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We need a plausible mathematical geometric objective algorithm, or at least limits to tortuousness.

I've been thinking the same thing. Just a few thoughts off the top of my head:

Draw districts so that the sum of the length of all boundaries is minimized.
Follow political boundaries as much as practical, while still accomplishing #1. County boundaries take precedence, followed by municipal boundaries.
Ethnicity, gender, religion, sexual preference, political party affiliation, and even voter registration are NOT to be considered. Population is the only consideration.
For purposes of redistricting, US citizens are the only ones to be counted. Everyone else is a guest, and won't be voting anyway.

Unfortunately, it will probably require a Constitutional amendment to enforce this uniformly in every state. To sweeten the pot, I'll toss out a few more suggestions. Comments and counter-points welcome:

A state's electoral votes will be allocated per the "Maine model". The candidate achieving a plurality of votes in each Congressional district shall receive one electoral vote from that district. The candidate achieving a plurality of votes in the entire state shall receive two electoral votes.
A voting machine standard will be established. I'll suggest something like The Open Voting Consortium. It's an electronic voting system using commodity hardware, but it prints a paper ballot. You can see exactly the choices that you made. A bar-code is also printed on the page, and can easily be scanned by a machine to count the votes. If a recount is necessary, the paper ballots are available. And if you doubt the bar-code, a scanner is provided for you at the polling location to double-check your vote(s).
Standardize voter registration, to confirm citizenship and eliminate ineligible voters. Implement anti-fraud measures to detect things like hundreds of voters at the same address.
Standardize voter ID at the polls: photo ID required.
Ban absentee and mail voting except for demonstrated need (i.e. active-duty military, infirm/hospitalized, and pre-planned travel). Standardize early voting procedures to reduce the need for absentee voting.
Standardize voting procedures and security. Define chain-of-custody requirements for ballots that are similar to those for evidence in a criminal trial (no more "finding" ballots in the trunk of a car).

OK, this became a lot longer than I planned. It's more of a "stream of thoughts" than a thought-out strategy. What did I miss?