Test Fold

I recently posted a test model of a cone-based diagonal shift where the cone pointed inward toward the shift. This model is the reverse of that, where the cone instead points outward toward the shift.

The math of this variation is very similar to the inturned cone. For both variations, if the cone angle and the plane angle are the same, the convergence point is exactly the same distance from the top edge of the ellipse. But, when the paper comes back to being a cylinder, the effective distance of the horizontal shift is very different. When the cone is inturned, the top edge of the ellipse is shifted toward the center of the base cylinder, so the shift looks small. Here, since the cone is out-turned, the top edge of the ellipse instead sticks out quite a ways past the edge of the narrow base cylinder, so the amount of total shift horizontally looks much larger.

One more diagonal shift variation, with a different form of playing with the shape of the connecting tube of paper. Here, the shift is based on a cone of paper instead of a smooth cylinder.

Like my usual cylinder-based diagonal shifts, this model has a flat plane of paper connecting the top and bottom halves of the model, and that flat plane is essentially an ellipse. Because the cone shifts the convergence point much closer to the center of each cylinder, the two halves don’t have nearly as much of a horizontal shift between them.

A second photo, showing an example with just one half of the diagonal shift, showing the internal construction more clearly:

This test fold is a minor re-work of one from 2014 that combines ideas from my Diagonal Shift series with the flat vertical planes of my earlier Intersections work.

At the time, this was an idea I was interested in developing further, but the motif was annoying enough to fold that I didn’t do anything with it at the time. With several years more experience folding similar models and some very minor re-engineering of the internal structure, I got this to a point where I can use it in more complex models.

A few more views:

I’ve played quite a bit with ways to curve the axis of paper tubes, starting with simple pleats and extending to the crimp-bends that I used in my curved-neck vases and an assortment of geometrically distorted models. The crimp-bends have some advantages, but are a lot of work to set up for tubes with vertical pleats.

These test folds are a return to simpler pleats, testing how easily I can fold a series of pleats along the length of a tube. The first one is a test of changing the pleat axis along the length of the model, creating a slight helical twist.

The second keeps the same axis but changes in width along the length of the tube.

As a follow-up to my recent test fold of diagonal shifts in square tubes, this test fold joins one square tube with one circular (or actually 16-sided) tube.

I’ve played with twists that join tubes of different shapes before, and this design extends that to a shifted twist. As in the previous test fold, the center of the twist relies on the flexibility and stretch in the paper to actually collapse correctly.

I’ve played a bit before with twists in non-round tubes and using them in models. These test folds are my first attempt to combine those with my diagonal shifts, giving two square tubes that are offset from each other. The offsets of the two square tubes are the same in both test models.

In both versions (as in my previous tests), the points on each face of the square tube all fall along a parabola. Because of the offset between the tubes, the vertex of the parabola is aligned with the focus of the twist (the point at which the paper converges).

In the first version, I tried a simpler version of the math that places the focus of the twist at the same height as the vertex of the higher parabolas. In this version, the shortest segments of the twist go through the focus essentially horizontally, giving some vertical separation between the two tubes.

In the second version, the focus is shifted down to eliminate that vertical gap, or at least greatly reduce it. This gives something that looks a lot more like the twists I used in my diagonal shift series.

Somewhat inspired by some of Robert Lang’s recent work, this model has a twist that transitions from a 16-sided ‘circle’ to a square. Interestingly, the bottom edges of the square tube are approximately parabolic segments. The center doesn’t quite collapse cleanly, but it’s close enough for practical purposes. This twist is a bit more challenging to fold than the ones I’m used to. I might try incorporating some multi-shape twists like these into some of my more complex models.

This is a new variation of my recent crimp-bend design from several months ago. In the original design, I had two sine waves that lined up, with the paper between them crimped to create an internal flat plane. For this variation, I’m using one straight line and one sine wave to define the top and bottom of the bend instead of two sine waves. Because the flat and diagonal planes have different lengths, part of the flat crimped plane is visible.

I’ve been working on diagonal shift designs for a couple years, but up to this point all of the fully paper diagonal shifts have been ‘uphill’ shifts. With the basic crease pattern I’ve been using, a specific height of the sine wave naturally gives a matching distance for the uphill shift.

After designing my crimp-bent tubes, I realized I can add one bend immediately above the diagonal shift and one bend directly below. This reorients the shift so it’s angled downhill instead of uphill. Because the shift and the bend use the same sine wave, it’s not obvious unless you look very closely that there are extra layers of paper there.

Most of the origami I’ve done for the past several years has been based on shaping tubes of paper in various ways, whether that’s by adding curves or intersecting the tubes with vertical or diagonal planes. One thing I’ve wanted to figure out for a while is how to create bends and curves in the paper tubes. I’ve explored some simpler approaches to that problem before, but this is the first approach I think I can realistically use as a part of a more complex model.

The concept here was inspired in part by my diagonal shift design. The top and bottom edges of the bend are both sine waves, which get folded such that they touch each other along the bend line. Inside, each gore has a small crimp to create a partial flat plane visible inside the model. The crimps all have slightly different angles, but the mathematical to find those angles is the same that I used for the diagonal shift.

Here’s the inside view for a simple tube of paper:

Things get a bit more complicated when there is already overlap of the paper along the outside of the tube, but the concept is the same. It’s harder to see, but there’s a similar partial plane of paper inside this one, too:

It’s a decent bit of work to measure and score all the appropriate lines for these, especially for the narrower tube, but the folding went more smoothly than I expected. Especially with a bit of wet-folding, all the crimps seem to form fairly easily. I have quite a few ideas of how I’d like to incorporate these into a variety of more complicated designs.