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> For example, we don't need to think in terms of Q[sqrt(3)]; I see that as a distraction.

Except for replacing Q[sqrt(3)] by a suitable ring extension of Z, I see no possibiliy to simplify the argument. So what kind of simplification do you have in mind?

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You're speaking as if floating point representations of numbers have zero utility. One approach here could be numerical. An algorithm could work with vectors represented as floating points. Another approach, which depends on the relative angles and lengths of track pieces, may be to encode each piece as an (angle, distance^2) pair. Many angles can give us an exact distance^2 values.

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