Algebraic topology of the Tetris game and binomial coefficients

Abstract

etris is a well-known video game created in the late 80s. In the game, players complete lines by moving differently shaped pieces called tetrominoes, which descend onto the playing field. The tetromino is a union of four unit squares connected edge by edge. If the number of the unit square is not emphasized, a figure is called polyomino.

We apply algebraic topology to study the tilings of a region in the plane or a torus by the given set of polyominoes. In order to do that, we introduce a specific combinatorial structure, the simplicial complex of the tiling. These complexes reveal many features of placements of tiles from the given set into the given subset of the grid without overlappings. We will prove that they have high connectivity and, in some cases, have the homotopy type of wedge of spheres. We obtain exciting identities involving binomial coefficients by interpreting the Euler characteristics differently.