Everyone knows about The Mobius Strip, the loop of paper with a twist in the middle that has only one surface. Now imagine taking a disk and gluing its edge to the edge of the Mobius strip. This will give you a Boy Surface, and one of the weirdest planets conceivable.
Imagine coming up on a planet, the first you've seen in weeks out of the window of your fabulous spaceship. At first, the planet looks like any other planet. You're coming up on the pole, and the rest of the planet slopes down in a vaguely roundish way from there. But as you get close, you see some strange features. You see that, instead of the downward slope from the pole leading to a regular globe, the planet has three 'wing like' structures that fan out. These wings twist around each other, giving the planet a hollow space in the center. How very strange. You set down on the planet, stick a flag in the North Pole, and get ready to do some exploring.
Starting at the North Pole, you follow your 'pole detector' and look for the South Pole. After moving along the surface of the planet for some time, the detector goes off telling you you're at the South Pole, and you take a look around. Immediately you realize that, although you didn't enter any caves, and only moved along the surface of the planet, you're somehow now inside the planet. What's more, when you look down at the ground, you see the bottom part of your flag, the one you stuck in the North Pole, sticking out next to your feet. In this crazy world, the North Pole is the South Pole, and the outside of the planet is the inside of the planet.
You are on Boy's Surface, discovered by Werner Boy in 1901, and represented as a strange planet by John Pierre Petit in his book, Le Topologicon, in 1902. Each of the 'wings' can be traced by a Mobius Strip. The planet is a four dimensional object that penetrates itself without causing any holes or edges. The best mathematical analysis of it was done by Bernard Morin, a mathematician who had been blind since childhood. In short, everything about Planet Boy pretty much sounds like it should be in a nutty, nerdy, science fable fantasy, and I like to think it's really out there.
Via Wolfram Math World and Berkeley.