|Gregory Galperin (Professor, EIU Mathematics Department)|
The multidimensional cube: its geometric, arithmetic, combinatorial, and "electrical" properties, and paradoxes
Abstract: Abstract: A generalization of a square is a standard 3-dimensional cube K3, whose generalization, in turn, is a _tesseract_, a 4-dimensional cube K4 that can be obtained from K3 by moving the standard cube along the fourth dimension. A d-dimensional cube Kd is obtained from the cube K(d-1) in the same manner. After drawing the tesseract on the blackboard and calculating the number of its vertices, edges, plane faces, and three-dimensional faces, we present a simple formula for the number of faces of all dimensions from 0 to d (for each d = 1, 2, 3, ...). A natural modification of this formula, applied to an arbitrary polyhedron, leads to relationships between the number of polyhedron's faces of all dimensions, one of which is the famous Euler's formula for polyhedra.
The rest of the hour will be devoted to some beautiful formulas associated with different arithmetic, geometric, and "electrical" properties of the d-dimensional cube. Two paradoxes of the multidimensional cube in large dimensions (starting with dimension 10) will also be presented.
The talk is based on elementary notions of geometry and algebra. It will be absolutely clear for undergraduate students. Students of all levels are encouraged to attend the talk.