# Math Help - Is this surface possible?

1. ## Is this surface possible?

Does there exist a surface such that if one stands on the surface at any given point and then freely chooses a direction in which to walk then the ensuing straight line path on the surface will necessarily reach every point on the surface?

Effectively, does there exist a surface such that if one was lost on the surface then one could guarantee to get home (a point somewhere) by simply choosing any direction and walking straight in this direction?

2. Good question! Is that an assignment question or just something you've wondered yourself?

You'd first have to define what a "straight line" is on this surface. Usually, such a "straight line" is called a geodesic.
Here is a trivial case: if the surface is not path-connected, then it certainly is impossible to go from a point in some path component to some point in another path component.

But if the surface is path connected, I don't know!

3. If you walk a line on a torus which has an irrational slope, then you'll eventually get arbitrarily close to any given point. And, of course, if you pick a direction "at random", it's pretty well guaranteed to be irrational. So "almost all" directions eventually get you "almost" to each other point.

Definitely weaker than what you asked for, but in the ballpark.

A surface with the full strength of your hypothesis would have the property that every geodesic is a space-filling curve. I'd be surprised if that exists, but I can't come up with a quick reason why not.

4. Thanks for the replies. It's just something I was wondering about stemming from observing people disagreeing on which direction to walk in order to get to the shop, so I thought that it would be interesting if the earth was a shape such that it would not matter in which direction they chose to go, they would always reach the shop.