Here follows an example inspired by some great mathematical work of *Jean-François Burnol*.

All graphics are setup with `PSTricks`. All animations are generated with the `animate` package of
*Alexander Grahn* and finally converted to `svg` format. The calculations within the animations are mostly done with
`xintexpr` (part of the mighty `xint` bundle) of *Jean-François Burnol*.

Consider a cubic equation with real coefficients $P=0$ with three distinct roots. These are the abscissas of the vertices of an equilateral triangle whose radius is equal to the horizontal distance between the local extrema of the graph of the given cubic polynomial. The animation shows this fact in varying the second term of the equation $P = u$, which corresponds to the intersection of the graph with the horizontal line $y=u$.

A much more complicated thing to find only *rational roots* solved and animated by *Jean-François Burnol*. All calculations within this animation are done with `xintexpr` of *Jean-François Burnol*.

**Note:** This animation can't be fluently due to only rational roots.

Even much more complicated thing to find only *integer roots* solved by *Jean-François Burnol*.

As an example we chose the cubic: $P(X)=(X-10)(X+1)(X+9)=X^3-91X-90$

The roots of the first derivative are not integers.

**Note:** This animation can't be fluently due to only integer roots.

Even much much more complicated thing to find only *integer roots* and integer roots of the first derivative solved by *Jean-François Burnol*.

As an example we chose the $\ZZ$-nice cubic: $P(X)=X(X-9)(X-24)=X^3-33X^2+216X$

This polynomial gives the minimal distance of the abscissas of its local extrema which is 14.

**Note:** This animation can't be fluently due to only integer roots.

Nice cubic polynomials: symmetry and arithmetic of the Lagrange resolvent

by *Jean-François Burnol* and *Jürgen Gilg* (in English)

La voie-sans-effort™ vers le cubisme

by *Jean-François Burnol* (in French)

Polynômes cubiques plaisants

Excerpts of the documentation by *Manuel Luque* (in French)

Polynômes cubiques plaisants (2)

Excerpts of the documentation by *Manuel Luque* (in French)

The $\TeX$ source files can be accessed at

PolynĂ´mes plaisants (figures et animation avec PSTricks,
calculs avec xint, en particulier les fractions)

Many thanks to *Jean-Michel Sarlat*