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Diversion: Folding Paper

When I’m blocked on something I often reboot by brain grabbing a piece of paper and doing some “free folding” origami-like exercise. Usually I settle into some variation on the Miura Fold and riff on that for a while.

This time I decided to systematize. The results were:

Center is a standard Miura, seven segments long. Left and right are my variations, also seven segments. The chess men are for scale.
And from the side.

A Miura fold repeats the pattern for as long as you have paper. The paper is initially folded with three parallel folds — valley, mountain, valley. This defines the “long axis” of the fold. The remaining folds create a series of triangular folds that guide refolding the three parallel folds. Here’s a YouTube video that illustrates the process well.

The fold in the middle of the images above is what happens when you do the Miura Fold properly and compress it down, you get a parallelogram. (See the first page of notes below.)

But I’m not a particularly attentive folder and I sometimes make a fold in the wrong direction. I like to think of this as a Bob Ross “happy accident” and call it a Miura-Ross fold. An example is in the notes below and on the left in the images above. The “happy accident” was in the fourth segment. The result is that the fold went from being flat to having a kink. The fully developed fold goes from basically tiling a plane to “tiling” two intersecting planes. (My math is not up to making a more accurate statement.

Representing the triangular folds in a Miura Fold as triangles viewed from the side. The kink at 4 will unfold out of the “plane” defined by triangles 1,2, and 3.

Since most of my accidents tend to be unhappy excursions into a higher state of entropy, I decided to create some rules for where to put the kinks in a Miura Fold. Here is an arbitrary set of rules for the seven segment length:

  1. Select an integer sequence. I’m using Fibonacci: 1,1,2,3
  2. Each term indicates the number of triangular segments to fold before having a “happy” accident” and putting a kink into the fold.
  3. Kinks cannot reflect back on any other previously folded segment.

      The result is the fold on the right in the images above. The notes below show the side view of the fold kinking into a zigzag.

      Creating a little order out of chaos by setting rules for how to fold and when to kink. Note that there are two kink options between fourth and fifth segments. The rule is to fold away from creating an overlap.


      1. These are arbitrary rules. Once you agree to starting with a Miura Fold, everything else is up to you.
      2. Miura Folds can be extended with more parallel mountain, valley, mountain sequences. Just google “miura herringbone”
      3. Variations on Miura Folds have often been used for folding structures that transport compactly but deploy to large frameworks. The triangles are inherently stable. At some point I’ll use these ideas for some structural elements of a project.

        Comments are welcome. Please, just remember to “be excellent to one another.”

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