It is very well known from Geometry that a regular polygon with seven sides is impossible construction by the unmarked straight edge and a compass, even though there are much APPROXIMATION methods such as using marked straight edge or using Origami, numeric approximations ... etc, where all those methods are very good approximations and not so different from using a directly suitable Protractor

But when you solve a polynomial say as $x^7 + 1$, the representation of complex roots seems to construct exactly that given polygon with seven regular sides

Link: https://www.wolframalpha.com/input/?i=x%5E7+%2B+1

However, this also seems applicable to any polynomial $x^n + 1$ with any degree $n$, where $n$ is not of the form of Fermat's polygon integer number

So, the question is how valid upon the complex roots solution to those polygons that represent the exact construction for polygons not of the form of Fermat's polygon numbers?

Or is that only imaginable or approximated constructions like imaginable complex roots? wonder!


closed as off-topic by Andrés E. Caicedo, Lord Shark the Unknown, José Carlos Santos, Chris Godsil, Shailesh Dec 17 '17 at 0:25

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  • 1
    $\begingroup$ $x^7+1=0$ has seven complex-number roots and those roots are easily plotted. But not all seven of those roots are constructable. $\endgroup$ – steven gregory Jul 9 '17 at 13:03
  • 3
    $\begingroup$ Absolutely any diagram you can look at on your computer screen is going to be approximate, since your screen does not have infinite resolution. If Wolfram Alpha (and any other plotting software for that matter) can approximate the roots to a precision much finer than the width of a pixel, how could you tell the difference between it and a theoretically exact value? $\endgroup$ – Rahul Jul 9 '17 at 13:09
  • $\begingroup$ @Rahul indeed every real (representation of an) object is an approximation of some ideal static object, not only due to limitation of pixels, also because due to quantum mechanics :p $\endgroup$ – Masacroso Jul 9 '17 at 13:18
  • $\begingroup$ Wolframe\Alpha can also construct EXACTLY the regular Pentagon or any constructible regular polygons, exactly like Euclid had done thousands of years back, without knowing anything about complex numbers, so the point is that can be made APPROXIMATELY by a suitable protractor for any non-constructible regular polygons, so where is really the addition to our modest knowledge added by those called complex numbers then? wonder! but the scene of mathematics is basically about EXACTNESS and never was about APPROXIMATIONS since the later doesn't require any big talent at all, but only little skills $\endgroup$ – bassam karzeddin Jul 9 '17 at 13:22
  • $\begingroup$ @Rahul Mathematics objects are mainly numbers only, where some of them can be proven by real existence as the constructible numbers by Geometry which is the physical reality around us, but other numbers that can be deduced and never be proved existing except in mind only for our own narrow needs or purposes that is irrelevant to any mere facts $\endgroup$ – bassam karzeddin Jul 9 '17 at 14:15

Of course a 7-gon is easy to construct using the roots of polnomial $z^7+1$. The classical problemm is to construct a 7-gon using straightedge and compasses. This means (in modern language) we may use only "constructible" numbers, in this sense.

  • $\begingroup$ Do you mean true and exact construction as known for existing real constructible numbers or imagined construction based on those complex roots or approximate but never exact construction which isn't any new? wonder! or else what is exactly cos(2*Pi/7), does it fit exactly in any Pythagorean triangle? wonder, however, I had proved this angle (2*pi/7) is a non-existing angle in any imaginable reality, but only an assumed existing angle in human minds only, for sure $\endgroup$ – bassam karzeddin Nov 12 '17 at 8:40

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