Test Fold

Following up on my recent exploration of geometric distortions to square tubes, this test fold takes that exploration in a new direction. In my previous explorations of designs where the twist has more sides than the tube, I always had the issue that to make the paper lengths match up, the twist was never flat. That led to some limitations when trying to design models using those twists – the twist took up a lot of height in the model, and it was harder to make sure the model would be stable long-term and not want to tilt and drift as it aged.

This is the first successful test fold of a new approach to twists with mismatched numbers of sides. The twist is flat because the extra paper along each edge is tucked inside the model. The concept isn’t that much more complicated than what I’ve been doing in other designs, but it took quite a bit of geometry and algebra to get all of the paper lengths correct.

This is obviously only half of the full twist, but a second half could easily be stacked on top of it. Similar to some of my older explorations with the non-flat twists, it should be possible to stack different shapes on top of each other. I’ll have more to come on exploring this new direction further.

Continuing from my recent post, I’m exploring test-folds of geometric distortions to square tubes instead of the cylinders I’ve usually worked with. These two designs are very closely related to some designs I explored about 10 years ago and used in a diagonal shift vase about 5 years ago. These test-folds are actually based on pleats, not twists. The pleats can be trickier to work with because they don’t naturally hold themselves locked in place after you fold them, and they can tend to collapse partway but not all the way. For the first test-fold where the square tubes are aligned, either a pleat or a twist would work and give a very similar result.

The distances between the central convergence point and the tube work out such that the points on each flat face fall on a parabola. That’s less obvious here than for some of my previous examples where I had a 16-sided twist in a square tube.

The second test fold, with the square tubes rotated 45 degrees, only works with pleats. I attempted a version using a twist, and the paper lengths just don’t work – there are certain point pairs than need to be farther apart in 3D space than the length of the paper connecting them, which doesn’t work. But the pleated version works just fine. The rotated version is also fairly unstable because the top and bottom tubes are only really connected at the central convergence point and don’t rest on top of each other anywhere else. So as is, this wouldn’t work well in a vase because the top half would tend to tilt off to one side over time.

I’ll have more explorations that go into newer territory soon.

This test-fold is essentially a recreation of one from 2015, re-learning how I folded the shape. From the original post:

Creating the twist is actually fairly straightforward – I would be surprised if it’s not a re-invention of someone else’s idea. I used the same angle between each box here, but any angle from 0 to 90 degrees would work with very minor changes to the crease pattern. It could also work for shapes with more sides, not just squares. Collapsing the stack of twists is a bit tricky and uses a lot of paper, so I’m not sure I would want to try a stack too much taller than this.

With a few more years of practice, I re-created the same basic design with cubes stacked at 45 degree angles, giving it a nice symmetry. I’m starting to post a series of test folds exploring variations of twists on square tubes and other simple geometric forms instead of the more cylindrical forms I’ve usually folded, so this made a nice, simple entry point to thinking in those directions.

Before my diagonal shift series went on hiatus for several years, I was playing with cone-based diagonal shifts, including ones where the cone pointed inward as it approached the shift plane and ones where it splayed outward. This test-fold combines an in-turned cone on the bottom with an out-turned cone on the top, creating the appearance that a singe cone continues through both halves of the model. This continues a theme of mixed-shape twists that I had started several years ago.

This was a test fold for one of the pieces currently on display at the Urbandale Art Gallery – photos of the finished piece coming soon!

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.