On a number triangle
P. E. Shestov · Discrete Mathematics and Applications · 2006
In this research, we construct combinatorially continuous (neighbour-to-neighbour) mappings of a 64-pixel triangle to a 64-pixel cube and square. The pixels constituting the triangle, cube, and square are, respectively, triangles, cubes, and squares themselves, which form a partition of the initial object. On these objects, various neighbouring relations are considered. With the use of a computer, we construct a mapping of a triangle onto a cube such that any triangular pixels with common side are mapped to overlapping cubical ones. Also with the use of a computer, we establish nonexistence of a mapping of a triangle onto a cube such that any triangular pixels with common side are mapped to cubical ones with common side. Without help of a computer, we construct a mapping of a triangle to a square which maps overlapping triangular pixels to overlapping square ones.