I dunno. Slate on a roof never has any feet slowly polishing it down. One client has slate in his kitchen. Has been there for about 20 years. All of the edges that give slate it's traction have been polished down. Looks sort of like a cake left out in the rain. Weird analogy, can't think of another.

I'd look into travertine.

Let me add that unpolished travertine is outstanding for outdoor steps. One client has that and even in the rain gives amazing traction. Probably not good in a shower as soap and skin (yech) bits would clog up the pores. Polished and sealed in a shower is good.

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1986 300SDL, 362K
1984 300D, 138K