  The concept of bisection of a harmonic oscillator or hydrogen atom, used in
the past in establishing the connection between U(3) and O(4), is generalized
into multisection (trisection, tetrasection, etc). It is then shown that all
symmetries of the N-dimensional anisotropic harmonic oscillator with rational
ratios of frequencies (RHO), some of which are underlying the structure of
superdeformed and hyperdeformed nuclei, can be obtained from the U(N) symmetry
of the corresponding isotropic oscillator with an appropriate combination of
multisections. Furthermore, it is seen that bisections of the N-dimensional
hydrogen atom, which possesses an O(N+1) symmetry, lead to the U(N) symmetry,
so that further multisections of the hydrogen atom lead to the symmetries of
the N-dim RHO. The opposite is in general not true, i.e. multisections of U(N)
do not lead to O(N+1) symmetries, the only exception being the occurence of
O(4) after the bisection of U(3).
