For the past few weeks, baseball fans have been worshipping at their chosen shrine with a fervour that verges on the religious. This is not surprising in view of the origins of the word “fan”. It was in reference to baseball, in fact, that the word was first used, in the 1880s, before being extended to other sports and then to the theatre and other activities. It is a shortening of “fanatic”, derived from the Latin word fanaticus meaning “pertaining to a temple” (the Latin word for “temple” being fanum). But fanaticus also had an extended meaning, “inspired by orgiastic rites, frantic with religious enthusiasm”. I don't know if a baseball game can be described as an orgiastic rite, but fans can certainly get frantic.
To cool their ardour, they might want the other kind of fan, but that is a different word entirely, from the Latin vannus, originally a type of basket for winnowing grain by tossing it in the air. This word came to be applied to a hand held device used for agitating the air.