Abstract
One of the fundamental questions of a three-dimensional geometer is how to imagine a four-dimensional object. And yet he draws pictures of three-dimensional objects in the two-dimensional paper.
Moreover, would a two-dimensional geometer understand our sketches? Based on analogies, we give an overview of methods of examination of four-dimensional objects. We emphasize visualization as the main element of perception of four-dimensional space.
For this purpose, we describe the double orthogonal projection of the four-dimensional space onto two mutually perpendicular three-dimensional spaces as a generalization of the classical Monge's projection. In such a projection, we construct a four-dimensional playground for convenient synthetic creation of four-dimensional objects.
All our constructions are easily accessible with the 3-D modeling software GeoGebra. Furthermore, we apply the method of projection to an intuitive investigation of various four-dimensional mathematical phenomena - polytopes, four-dimensional quadrics, three-sphere and its stereographic projection, complex plane, and the Hopf fibration.