The G. Thorpe's surface in a four-dimensional space

Background. At the present time, the basic provisions of the theory of surfaces in multidimensional Euclidean spaces are developed. But there are not many examples of specific surfaces of these spaces. Each of multidimensional surfaces is of great interest. The surface defined by John Thorpe, named four-dimensional space torus, is not an exception. In this paper we investigate the J. Thorpe's surface. Materials and methods. The authors used the method of section and surface projections to smaller dimension subspaces. These methods led us to unusual results. Results. For general geometrical reasons, it turns out that the equations by J. Thorpe define the sphere in a four-dimensional space. But it appears that the equations of the J. Thorpe's surface determine a surface, the properties of which differ from the properties of a sphere. This statement is justified in the present work. Conclusions. As a result of the research, the following deductions have been received: the surface defined by J. Thorpe is neither a torus nor a sphere. It is a surface with other characteristics, and it can be called “J. Thorpe's surface”.