4 ms·

Manufacturing errors? I can't tell if this is plastic, but if it is, surely a machinist can do better with metal.

kenferry 13y ago
They're just saying that building it is not a proof - the computer simulation is more exact than the model, and even the simulation is subject to rounding errors in what is considered "touching". Sounds like they used the computer to search for a proposed solution that _could_ have been a false positive due to rounding error, then verified it through exact mathematical methods.
delinka 13y ago
By "errors" they mean "the minute imperfections created by molding, machining, carving, etc." The rounding errors in the computer model would produce smaller physical artifacts than the manufacturing method would.
- tehwalrus 13y ago
  Surely mathematicians would use rationals[1] rather than floating point numbers, thus eliminating rounding errors at the expense of performance? [1] https://gmplib.org/ https://gmplib.org/
  - ithkuil 13y ago
    what if irrational numbers are involved?
    - sbarre 13y ago
      you couldn't argue with that
  - lifthrasiir 13y ago
    It is not a matter of rational numbers, since the solution is likely to be an irrational number. The paper [1] describes a system of 20 polynomial equations with 20 variables (Equations 10--12) and solving them is no trivial task. To the end, authors first numerically found candidate solutions up to some ten decimal digits [2] and used specialized tools to prove (!) that there exists a real solution sufficiently close to given numerical solution. It turns out that there are several ways (and corresponding implementations) to do that, e.g. alphaCertified [3]. [1] http://arxiv.org/abs/1308.5164 http://arxiv.org/abs/1308.5164 [2] 10^-11 to be exact, according to the paper. It would be much easier to find a near-solution with larger errors (say, 10^-5). The physical model of such near-solution would be indistinguishable to the correct model. [3] http://www.math.tamu.edu/~sottile/research/stories/alphaCertified/ http://www.math.tamu.edu/~sottile/research/stories/alphaCert... (that actually does use GMP and MPFR for the obvious reason)
    - tehwalrus 13y ago
      Interesting. I would have thought that one could solve such things exactly by representing each unique known irrational that arises (root 2, pi, etc) by its own rational multiplier, and then overloading the relevant equality checks. Of course, you'd need to anticipate/implement each irrational type that might arise (roots, the geometric transcendental pi, and so on.) [Leaving the next sentence in, for comedy value. I typed it and then realised how ridiculous it is - apologies!] Essentially it's just teaching the computer to do the algebra for you, isn't it?
      - lmm 13y ago
        How would you test whether two generic irrational numbers were equal? Obviously you can numerically approximate them and if you see any difference in the numerical approximation then they must be different - but if they seem the same up to e.g. 10 decimal places, what do you do next?
        tehwalrus 13y ago
        I would generally assume that sqrts of primes don't overlap, so you can always do exact comparisons of rational coefficients? It wouldn't be perfect, but it would do as well as a person with pen and paper and a hundred years. [[ Approximate algorithm, in case I'm not being clear: you need sqrt(35), represent it as 1×sqrt(5) × 1×sqrt(7). You simplify each expression evaluated down to roughly what you'd write on paper in RAM, and then you do exact comparisons - is that the same coefficients of the same number of the same prime sqrts? For greater than/less than you cast them to a hundred-sig-fig float, and if those are still equal, keep going down the rabbit hole. Obviously the RAM requirements for complicated numbers would be large, but that's the same with rationals - this is just taking it to the extreme. I'd be very surprised if some function of sqrts was exactly the same as some other function of different sqrts. ]]
        lmm 13y ago
        Sines of certain fractions of pi are equal to certain ratios of square roots though. And once you get to more complicated functions it's very hard to prove anything (e.g. I believe it's still not even proven that pi^e is irrational, never mind transcendental). Could we construct a countable field (for to be able to write down the numbers involved the field would have to be countable, i.e. only an infinitesimal fraction of the reals) that contained everything we need to do this kind of geometry in? Maybe. But I can just as easily believe you could construct a way of arranging objects that required a turing-complete computation to determine whether two objects met, in which case the halting problem makes this impossible.
darklajid 13y ago
Plastic? I'm confused. The article states: "Finally, they built a _wooden_ model to demonstrate their answer"

They're just saying that building it is not a proof - the computer simulation is more exact than the model, and even the simulation is subject to rounding errors in what is considered "touching". Sounds like they used the computer to search for a proposed solution that _could_ have been a false positive due to rounding error, then verified it through exact mathematical methods.

delinka 13y ago

By "errors" they mean "the minute imperfections created by molding, machining, carving, etc." The rounding errors in the computer model would produce smaller physical artifacts than the manufacturing method would.

tehwalrus 13y ago
Surely mathematicians would use rationals[1] rather than floating point numbers, thus eliminating rounding errors at the expense of performance? [1] https://gmplib.org/ https://gmplib.org/
- ithkuil 13y ago
  what if irrational numbers are involved?
  - sbarre 13y ago
    you couldn't argue with that
- lifthrasiir 13y ago
  It is not a matter of rational numbers, since the solution is likely to be an irrational number. The paper [1] describes a system of 20 polynomial equations with 20 variables (Equations 10--12) and solving them is no trivial task. To the end, authors first numerically found candidate solutions up to some ten decimal digits [2] and used specialized tools to prove (!) that there exists a real solution sufficiently close to given numerical solution. It turns out that there are several ways (and corresponding implementations) to do that, e.g. alphaCertified [3]. [1] http://arxiv.org/abs/1308.5164 http://arxiv.org/abs/1308.5164 [2] 10^-11 to be exact, according to the paper. It would be much easier to find a near-solution with larger errors (say, 10^-5). The physical model of such near-solution would be indistinguishable to the correct model. [3] http://www.math.tamu.edu/~sottile/research/stories/alphaCertified/ http://www.math.tamu.edu/~sottile/research/stories/alphaCert... (that actually does use GMP and MPFR for the obvious reason)
  - tehwalrus 13y ago
    Interesting. I would have thought that one could solve such things exactly by representing each unique known irrational that arises (root 2, pi, etc) by its own rational multiplier, and then overloading the relevant equality checks. Of course, you'd need to anticipate/implement each irrational type that might arise (roots, the geometric transcendental pi, and so on.) [Leaving the next sentence in, for comedy value. I typed it and then realised how ridiculous it is - apologies!] Essentially it's just teaching the computer to do the algebra for you, isn't it?
    - lmm 13y ago
      How would you test whether two generic irrational numbers were equal? Obviously you can numerically approximate them and if you see any difference in the numerical approximation then they must be different - but if they seem the same up to e.g. 10 decimal places, what do you do next?
      - tehwalrus 13y ago
        I would generally assume that sqrts of primes don't overlap, so you can always do exact comparisons of rational coefficients? It wouldn't be perfect, but it would do as well as a person with pen and paper and a hundred years. [[ Approximate algorithm, in case I'm not being clear: you need sqrt(35), represent it as 1×sqrt(5) × 1×sqrt(7). You simplify each expression evaluated down to roughly what you'd write on paper in RAM, and then you do exact comparisons - is that the same coefficients of the same number of the same prime sqrts? For greater than/less than you cast them to a hundred-sig-fig float, and if those are still equal, keep going down the rabbit hole. Obviously the RAM requirements for complicated numbers would be large, but that's the same with rationals - this is just taking it to the extreme. I'd be very surprised if some function of sqrts was exactly the same as some other function of different sqrts. ]]
        lmm 13y ago
        Sines of certain fractions of pi are equal to certain ratios of square roots though. And once you get to more complicated functions it's very hard to prove anything (e.g. I believe it's still not even proven that pi^e is irrational, never mind transcendental). Could we construct a countable field (for to be able to write down the numbers involved the field would have to be countable, i.e. only an infinitesimal fraction of the reals) that contained everything we need to do this kind of geometry in? Maybe. But I can just as easily believe you could construct a way of arranging objects that required a turing-complete computation to determine whether two objects met, in which case the halting problem makes this impossible.

darklajid 13y ago

Plastic? I'm confused. The article states: "Finally, they built a _wooden_ model to demonstrate their answer"