Seeing is believing. Our novel virtual reality system, Polyvision, applies this old saying to the fourth dimension. Various shadows of an object in a four-dimensional (4D) space are simultaneously projected onto multiple three-dimensional (3D) screens created in a virtual environment to reveal its intricate shape. The understanding of high-dimensional shapes and data can essentially be enhanced when good visualization is complemented by interactive functionality. However, a method to implement an interface for handling complex 4D transformations in a user-friendly manner must be developed. Using our Polyvision system, the user can manipulate each shadow as if it were a 3D object in their hand. The user’s action on each projection is reflected to the original 4D object, and in turn its projections, in real time. While controlling the object’s orientation minutely on one shadow, the user can grasp its global structure from multiple changing projections. Our system has a wide variety of applications in visualization, education, mathematical research, and entertainment, as we demonstrate with a variety of 4D objects that appear in mathematics and data sciences.