On a number triangle

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.
This research was supported by the program ‘Algebraic and Combinatorial Methods in Mathematical Cybernetics’ of the Department of Mathematics of the Russian Academy of Sciences, project ‘Algorithms of Discrete Geometry.’