An introduction to torus and a related fundamental questions on quotient mappings

Let us consider the following mapping on the square [0,1] $\ X$ [0,1] as follow :

($\ x$,0)~($\ x$,1) and (0,$\ x$) ~ (1 ,$\ x$) . The quotient space defined on it is called the two dimensional torus . Can anyone tell me how to find out the shape of a torus from this description ?

I have read that when we define an equivalence relation on a line such that two of its endpoints are mapped to the same quotient set then the resulting shape is a "circle "and similarly when we define an equivalence relation on a rectangle such that the points on one of its side are taken with their corresponding point (obtained as the point of intersection of the perpendicular dawn form the point to the parallel side ) on the other parallel edge then we obtain a cylinder.

what if I like to know the shape when we map the points on a side AB of a rectangle ABCD, to the points on a perpendicular side,BC, such that a point $\ p$ on AB is related to point $\ q$ on BC , if $\ q$ lies on the line passing through $\ p$ and making an angle of $\ \pi$/6 with AB or let's say if I want to know how to define a mapping on a cylinder that converts it into a cone or any other question of that kind .

As we can see in the description of 2-d Torus the mapping was described only about points having coordinates of the form ($\ x$,0) or ($\ x$, 1) or (0,$\ x$) or (1,$\ x$) but I guess in the resulting shape other points did get affected (hope I am right in saying so ) as the shape got changed . I would like to know why did they get affected in this way to form a cylinder and why not a shape like a crushed cola can ? What constraints do the other points adhere to ?

WAITING DESPERATELY FOR AN ANSWER . I WISH I HAD SUFFICIENT REPUTATION TO OFFER A BOUNTY

PS: I am afraid this question might get classified as a broad one. So , I would like to say that in general I am trying to know in what way shapes are modified under different quotient maps , how to find a mapping for a particular shape transformation , and also how does a torus look like.

PPS : Please do inform me if there is anything repulsive in the way I have put this question . In case you are passing by and know of the name of some concept which is used for this kind of question , kindly cite the name of the concept .

• As a start, you might have a look at Jeff Weeks' computer games, which include several intuitively-compelling games on the torus. – Andrew D. Hwang May 14 '16 at 11:35
• @AndrewD.Hwang : For some reason , I was unable to download these games .So , could you help me with a mathematical answer ? – itp dusra May 15 '16 at 8:26

1 Answer

The question is intuitively reasonable (+1), but on closer inspection may not be as well-posed as it first appears. Here are some quick introductory points by way of explanation.

The "shape" of a topological space can mean multiple things: How the space is internally connected (topology); how distances and angles are measured internally (intrinsic differential geometry); how the space bends inside a larger-dimensional space, such as a curve or surface in Euclidean space (extrinsic differential geometry); and so forth.

The language of the question suggests you're thinking of topology: What, topologically, does a torus look like? What about other spaces obtained from a square by boundary identification?

Here are three answers to "What does a torus look like?", chosen to be rather widely varying:

1. A torus looks like an inner tube, obtained by bending the axis of a rubber cylinder around a circle, bringing the ends together. It turns out this operation can be performed more nicely in four-dimensional space; just as a flat square of paper can be rolled into a cylinder without tearing, folding, or crinkling the paper, in four-space the resulting cylinder can be rolled into a torus without tearing, folding, or crinkling. In three-space, the best you can do is to flatten the cylinder into a rectangle, then "roll up the long direction", bringing the ends together. The result is a two-ply cylinder (or a four-ply square) with no free boundary.

2. A torus looks like the universe of certain video games: When one's avatar "leaves" the screen, it immediately reappears on the opposite edge. If you imagine the rectangular screen rolled up joining the left side to the right and the bottom to the top, the movement of one's avatar is continuous; the "jump" between opposite sides of the screen is an illusion necessitated by cutting open the toroidal universe.

3. A torus looks like an infinite plane tiled by squares (or, more generally, by congruent parallelograms). The catch is, there is only one square: What appears to be a symmetric array of spirals below is "really" a single spiral. A "point" is this space is an infinite lattice of points in the plane, such as the set of purple spots. A video game avatar in this universe who starts at a purple spot $A$ and walks in a straight line toward another spot $B$ returns to its original position when it reaches $B$.

As these examples show, "What does a quotient space look like?" can have several correct answers that do not appear equivalent.

Asking "How does the shape of a square change when boundary points are identified?" is difficult to address as stated. Locally and intrinsically, "the only changes occur on the boundary of the square, not at interior points".

I suspect, however, that you're asking how to represent a quotient in three-space (extrinsically). In general, that's visually tricky. Quotient spaces obtained from a square by boundary identification include the Klein bottle, the dunce cap, and the real projective plane.

A Topological Picturebook by George K. Francis contains many examples with gorgeous, hand-drawn illustrations. If you can find a library copy, I recommend the book highly.

• @AndrewDHwang : Tons of thanks for your reply ,first of all . I know of a device used to generate magnetic fields named as toroids . I am sure that you are aware of it . when you mean the "inner tube" , do you mean the interior of the ring of toroid or the coils of it . I strongly guess that you mean the former, please clarify . – itp dusra May 16 '16 at 16:16
• @AndrewDHwang : Now the third definition , it was kind of mystically intriguing when you said that "the symmetric array of spirals is just one spiral " and "when I read that "the video game avatar reaches back to its inital place after going around a spiral ". – itp dusra May 16 '16 at 16:20
• @AndrewDHwang : Do you mean that each of these square tiles , owing to their completely similar characteristics are viewed as a single square tile . The points of the squares are in an equivalence relation ,is it so ? – itp dusra May 16 '16 at 16:22
• @AndrewDhwang : Could you illustrate the process of deciding how a quotient space looks like by solving for atleast one of the cases that I have put on this question ? – itp dusra May 16 '16 at 16:25
• @AndrewDHwang : I am still not clear about "how these changes in surface affect other points ". Like in the case of a square ABCD, getting converted into a cylinder , when AB and CD were glued , BC didn't remain a straight line anymore but turned into a curve . – itp dusra May 16 '16 at 16:30