What is the symbol of this torus?

[Heidi]

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?

 

[Mark44]

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).

 

[Heidi]

Of course it was a torus and not a taurus :) even if we have bullfights in the south of France (i do not like).

 

[pasmith]

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].)

 

[WWGD]

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. ;).

 

[Heidi]

I saw the notation $R^{16} / E8 \oplus E8$.
Do you know what is the lattice in the quotient?

 

[WWGD]

Maybe the Lie group E8? Doesn't seem like it though.

 

[Heidi]

Why do i see not answered (a "-" in front of the number of answers)?

 

[pasmith]

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].