Is it possible to derive a mathematical physical form for Euler’s formula in the reality of complex harmonic oscillator?

Abstract Euler’s formula is a pure mathematical formula. In spite of its nonphysical nature, the form of this formula has a genuine fundamental role in quantum mechanics, but with no physical reason for this role. Philosophically, Heisenberg has proposed Kantian ideas in physics like “nature in itself”, unobservable nature, and “nature as appears”. He proposed a possibility of mathematical formulation for a system in nature in itself. In the present work, a mathematical model of rolling circles in real space has been proposed as a model for a system in nature in itself. Observation process acts as a transformation of the system to a model in observable nature. Partial observation transforms the real position vector of a point in that system to a form similar to the Euler’s formula. That causes the appearance of the imaginary unit. Interestingly, the transformed real kinematical forms exhibit similarity with the forms of relativistic quantum mechanics.