I was joking with a colleague today about replacing Mathematica (a symbolic math program) with an abacus, and it started me wondering: how would one go about making a "symbolic abacus"?
Let's just limit it to finding roots of polynomials, for example. Could you make a mechanical device that lets you slide/fold/twist "x^2 = x+a" and then slide/fold/twist to get "x = (+/- sqrt(4a+1) + 1 )/2" ?
(Edit: To emphasize symbolic, not numeric solutions)
@catselbow You can generate roots for solvable polynomials just by straight edge and compass constructions; the general n-th degree polynomial equation can be solved in terms of modular forms ( see Thomae's formula for instance) so one might try to translate those functions into a physical machine, similar to how square roots can be dealt with by compass constructions.
@catselbow For quadratic equations you could imagine a machine consisting of a second-order differential equation (like a damped spring with tunable parameters) and maybe some external force. The frequencies of this system would be solutions of a quadratic equation.
@ramankhutu
I don't think that's what I'm thinking of. It's straightforward to make an analog computer to give you a numerical answer to an algebraic problem.
I'm wondering about a mechanical device that gives an ALGEBRAIC answer.
Maybe the only answer is "build a mechanical computer to run Mathematica", but I'm wondering if there are simpler devices, with a more limited but still interesting scope.
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