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
, R radius, and the reduced density is
,
central density. Reduced density profiles are plotted for several polytropic indexes within the range, 0
≤n
≤5, disclosing two noticeable features. First, any point of coordinates, (w, v), 0≤w≤1, 0≤v≤1, belongs to a reduced density profile of the kind considered. Second, sufficiently steep i.e. large reduced density profiles exhibit an oblique inflection point, where the threshold is found to be located at n=n
th=0.888715. Reduced pressure profiles,
,
central pressure, Lane-Emden fucntions,
, and polytropic curves, q=q(v), are also plotted. The method can be extended to nonspherical polytropes with regard to a selected direction,
. The results can be extended to polytropic spheres made of collisionless particles, for polytropic index within a more restricted range, 1/2
≤n≤5 .