The Schrödinger 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. This quantum kaleidoscope is motivated by our method of classical hydrodynamics images in a wedge domain, described by
q-calculus of analytic functions with
q as a primitive root 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 these states can be generated for an arbitrary
n by the Quantum Fourier transform and can provide qutrits, ququats and in general, qudit units of quantum information. Relations of our states with quantum groups and quantum calculus are discussed.
The hydrodynamic flow in several bounded domains can be formulated by the image theorems, like the two circle, the wedge and the strip theorems, describing flow by q -periodic functions. Depending on geometry of the domain, parameter q has different geometrical meanings and values. In the special case of the wedge domain, with q as a primitive root of unity, the set of images appears as a regular polygon kaleidoscope. By interpreting the wave function in the Fock-Barman representation as complex potential of a flow, we find mod n projection operators in the space of quantum coherent states, related with operator q -numbers. They determine the units of quantum information as kaleidoscope of quantum states with quantum group symmetry of the q-oscillator. Expansion of Glauber coherent states to these units and corresponding entropy are discussed.