axis that is coplanar with the circle.

2. What is the symbol of a torus?

The symbol of a torus is a capital letter T with a small circle or dot above it, representing the cross section of the torus.

3. What does the symbol of a torus represent?

The symbol of a torus represents the shape of a torus and is commonly used in mathematics and physics to represent this geometric shape.

4. How is the symbol of a torus used in science?

The symbol of a torus is used in science to represent the torus shape in various equations and diagrams, such as in topology, electromagnetism, and fluid dynamics.

5. Are there any other names for the symbol of a torus?

Yes, the symbol of a torus is also known as the "torus cross section" or the "torus knot symbol".

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