A strange loop is a system that unexpectedly arrives back where it started. This sentence is false. This sentence has five words. As humans, we try to make sense of the world through cause and effect, and when this system is flipped or tangled, it becomes a strange loop. Here's some examples of strange loops.
Strange loops have existed forever. Examples like the chicken and the egg problem or the ouroboros, the ancient symbol of a dragon eating its own tail, are found worldwide.
The most famous strange loop is Gödel’s Incompleteness Theorems. Without going too much into depth, Gödel showed that we couldn't have a single mathematical system that is complete and consistent. Thus, there will always be a theorem in any system with a set of ground truths that can't be proven.
A quine is a computer program that takes no input and outputs a copy of its source code as the only output.
_='_=%r;print (_%%_)';print (_%_)
The Ship of Theseus paradox references an old Greek puzzle that describes a ship whose planks were replaced as they decayed. Eventually, every part of the ship had been replaced - was it still the same ship?
The Shepard Tone is a sound that creates the illusion that it is always increasing or decreasing in pitch.
Canon 5 from Bach's Musical Offering continues to rise in key, modulating through the entire chromatic scale until it reaches the same key in which it began.
The Klein bottle is a strange loop. If you travel along the one-sided surface, eventually you'll return to where you started but flipped upside down. (You can buy these on Amazon)
As well as the more common Möbius strip.