  The problem of uniqueness of universal formulae for (quantum) dimensions of
simple Lie algebras is investigated. We present generic functions, which
multiplied by a universal (quantum) dimension formula, preserve both its
structure and its values at the points from Vogel's table. Connection of some
of these functions with geometrical configurations, such as the famous
Pappus-Brianchon-Pascal $(9_3)_1$ configuration of points and lines, is
established. Particularly, the appropriate realizable configuration
$(144_336_{12})$ (yet to be found) will provide a symmetric non-uniqueness
factor for any universal dimension formula.
