In this paper, we present an interactive platform for visualizing and manipulating the digital twin of a human heart reconstructed from computed tomography (CT). The platform involves a pair of holographical glasses whose cameras are used to input the control parameters by hand gestures, a highend graphical workstation acting as the platform manager to render the data and control the computations, and a high performance computer cluster which is the workhorse for heavy computations required by the physics-based model of the heart. The numerical model of the heart is referred to as the digital twin of the biological heart, and using the proposed platform we can see and operate on certain part of the heart that is difficult to reach in the biological heart. Such a platform can be used as an off-line tool for surgical planning or as a near real-time tool during the operation. The main focuses of the paper are algorithms and software for manipulating the threedimensional elastic object through the holographic glasses using hand gestures, the control of the surface geometry, and the near real time computation of the motion and displacement of the elastic body on a parallel computer. The deformation of the heart is modeled by a hyperelasticity equation solved on a supercomputer so that physically meaningful motion is obtained in near real time. Preliminary results are reported for the heart of an actual patient.