  The Schr\"odinger cat states, constructed from Glauber coherent states and
applied for description of qubits, are generalized to the kaleidoscope of
coherent states related with regular n-polygon symmetry and the roots of unity.
The cases of the trinity states and the quartet states are described in
details. Normalization formula for these states requires introduction of
specific combinations of exponential functions with mod 3 and mod 4 symmetry.
We show that for an arbitrary $n$, these states can be generated by the Quantum
Fourier transform and can provide qutrits, ququats and in general, qudit units
of quantum information. Relations with quantum groups and quantum calculus are
discussed.
