It is easy to decompose the plane into half-open intervals of unit length (decompose the x-axis into the intervals [n, n + 1) and then translate vertically). Define the direction of a half-open unit interval to be the vector whose "head" in the closed endpoint and whose "tail" is the open endpoint. For the decomposition above, only one direction is realized (the vector with coordinates <−1, 0>). This month's problem is to find a decomposition of the plane into half-open unit intervals such that every possible direction is represented. Can you find an an analogous decomposition of space?
Source: Bill Taylor