> is there a fairer, algorithmic way to carve up a geographical area into election districts so we can apply the same solution to every state?

I would argue to throw out the constraint that each district must contain the same number of people. There is no reason for that other than the fact that "one man, one vote" sounds good when you hear it. It makes just as much sense as applying the same constraint to senators.

Representatives are supposed to represent a common interest group, not Democracy In The Abstract. If two people are a part of the same community with the same interests, why wouldn't they share a representative? The right approach is to identify natural districts first and then not care whether they're the same size.