Topology: Borsuk-Ulam theorem and its variants - A BRIEF MOVIE

Topology, the mathematical branch that assesses objects and their properties preserved through deformations, stretching and twisting, allows the investigation of the most general physical  and biological systems features. In particular, the Borsuk-Ulam Theorem (BUT) states that, if a single point projects to a higher spatial dimension, it gives rise to two antipodal points with matching description.  Physical and biological counterparts of BUT and its variants allow an inquiry of the world.   The opportunity to treat systems as topological structures makes BUT a universal principle underlying natural phenomena.

 

WATCH HERE THE (SHORT! JUST ONE MINUTE!) MOVIE

 
NOTE: we used here a 2D circle and a 3D sphere.  However, the concept can be extended in every dimension, either spatial, or temporal, or abstract.  For example, we can find one region on a 4D sphere that projects to two regions on a 5D sphere.Topology, the mathematical branch that assesses objects and their properties preserved through deformations, stretching and twisting, allows the investigation of the most general physical  and biological systems features. In particular, the Borsuk-Ulam Theorem (BUT) states that, if a single point projects to a higher spatial dimension, it gives rise to two antipodal points with matching description.  Physical and biological counterparts of BUT and its variants allow an inquiry of the world.   The opportunity to treat systems as topological structures makes BUT a universal principle underlying natural phenomena.
 
NOTE: we used here a 2D circle and a 3D sphere.  However, the concept can be extended in every dimension, either spatial, or temporal, or abstract.  For example, we can find one region on a 4D sphere that projects to two regions on a 5D sphere.