One Sided Surface in 4D

In the applet below, two repers - a pair of perpendicular segments - are randomly placed on one of 24 squares of the tesseract - a 4 dimensional cube or a hypercube. One 2D reper remains on that square for the duration of the experiment. The other reper can be moved to any of the 8 squares that have a common edge with the current one. (Obviously, there are 8 candidate squares, right?) The reper moves without rotation: if the two squares (the from-square and the to-square) were placed on the same plane, the reper would just glide from one to the other. The task is to take the moving reper on a ride at the end of which, back at the original square, the two repers will have different orientations.

The tesseract is the set of points

{(x,y,z,h): 0 x 1, 0 y 1, 0 z 1, 0 h 1}.

Its boundary cubes are defined by fixing value of one of the coordinates to either 0 or 1. This is why there are 8 of them. Each of the 24 squares is defined by fixing values of any two coordinates. There are 6 possible pairs and 4 possible values (00, 01, 10, 11) for each. Every square is assigned a 4 symbol name. X01H, for example, denotes the square for which y = 0 and z = 1.

Explanation