It's an amazing fact about 4-space that there exist orthogonal bases like the one Jakob describes, tilted so that each vector is perpendicular to 2 of the standard basis vectors, and at 45 degrees to the other 2. --Dan Jakob wrote: << What about orienting a tesseract with edge length sqrt(2) in the following manner. Let one vertex sit in the origin, and let (1,0,1,0), (-1,0,1,0), (0,1,0,1), (0,-1,0,1) be the coordinates of the four vertices that are (edge-)adjacent to the origin. Since the corresponding vectors are orthogonal and have the same length sqrt(2), this indeed gives rise to a regular tesseract. Now, consider the projection (x,y,z,w) -> (x,y). The vertex set of the tesseract is mapped to the set {-1,0,1} x {-1,0,1}, and the projection of each edge has direction (1,0) or (0,1). Thus the projection of the 1D-skeleton of the tesseract is a 2x2 grid.
David wrote: << Can a regular tessertact can be oriented in R^4 so that the orthogonal projection of its 1D edges onto a 2D plane have the shape of a 2 x 2 grid?
Sometimes the brain has a mind of its own.