Abstract
According to the Poincaré conjecture (1904) proved by Grigory Perelman (2002-2003) that any simply connected compact three- dimensional manifold without edges is homeomorphic to a three- dimensional hypersphere [1], to solve the problems of visualizing four- dimensional objects in three-dimensional space [2], it is proposed to choose a suitable manifold, in in this case, a ball, establishing a homeomorphism between objects located in different spaces by technological means of cartography. As a result of this work, it seems possible to build a dynamic video of the population distribution process on a map of the globe, which provides informational four-dimensional data flow, following the ideas embodied in 4D Anatomy [3]. The proposed technology opens up new ways of visualizing four-dimensional space This work was performed within the framework of the state assignment of the ICM MG SB RAS (project 0315-2019-0003).
Highlights
After a small transformation, which consists in transferring the parameter H along the fourth dimension to the left side of the equation, we can see on the right side of it a parametrically defined ball
1.1.1 Interaction of homeomorphic varieties. The diagram of such correspondence, which explains the interaction of homeomorphic manifolds, is shown in Fig. 1, for two-dimensional spaces of smaller dimension
The issues of the geometric representation of multidimensionality began to be given scientific attention from the middle of the 19th century, which was soon reflected in Einstein's four-dimensional space-time concept, linking the relationship between space and time
Summary
The formula for a hypersphere can look like this, where R is its radius: R2= X2+Y2+Z2+H2. After a small transformation, which consists in transferring the parameter H along the fourth dimension to the left side of the equation, we can see on the right side of it a parametrically defined ball
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