My friend says the Earth badge is a feather but I think it doesn't make sense for a ground type gym leader to use something having to do with a bird
I think of it more as a leaf. A leaf would sound better when it relates to a Ground Type Gym (if we're talking Gen 1), rather than a feather, which doesn't seem to have much of a relationship to the ground.
I always thought it looked like a feather. No clue why it would be a feather but it always looked like that to me.
I can actually see that, this was how I knew the badge. I thought like a farfetched feather or something but I really see tree! Spoiler
Logically speaking, it is called the Earth Badge as you pointed out. A feather makes no sense in that context, like at all. I used to think it was something like a fern.
I used to think it looked like a leaf, then a tree, then a flower. Maybe Giovanni designed his badge to confuse challengers?
OH MY GOD YOU'VE DISCOVERED THE SECRET I thought it was a...uh...carrot at first. A green one. I was eight, okay? It kinda looks like an birds-eye view of a emerald seam to me. Like, an open mine and looking at the gems in it. Or the crystals in a geode. But it looks like a shoot of grass or wheat or something, and since that is the least weird answer I came up with, that's what I'm going with. #MissingNo
I thought the Earth Badge looked like a feather at first, but yea, the picture on the second post makes it more like a leaf. Guess Game Freak couldn't find something better to depict the Ground type at the time (since they already had another design for the Grass type gym's badge.)
I have two ideas, one that is more likely than the other one. First off, it could be a leaf that has been crystallized over. I feel like that is how it would relate to the ground. The crazy idea would be that it was foreshadowing to the false tree Pokémon, Pseudowudo.
Both seem kind of plausible, but I think the first one would make somewhat more sense to me since I think of fossilization with that particular theory.
I'd call it a tree not a feather or a leaf. A leaf would be more associated with Grass than Ground as it's a part of some sort of foilage whereas a tree is growing directly out of the Ground which gives it it's relationship to Ground. Appearance wise it has a trunk and then goes into branches with leaves on them and finally blooms out at the top which is what would form the canopy in forest. Bulbapedia makes mention of the Sakaki Tree.
The hexagons at the top almost look like Zygarde eyes. Zygarde is a dragon/ground type so maybe it's supposed to look like that?
Oh wow for some reason I never stopped to consider that assuming it was a feather makes 0 sense when it's a ground type gym : o and even now that I stare at it I can't really see a leaf either... it's a strange one!