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Thinking about getting the bitcoin elliptic curve (secp256k1) as a tattoo. Would that be cringe?

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That's the curve over real numbers, it's not what Bitcoin's curve looks like, since Bitcoin's curve is over a finite field.

Yeah, I know. But this representation is aesthetically pleasing.

Would be way cooler to map the xy-plane onto the surface of your body and then get some of the actual secp256k1 points scattered over your body imo

also a real option

Sure... if you don't mind that it will slowly deform over the years and having to explain what it is to literally everyone that sees it.