The intrinsic beauty of polytropic spheres in reduced variables
  The concept of reduced variables is revisited with regard to van der Waals'
theory and an application is made to polytropic spheres, where the reduced
radial coordinate is ${\rm red}(r)=r/R=\xi/\Xi$, $R$ radius, and the reduced
density is ${\rm red}(\rho)=\rho/\lambda=\theta^n$, $\lambda$ central density.
Reduced density profiles are plotted for several polytropic indexes within the
range, $0\le n\le5$, disclosing two noticeable features. First, any point of
coordinates, $({\rm red}(r),{\rm red}(\rho))$, $0\le{\rm red}(r)\le1$,
$0\le{\rm red}(\rho)\le1$, belongs to a reduced density profile of the kind
considered. Second, sufficiently steep i.e. large $n$ reduced density profiles
exhibit an oblique inflection point, where the threshold is found to be located
at $n=n_{\rm th}=0.888715$. Reduced pressure profiles, ${\rm
red}(P)=P/\varpi=\theta^{n+1}$, $\varpi$ central pressure, Lane-Emden
fucntions, $\theta=(\rho/\lambda)^{1/n}$, and polytropic curves, ${\rm
red}(P)={\rm red}(P)({\rm red}(\rho))$, are also plotted. The method can be
extended to nonspherical polytropes with regard to a selected direction, ${\rm
red}(r)(\mu)=r(\mu)/R(\mu)=\xi(\mu)/\Xi(\mu)$. The results can be extended to
polytropic spheres made of collisionless particles, for polytropic index within
a more restricted range, $1/2\le n\le5$.
