Is there a shape, which when repeated, can create a Mobius strip?
Yep, there is. Really all you have to do is chop up a strip into squares, however many pieces you want and there you go, done. That was easy. Maybe I’m not asking the right question.
the better question
Is there an asymmetrical shape which interlocks with itself to create a continuous band on a non-orientable surface?
That one is a lot better, but it seems a little unrealistic that I would have started out wondering that instead of the first question.
To create a simple shape that tiles rectilinearly, you can start with a square, and any change you make to one side, you make the opposite change to the other side. So, if you squish in from the right, you squish out from the left. Continue until you have something interesting.
Listing bands have the additional twist (pun intended) that at some point, the top of one tile (or “scale” which is what I’m calling the individual components) is eventually going to have to fit in with the reverse of the bottom of another. Only bands made up of odd numbers of scales will work. This is probably easiest to understand if you consider that just one scale, twisted into a band would have to fit into itself this way, and one is an odd number. Even numbers simply twist too many times for an asymmetrical scale.
This particular band is made up of nineteen scales, shown above. They were fabricated from .03″ mild steel and allowed to rust naturally. The scales were designed so that just the right amount of twist and bend could happen with this size and material.
The base was constructed from fir and walnut. Displaying a band this large was a challenge, since it sort of collapses if it’s set on the floor. The base allows easy viewing from multiple angles, which is really necessary to get an idea of how complex the shape is. Fortunately there’s a sweet spot that allows just the right distribution of weight so that it’s balanced and sturdy on the base.
I didn’t do a rough draught for this one, which isn’t my usual process, but everything went smoothly anyway.