TITLE: Can you make a sphere out of a plane?
QUESTION [93 upvotes]: I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a very large sphere. (Stretching or shrinking of different places is OK.) 
What I first thought about doing was building a finite rectangular model of the world as like a mercator projection, and tessellating this model infinitely throughout the plane. 

Someone starting in the US could swim eastwards in a straight line across the Atlantic, walk across Africa and Asia, continue through the Pacific and return to the US. This would certainly create a sense of 3D-ness. However, if you travel north from the North Pole, you would wind up immediately at the South Pole. That wouldn't be right.
After thinking about it, I hypothesized that an explorer of this model might conclude that they were walking on a donut-shaped world, since that would be the shape of a map where the left was looped around to the right (making a cylinder), and then the top was looped to the bottom. For some reason, by simply tessellating the map, I was creating a hole in the world.
Anyway, to solve this issue, I thought about where one ends up after travelling north from various parts of the world. Going north from Canada, and continuing to go in that direction, you end up in Russia and you face south. The opposite is true as well: going north from Russia, you end up in Canada pointing south. Thus, I started to modify the tessellation to properly connect opposing parts of Earth at the poles. 
When going north of a map of Earth, the next (duplicate) map would have to be rotated 180 degrees to reflect the fact that one facing south after traversing the north pole. This was OK. However, to properly connect everything, the map also had to be flipped about the vertical axis. On a globe, if Alice starts east of Bob and they together walk North and cross the North Pole, Alice still remains east of Bob. So, going north from a map, the next map must be flipped to preserve the east/west directions that would have been otherwise rotated into the wrong direction.

Now the situation is hopeless. After an explorer walks across the North Pole in this Minecraft world, he finds himself in a mirrored world. If the world were completely flat, it would feel as if walking North will take you from the outside of a 3D object to its inside.
Although I now think that it is impossible to trick an explorer walking on infinite plane into thinking he is on a sphere-like world, a part of me remains unconvinced. Is it really impossible? Also, how come a naive tessellation introduces a hole? And finally, if an explorer were to roam the world where crossing a pole flips everything, what would he conclude the shape of the world to be?

REPLY [4 votes]: I'd like to add another visual which complements James K's answer.
What you need is this:

Topologically, the above pillow is just a sphere. It would be pretty easy to take a globe and put it on this pillow; you'd need to stretch and shrink things a bit, but not too severely.
Once you've done that, it's not too hard to cover the plane with copies of the pillow. Start with an infinite pile of globe pillows, and use a seam ripper to separate each pillow into the top half and the bottom half. Place one top half in the center of an infinite table, then surround it with 4 bottom halves, so that their edges match up the same way as on the original pillow. Then surround these bottom halves with 8 top halves, matched appropriately, and so on forever.
As your explorer walks across the quilt that you've created, they will have a very hard time distinguishing the quilt from the original pillow. They'll only notice a difference if they come across one of the corners, and if they're astute enough to somehow notice that, standing at the corner, they are now immediately surrounded by two copies of the world, not one.
To see what this would actually look like, look at the image in James K's answer again. One side of the pillow would have the northern hemisphere on it, and the other side would have the southern hemisphere. The equator would, of course, lie along the seam of the pillow. The orientation of the map has been chosen so that one corner is in the middle of the Atlantic Ocean, one is in the middle of the Indian Ocean, and the other two are in the Pacific Ocean.
That image shows that if you take two copies of the northern hemisphere and two copies of the southern hemisphere, you can form them into a square which tiles the plane in the usual way. The two copies of each hemisphere are 180 degree rotations of each other.