Now, I already did fanatic not that long ago when I looked at words meaning crazy, but quick recap: it’s actually from the classical Latin word festus, which means holiday. The word fan that’s short for that showed up in 1889 as a word for baseball enthusiasts. Now, for the rest of the fan words.
Fancy isn’t related to the above words. It showed up in the mid fifteenth century as fantsy, which is actually a contraction of fantasy. Fantasy showed up in the early fourteenth century meaning illusory appearance, or whimsical notion—the latter of which is what fancy used to mean before evolving to mean imagination, and then in 1751, meaning elegance. For some reason. Fantasy comes from the Old French fantasie/phantasie, from the classical Latin phantasia, fantasy, from the Greek phantasia, imagination. That word is related to phantazesthai, imagine, from phantos, phantom, from phainesthai, literally “it seems”, from phainein, to show or bring to light. And that one’s traced to the Proto Indo European bha-, to shine. Man, what a rabbit hole that one turned out to be.
Now let’s go back to the other kind of fan, the one you use to cool yourself off with. The noun of the word comes from the Old English fann, while the verb from was fannian, and the words actually had to do with sorting grain. Basically, a fann was something to shovel grain with, and a fannian was the process of doing so. The words come from the classical Latin vannus (there was no v in Old English), which is a fan for, again, sorting grain. One theory is that the word is related to ventus, wind, which would bring things full circle considering what we use fans for these days. But it also might be another one of these huge coincidences that seem to be eighty percent of etymology posts.
Tony Jebson’s page on the Origins of Old English