Let $\nu \left( x \right) $ be the least number of steps that is required to construct a constructible length $x$, using compass and ruler in the well known fashion. Now, define the distance $d\left(x,y\right)$, of two constructible numbers $x$ and $y$ as the $\nu \left(|x -y|\right)$. Is this a metric for the space of all constructible numbers?
Ruler-and-Compass metric
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abstract-algebra
metric-spaces
geometric-construction
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1I don't know. What do you think? – 2012-09-11
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0I think it is not. The third axiom for metric space should fail. – 2012-09-11
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0When you say ruler, do you mean that distances are marked on it? Or are you referring to a straight-edge? – 2012-09-11
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1Well, it seems to me that $\nu(1)=0$, in which case the answer is no. – 2012-09-11
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0@ Harold Hanche-Olsen: No, we assume that only 0 takes the zero step to construct. 1 takes 1 step to construct. – 2012-09-11
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1Actually, it depends on something subtle about knowledge. Years ago, i put in the final touch on proving Bolyai's construction in the hyperbolic plane of curvature $-1,$ that there are a countably infinite set of squares and circles, both constructible, with equal area, and showed there are no others. But I also showed there there are no one-way constructions there, it is not possible in $\mathbb H^2$ to square the circle or to circle the square. So there is something very tricky about knowledge that cannot be avoided. – 2012-09-11
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1Closer to what you are thinking about, T.-Y. Lam has pointed out that there is no canonical form in the constructible numbers. Given two expressions involving rational numbers, field operations, and square roots of positive elements, we can (I think) eventually decide whether they are the same number, I guess by showing they satisfy the same one-variable polynomial. – 2012-09-11