it seems like a map can be more or less distorted to its true globe counterpart. is there a way to quantify that and produce maps which optimize for least distortion (but maybe not other qualities, such as “being in one piece”)?

it seems like a map can be more or less distorted to its true globe counterpart. is there a way to quantify that and produce maps which optimize for least distortion (but maybe not other qualities, such as “being in one piece”)?
There are at least two kinds of distortion people usually measure: angle distortion, and length distortion. You can also compound these and at a larger scale measure area distortion.
You can actually get plain (unsliced) maps that minimize length or angle distortion in exchange for blowing up the other one.