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 .