In "Angle Trisection, the Heptagon, and the Triskaidecagon", published in the American Mathematical Monthly in March 1988, Andrew Gleason discusses what regular polygons can be constructed with compass, straightedge and angle trisector. At the end of that article he notes that the angle p-sectors required for a regular n-gon are the odd primes p dividing $\varphi(n)$.

For the heptagon, which only requires an angle trisector, he gives the minimal polynomial of $2\cos(2\pi/7)$ $$x^3+x^2-2x-1$$ and transforms it into the Chebyshev polynomial expression $$7\sqrt{28}(4\cos^3\theta-3\cos\theta)=7$$ leading to the final identity $$\sqrt{28}\cos\left(\frac{\cos^{-1}(1/\sqrt{28})}{3}\right)=1+6\cos(2\pi/7).$$

I am interested in the hendecagon (11 sides), which requires an angle quinsector (that splits an angle into five equal parts).

Is there a similar transformation between the minimal polynomial for $2\cos(2\pi/11)$ $$x^5+x^4-4x^3-3x^2+3x+1$$ and the relevant Chebyshev polynomial $$\cos 5\theta=16\cos^5\theta-20\cos^3\theta+5\cos\theta$$ and how do I find it? If I had such a transformation, I could construct an exact hendecagon with the quinsector.

I have tried Tschirnhaus transforms on the former polynomial, without success.


The question essentially asks about transforming solvable equations from one form to another.

I. Cubic

Using just a linear transformation, the general cubic $P(x)=0$ can be transformed to the form,

$$y^3+3ay+b = 0\tag1$$

with solution,

$$y_k = 2\sqrt{-a}\;\cos\left(-\tfrac{2\pi\,k}{3}+\tfrac{1}{3}\,\arccos\big(\tfrac{-b}{2\sqrt{-a^3}}\big)\right)\tag2$$

for $k=0,1,2$. Undoing the transformation establishes a relation between the roots $x,y$.

II. Quintic

Similarly, an appropriate Tschirnhausen transformation can transform a solvable quintic $P(x)=0$ to the Demoivre form (essentially the Chebyshev polynomial mentioned by the OP),

$$y^5+5ay^3+5a^2y+b = 0\tag3$$

with analogous solution,

$$y_k = 2\sqrt{-a}\;\cos\left(-\tfrac{2\pi\,k}{5}+\tfrac{1}{5}\,\arccos\big(\tfrac{-b}{2\sqrt{-a^5}}\big)\right)\tag4$$

for all five roots $y_k$. A cubic Tschirnhausen gives us three degrees of freedom to transform a solvable quintic to Demoivre form.

III. Transformations

For $p=7$:


$$\color{blue}{y=3x+1} = 2\sqrt{7}\cos\left(\tfrac{1}{3}\,\cos^{-1}\big(\tfrac{1}{2\sqrt{7}}\big)\right)=4.7409\dots$$

then $x,y$ solves,

$$x^3+x^2-2x-1=0$$ $$y^3-21y-7=0$$

For $p=11$:

Let $\phi=\tfrac{1+\sqrt{5}}{2}$ be the golden ratio.


$$\color{blue}{y=x^3-\phi\,x^2-\tfrac{7+\sqrt{5}}{2}x+\tfrac{5+4\sqrt{5}}{5}} = 2\,\phi\sqrt{\tfrac{11}{5}}\cos\left(-\tfrac{6\pi}{5}+\tfrac{1}{5}\,\cos^{-1}\big(\tfrac{-89-25\sqrt{5}}{44\sqrt{11}}\big)\right)=-4.7985\dots$$

then $x,y$ solves,

$$x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1=0$$ $$y^5-5ay^3+5a^2y+b=0$$

where $a=\tfrac{11}{5}\phi^2,\;\;b=\tfrac{11(125+89\sqrt{5})}{250}\phi^5$.

Thus as you can see, the transformation (in blue) which relates the quintic roots $x,y$ is more complicated than the cubic version, but is nonetheless doable in radicals.

  • $\begingroup$ @Parcly Taxel: I hope the cubic relation between the quintic roots $x,y$ is enough for your question. $\endgroup$ – Tito Piezas III Aug 26 '16 at 20:11
  • $\begingroup$ The construction from here is doable with the quinsector, except possibly for the transformation from y to x, which looks like it needs an angle trisector. However, if I interpret my quinsector as having six rods (0 to 5) and the trisection as using only rods 0 to 3, then it is valid. Enough for an accept. $\endgroup$ – Parcly Taxel Aug 27 '16 at 2:25
  • $\begingroup$ @ParclyTaxel: I considered a $7$th deg analogue in this post. $\endgroup$ – Tito Piezas III Aug 27 '16 at 2:48

If the object is to construct a regular hendecagon, it can be done more simply without going through an angle quinsection. Benjamin and Snyder proved the existence of a construction using a marked ruler and compasses in 2014 (BENJAMIN, ELLIOT; SNYDER, C. Mathematical Proceedings of the Cambridge Philosophical Society156.3 (May 2014): 409-424.; http://dx.doi.org/10.1017/S0305004113000753 ).

The basic premises of the construction are as follows:

1) It's based on the properties of "conchoid-circle" constructions where we position the marked straightedge to paas through a fixed point $P$, with one mark on a line $l$ and the other on a circle $K$.

2) With this type of construction, we define a "signed distance" $z$. This is the distance between $P$ and the mark on $K$, with a negative sign if that mark lies vetween $P$ and the other mark which is on $l$, a positive sign otherwise.

3) Then $z$ satisfies a sextic equation whose coefficients satisfy certain relationships called the "verging theorem".

4) The quintic equation given here for the hendecagonal cosines is converted to a sextic equation that satisfies the verging theorem by (4.1) defining $z=ux$ for some scale factor $u$, and (4.2) introducing an additional root $\eta$ with the appropriate value relative to $u$. Then, all the geometric parameters needed to determine $l$ and $K$ may be expressed and constructed in terms of this scale factor $u$.

5) Now to the heart of the matter. It looks like we have to solve a seventh-degree equation for that parameter $u$. But, "a miracle occurs" (the authors' own words); the equation for $u$ is reducuble and all we are left with is cubic factor equation (with integer coefficients), which can be solved by an auxiliary marked ruler construction.

6) So $z=ux$ has a construction with a marked ruler and compasses because it solves a sextic equation that satisfies the verging theorem, and $u$ has such a construction as well because it is obtained from a cubic equation in $Z[u]$; and so $x=2 cos(2\pi m/11)$ has one too.

7) Now for the parameters. For $u$, choose the one real root of $u^3+2u^2+2u+2=0$. For the construction of $z=ux$: $P=(0,0)$, $l$ is the line $x=-u-1$ where the length unit is the distance between the marks (conventional in this type of construction), $K$ is centeted at $(u(u-1)/2,-(u^2+3u+1)/2)$ and passes through $(-u-2,0)$. One orientation of the ruler is along the $x$ axis, that is the "extra" root of the sextic; the other roots on $K$ with proper distance signs, see (2), give the roots for $z$. Note that the authors do not give the formulas that way, I did some algebra of my own to get everything in terms of $u$.


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