isomorphismes

Yes! Video games are the best way to explain basic topology. 

1. • You know those special levels in Super Mario Bros. where the screen doesn’t move with you — you’re just in a “room” and if you go off to the right you arrive back on the left?
   • That was just a convenience for the game programmers.
   • But think about this: what’s the difference between a straight line that meets back to its other end and you loop over and over and over the same spot while running forward — and a circle?
   • (Answer: there is no difference. If we wanted to imagine Mario being a 3-D person, we could — he’d be running around a cylinder (this room he’s jumping to get the coins in would just be maybe 2-3 shoulders wide and dug in a circle underneath the ground. Or the brick platform he’s running on equally wide and it’s built in a cylinder above off the ground. We just don’t see that he’s constantly adjusting to bear a few degrees left as he’s running.
2. • How about Star Fox battle mode?
   • If you drive off the north of the screen you end up at the south, and if you fly off the east you end up on the west.
   • 
     At first I thought this just meant we were flying around an entire planet Like The Little Prince’s moon or so. (The non-map visuals—the main flying visuals—could go along with this story, since there’s suelo below and cielo above.)
   • But on further consideration this can’t be the case.
     
     Think about a globe of the Earth: east and west are connected contiguously — but the North Pole is as far away from the South Pole as you can get.
      
   • Think about running away from your enemy to the northeast corner and disappearing very quickly off the north to the south, then disappearing just as quickly from east to west. What allows you to do this quick of a dodge?
   • Think about if the North Pole and the South Pole WERE equivalent. Picture the Earth and start “sucking the two towards each other”.
      
   • You would end up with….
     
   • I’m still thinking about what follows. But I think this is true: at the “corners” you are on the “inner tube” of the torus.
     
      
   • At least on some torii it would be shorter to sail ‘round the world by sailing [1st] grip-around from outer to inner circle, [2nd] ring-around the inner circle, and [3rd] grip-around from inner to outer again.
     
     Actually I’m not sure about that. Need pencil + paper + time.
     
     Or even helixing around the rubber tube.
     
   • So imagine if the North Pole = the South Pole. And you wanted to fly from New York to Melbourne. Currently one does this through Los Angeles → Honolulu → Jakarta | Darwin. But if the North Pole = the South Pole, you could fly from Melbourne over Antarctica (I guess it wouldn’t be ice because Earth/sun dynamics would change … but never mind how the planet would be completely different, pretend there are still cities in Melbourne and NYC lat:long), come out near Greenland, and fly over Reykjavik → Newfoundland → Maine → NYC. 

Top pic via deifying.