TITLE: Geometric interpretation of a quintic's roots as a pentagon?
QUESTION [6 upvotes]: According to this subsection, "Geometric interpretation of a cubic's roots", given,
$$F(x)=x^3+ax^2+bx+c=0$$
with three real roots, "...then the roots are the projection on the $x$-axis of the vertices $A, B, C$ of an equilateral triangle. The center of the triangle has the same abscissa as the inflection point."
$\hskip2.7in$ 
$$\text{Fig.1}$$
Questions: 

In general, given $F(x) = 0$ of degree $n>2$ with $n$ real roots, are the roots the projection on the $x$-axis of the vertices of a regular n-gon?
If so (and at least for deg $n=4,5$), what features of the $n$-gon (like $\theta$, center's abscissa, etc) can be given a closed-form in terms of the roots?

REPLY [2 votes]: I'll use a separate answer to address OP's request for handling the specific case where the points are the roots of this quintic:
$$x^5 + x^4 - 4 x^3 - 3 x^2+3 x+1 = 0$$
We'll polygon-ize by imposing an order on the roots from least to greatest; that is we take this to be the coordinate matrix:
$$P := \left[\begin{matrix} 
-1.918\dots & -1.309\dots & -0.284\dots & 0.830\dots & 1.682\dots \\
 0 & 0 & 0 & 0 & 0\end{matrix}\right]$$ 
Here's the graph of the polynomial, showing the (tinted) roots and the (black) $\{5/0\}$ component at the roots' centroid (aka, their average).

Following the process outlined in my other answer, we get an affinely-regular origin-centered $5$-gon and $\{5/2\}$-gon; each of these decomposes into two actually-regular components, as shown:


You'll notice that in this special case, where $P$'s vertices are collinear, the regular components of each type are mirror images of each other, in order that their $y$-coordinates in the vector sums cancel. If we choose to introduce projection to zero-out $y$-coordinates, then we can eliminate one of each of the regular components, and double the size of the remaining one. Here's one possibility:

In the above, I took the opportunity to re-center the regular components with the centroid. The extra dots floating around are the vector sums of corresponding vertices of the components; projections of these onto the $x$-axis give the polynomial's roots.