And yes, the 1701 really has a bowling alley.
Bowling is kind of a clutch sport for a Federation starship, if you think about it.
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Bowling alleys are long and narrow, which means you can fit one in by shaving a little off a lot of rooms instead of a lot off a few rooms. They’re infinitely scalable, too; you can have a 2-lane alley, or a 200-lane alley, whatever fills your space. You can even stack them; if you’ve got two three-lane-wide spaces on adjacent decks, just add a staircase.
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Bowling accomodates a wide range of player counts; a lane might be one guy whose friends are all on-shift, trying to nail a perfect game, or it might be a scheduled five-member team competing against four other teams. There’s not a lot of “big room” sports that take up less space per player than bowling. Especially in the future, where the machinery is probably one micro-repulsor emitter in the ball return and a tiny antigrav in each pin.
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Bowling is easy to teach: even aliens who’ve never heard of it can pretty easily pick up “roll this ball across that floor to hit those sticks”.
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Somewhat relatedly, bowling is very social; if you’re on a team, you’re not actually playing n-1 shares of the time, so you can chat with your teammates, other players, etc. Good for both crew bonding, and for diplomacy.
Damn good sales pitch, Ferengi.
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In TOS they didn’t have holodecks.
A space ship might fit bowling lanes but not a whole baseball field.
Also, bowling was waaaay more popular at the time TOS was airing than it is today
“Fucking Gary. That creep can roll, man.”
Bowling is fun
So is baseball.
This isn’t facebook, please don’t willfully spread disinformation.