The unification of Pythagorean theorem for electronic orbitals with Kepler's law for planetary orbits

Abstract

In the context of two-dimensional spacetime within a helium atom, both 1s electrons are characterized by wave functions that observe duality equation. They are symmetric, orthogonal and interwoven, forming a dynamic rope structure at any moment. Instead of elliptical orbit of planets around the sun, electronic orbitals take the form of matter state transformation cycle. While the kinematic movement of planets is governed by Kepler's first law, electronic transformation obeys Pythagorean theorem, both being equivalent in physical principle. The atomic spacetime is a continuous medium of electron clouds in synchronized differential and integral processes that are implemented by smooth trigonometry. In order to integrate this new approach with conventional physics, the author translates the pattern of electronic motion in the atomic spacetime into spherical volume undulation in Euclidean geometry and calculates the probability density of an electron within the sphere from the classical perspective. From the primary wave function of a 1s electron, the author also tries to derive the mathematical expression of central force that guides the surrounding bodies along the orbits. The result is exciting and surprising that questions the exactness of the venerable Coulomb's law.