You’ve all seen a ship in the bottle. Some old geezer spent hour figuring out how to get the hull into the bottle then very carefully used a string to raise the masts and rigging before sealing the bottle. Today they call them impossible bottles. How did they get a deck of cards in there? I can figure it out when there’s no cellophane on there. Breaking a seal on a pack of cards just requires a hair dryer. Then it’s just a matter of slipping the empty box, slipping the cards back in one by one, and putting that seal back on. I’m sure there’s a trick for the cellophane too. Same thing with the tennis ball. It’s just a matter of using an inflation needle from a sports ball and sucking the air out then re-inflating it once it’s inside. Part of the whole trick of these things is they’re inside the bottle which makes it hard to examine a tennis ball for tiny holes especially when they’ve been arranged so you can’t see them.
Those bottles are like bar bets. Never take the bet that someone else can put more in a bottle than you can because there’s a trick that makes you a sucker. You have to go a long way to find a real Impossible bottle. They exist it’s called a Klein bottle. It’s simple just follow the math.
The “figure 8” immersion (Klein bagel) of the Klein bottle has a particularly simple parameterization. It is that of a “figure-8” torus with a 180 degree “Mobius” twist inserted:
The parameterization of the 3-dimensional immersion of the bottle itself is much more complicated. Here is a simplified version:
for 0 ≤ u < 2π and 0 ≤ v < 2π.
Is your head hurting by now? Mine too. Even the words hurt. It’s a non-orientable surface with no identifiable “inner” and “outer” sides. a closed surface with only one side; formed by passing one end of a tube through the side of the tube and joining it with the other end. It’s a Möbius strip in three dimesnions. Basically if an ant starts crawling on the outer part of the bottle it will eventually get back to the outer part again only upside down. It’s a real Impossible Bottle and you can get them in beer steins now too.