“The Rule of Three” is something that gets thrown around a lot in many fields. I’d say they’re especially prevalent in my fields of writing and theatre/improv, but it appears everywhere.
In fact, saying “the rule of three” is immediately misleading because there are so many rules of three. That’s simply because 3 is a magic, holy number. You have the Christian trinity, the 3 act structure, the 3 Musketeers, you name it.
To put it very briefly and very simply, 3 is the perfect number to establish a pattern or a group without overwhelming the audience of the message. It’s easy to describe the function of God the Father, the Son, and the Holy Ghost, but as soon as you add a fourth, it starts getting exponentially harder to remember which is which. It’s the same thing with the 3 Musketeers.
If you have 7 of something in a story, you can’t reasonably expect to teach the audience about each individual thing and expect them to hold interest. Their attention span will only last for about 3 or so.
The logician in me thinks its unfair that 3 gets so much praise for being the holy number. It’s a self-fulfilling prophecy, you see.
So this is to say that when you’re writing something, whether its a story, a piece of worldbuilding, or a plot, you know whatever—use the rule of three. Don’t put five characters in a story that works with three. Don’t put 2 try-fail cycles unless you have a good reason for doing so. When you write a passage using lots of repetition, communicate the idea thrice. (I was about to quote Marc Antony’s usage of the phrase “honourable men” in his famous monologue, but he actually states it 5 times, not 3. Damn Shakespeare and his 5’s!)
You get the idea. But this rule is a bit different in improv (and comedy writing in general). In fact, it’s a lot more specific than just “use the number of three because it’s a good psychological tool”.
When you’re writing a sketch, or performing in an improv scene, the first time you use a joke it’s funny for it’s own sake. When you later repeat this joke it becomes a callback—it’s funny because of the reference. But when you use this joke again, you can’t simply repeat it, because then it becomes repetitive in its own sake. You’re milking it. Instead, you find a way to turn the joke on its head and present it in a way the audience doesn’t expect. In The Three Stooges’ “Slowly I Turned” sketch, you see lots of uses of the rule of three. Funny enough, the actual punchline of the joke is used 4 times, but the last two times its used in a way that isn’t expected.
All this is to say, in conventional writing the number three is a good rule of thumb to know how many times to use a trope or with establishing rules/characters, and in comedy it’s a good way to get punchlines, but only if you subvert expectations on the last use.