How can only one edge and one side contain a volume? Like this! Each point on the single edge shows up at two distinct places on the side, once on the left and once on the right, "as the path is travelled."
This Mobius loop has a square cross section and 7/4 twist. The side is 1/4 inch wide and the loop's centre-of-volume radius is 1/2 inch.
3.362 w x
3.4 d x
1.324 w x
1.339 d x