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Tag: listing band

Listing bands (a.k.a. Möbius strips) are non-orientable surfaces made up of components which repeat in a way that show no beginning or end. The reason this is cool is that it applies a p1 symmetry to a paradromic ring.

But what does listing have to do with anything? Apparently in 1858, Johann Benedict Listing discovered the same properties of a one-sided surface as Möbius did, but didn’t publish his findings. So, 1) to honor his work and 2) to distance my work from what people know about the ever-elusive non-orientable surface, I’m using the term Listing band. (Although now it occurs to me that The Animaniacs were a bit of a Listing band themselves.)

  • triple knot listing band

    triple knot listing band

    A knot can be infinite in terms of translation if its left edge lines up with its right edge. That means that a knot can surface a cylinder. But, what about a non-orientable surface?

    If a knot (segment) is created so that its upper right lines up with its lower left, and the segment is repeated an odd number of times, the whole unit can surface a one-sided surface.

    Here is a maquette of a solid (flattened) version of a knot as a Listing band.

    This one shows the 3 individual knots more distinctly:

    Once again, look at those shadows! In this example, I could have chopped up the knot to give the illusion of lines going over and under other lines, but the end goal was to make something solid, where this one was simply printed on acetate. I actually started wondering if I could somehow encase a material in something clear so I could make a much heftier version and still have the over-under implication, but I ended up going with something much more down to earth. Metal!

    The pieces were designed with three holes at the end of each segment so I could pop rivet everything together and everything would self-align.

    I liked the colors I picked for the maquette, but I wasn’t about to paint anything when metal can look so amazing and varied just by using harsh chemicals. After degreasing and shining everything up, I applied a clear sealant for the blue line, gun bluing for the black, and rust juice for the red. Note the lack of photos of bluing the metal because I was scared to death I would poison myself.

    I waited to do the final assembly until just before I was ready to display it to avoid the possibility of damage. It’s very delicate for something made out of braided metal.

    the final piece installed along with the bonus of really nice shadows
    probably the final chapter in this piece’s life: turning it into a chandelier for our dining room
    another added bonus: a serendipitously projected rainbow on the wall
  • translucent listing band slice stack

    translucent listing band slice stack

    computer rendering of the encased band

    Perfectly symmetrical listing bands are difficult to display because of one of those things that bring us all down, like gravity. But, if you could slice one (or really anything that’s hard to balance) up, you could stack the slices and create the illusion of a dimensional solid.

    This a test using glass. It’s unreasonably heavy.

    Being able to view this from multiple angles makes a lot more sense:

    The first version, shown in this video, is made up of 40 pieces of glass, each with a cross-section of a simple listing band. The mind does its wonderful job of smushing and the illusion of a thing, as opposed to just a stack of lines, appears.

    test pieces shapes painted on, lounging together on a couch

    The final piece was 80 10″ x 10″ pieces of Plexiglas (which I learned allows for more optical transmission than glass) each with a different shape painted on its surface. The frame was made from busted Chinese chestnut, which I think contrasts nicely with the pristine precision of stacks of thin lines.

    a couple of tests for the wood enclosure

    This piece sits outside the main gallery because I wanted it to be an introduction to the rest of the pieces: its design is much simpler but its implementation invites the viewer to look closely to find out what’s really going on.

    One thing I find fascinating about this shape is that it is the same as the symbol on the window. It just looks completely different from every angle.

  • collapsible scale listing band (deconstructed)

    collapsible scale listing band (deconstructed)

    The two things you don’t want wood to do are bend and twist. The two things listing bands do are bend and twist. Is there maybe a compromise?

    a maquette of the original design. this thing is very small.

    If alternating cuts are made in opposite sides of a piece of wood (or any rigid material) the material can more or less bend. Of course, the material itself isn’t bending as much as the overall bend, but it averages out to a much greater bend that would be possible without the cuts.

    The same applies to twisting.

    this is more or less half of a band, but it shows what’s necessary for a twist-bend combo
    a gathering of tests made from various materials, enjoying a picnic on the lawn
    my notes for this project

    The horizontal spaces allow for twisting and bending to happen.

    Notice how the top and bottom fit into one another. This maintains the vertical translation during scale chirality alternation.

    A wire was to be placed as a connector as shown here.

    I thought a nice touch would be the floating circle in the “head.” Because drilling holes that lined up through a piece of wood thin enough to twist-bend would require a Hanukkah miracle, another solution was to rout out a curved groove half way through mirror images of the scales. Then the wire could be placed in the groove and the wood laminated together.

    I CNCed a template which I used to trace on the wood…

    …drilled some holes to make life easier…

    …and went to the band saw. There were casualties.

    the robot gingerbread man infirmary

    Then I used an edge trim bit and the original template to clean things up. I actually designed the whole thing based around the clearance I’d need for the bit to clear the interior zig-zag shapes which was nerve-wracking to say the least.

    Again, because of the thickness of the wood, I had to treat it very gingerly. To rout the grooves, I made router passes of 1/256 of an inch at a time. It took a while.

    templates for holding the little guys still while I routed the slots for the wire, along with one fancy boy made out of cedar

    But it was worth it!

    When I saw the way the scales looked linked together, I liked it so much that I wanted to show them like that instead of contorted into a band.

  • axonometric scale listing band

    axonometric scale listing band

    the question

    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.

    an answer

    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 individual scales. 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.

    one finished scale and its idealized shape
    notes and sketches
    wire used to links scales together

    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.

    In lieu of a maquette, I did a bunch of calculations, which is usually a recipe for disappointment. Luck was on my side this time.