I intend to make elements wear the features of the entire set which includes them. The k(k-1)n-1/2 diagonally colored squares, on which no colors are paired, seem to serve this purpose. Here I fill a Möbius band with these squares (tiles) in case of k=3, n=6.
Möbius strip patterned by 48 different striped squares
100 x 100 x 100 mm
Diagonally striped tiles of this arrangement create concentrically striped squares. The number of squares is the number of the all possible triplets of three symbols (no symbols are paired): 3*2*2 = 12. The surface of the Möbius strip are diced with this different 12 squares. The edge of the strip is diced with another whole set of such triplets. This arrangement would be realized on tori as well.