<sup>This is a repost of a question I posted at [MSE][1].</sup>



Mark L. Irons' paper [The Curvature and Geodesics of the Torus][2] gives a concise overview of the [geodesics][3] on the [torus][4]: 

- There are five clear-cut families of geodesics.
- Most of the geodesics are "ergodic": aperiodic and covering either the entire surface - by spiraling endlessly around - or substantial parts of it.
- Some of the geodesics are "boring": the meridians, the inner and the outer equator
- A few of them are "æsthetically pleasing": returning to their starting point after just a few circuits. 

> Does the structure of geodesics change when
> twisting the "hose" before gluing its ends?

![alt text][5]

E.g., there might be no equators anymore because after twisting the (two) equators lost their "ends".


  [1]: http://math.stackexchange.com/questions/219499/geodesics-on-the-torus
  [2]: http://www2.rdrop.com/~half/math/torus/torus.geodesics.pdf
  [3]: http://en.wikipedia.org/wiki/Geodesic
  [4]: http://en.wikipedia.org/wiki/Torus
  [5]: http://upload.wikimedia.org/wikipedia/commons/6/60/Torus_from_rectangle.gif