Pythagorean stretch – advanced topic (no.1)


Post is a part of a larger series (Advanced concepts):

This blog post assumes that you are familiar with the Pythagorean stretch concept, or at least that you have read (and hopefully understood) previous blog post on that topic (the Pythagorean stretch).

In the above-mentioned blog post, I have shown you how to move a Pythagorean stretch element along the diagonal. In this blog post, I will show you what could happen if we move a Pythagorean stretch element toward the non-square polygon.

But, before we proceed, let’s define what non-square polygons are.

They are precisely what their name suggests. They are polygons whose shape is not square. Most of you probably know that the Box pleating technique allows the polygons to be of almost any shape, as long as the circle of a particular size can be inscribed in it. You see, the size of a future flap is defined by the radius of the circle inscribed in it. Meaning, if a polygon is, for example, long and narrow, the length of the folded flap will be defined by its shorter side.

Since the polygon is not a square, we are allowed to move the circle inside the polygon. We are allowed to reposition it if we wish. The only restriction is that we have to keep the complete circle inside the polygon. But, this is obvious, I assume.

Origami Pythagorean stretch explained
Figure 1

This feature could be beneficial during a Pythagorean stretch element movement. I hope you remembered from the previous post, the limit to which we can move the Pythagorean stretch element is defined by the location of the circle inside the polygon. Or, more precisely, the limit is defined by the location of the circle centre. In other words, if we can reposition the inscribed circle in such a way that its centre is not on the Pythagorean stretch element movement path, then we can move the Pythagorean stretch element further than in usual cases.

Example No. 1

To make the whole concept more understandable, let me show you an example.

Origami Pythagorean stretch explained
Figure 2

If you examine figure 2, you will realise that one of the polygons is a so-called non-square polygon. Meaning, the inscribed circle is allowed to move between points A and B. In other words, it can assume any position in-between these two points. 

That being said, let’s define a Pythagorean stretch element that will solve the problem of overlapping polygons.

As always, let’s start by analyzing an overlapping area. As you can see, one of its sides is an even number, therefore we can construct a Pythagorean stretch element in the form of a parallelogram.

Origami Pythagorean stretch explained
Figure 3

Do you see what has happened? A Pythagorean stretch element is too big.  Or is it? It crosses the ridge creases of an upper-right polygon, which is a problem we will have to deal with eventually.  But, and this is essential, the Pythagorean stretch element does not reach a circle’s centre simply because a circle is moved to the right (point B). So, we can conclude that something like this is perfectly legal (figure 3). 

Disclaimer: technically speaking, the Pythagorean stretch element could be moved downward, and the configuration would be correct even if the circle centre remains at point A. But, we will not do that since I would like to show you what is going to happen if the Pythagorean stretch element collides with the ridge creases of another polygon. So we will assume that the Pythagorean stretch element will stay where it is now, and the circle would be the one that will be moved.

So if for some reason, we do not want to move the Pythagorean stretch element downward, then we have to rearrange ridge creases of the upper-left polygon and the Pythagorean stretch element since they cannot intersect one another.

This example is a simple one, so the solution is a simple as well. 

What we have to do is to connect points C and D and erase everything above that line. Everything above that new ridge crease.

Origami Pythagorean stretch explained
Figure 4

This way, our Pythagorean stretch element is a little bit modified, but that is OK. Everything is by the rules. 

So, whenever you see a crease pattern with a similar shape, you will know that this is just a Pythagorean stretch element that is in collision with the ridge creases of one of its polygons. Of course, that is possible only if the polygon in question is a so-called non-square polygon.

Now, we could add all axial and hinge creases, only to show you that this solution is indeed in line with the rules.

Origami Pythagorean stretch explained
Figure 5

Example No. 2

Let’s examine what would happen if we move the Pythagorean stretch element a bit further into the non-square polygon (figure 6).

Nothing much, except the collision between ridge creases of a Pythagorean stretch element and a non-square polygon, becomes more complex, and as such, asks for additional explanation.

Origami Pythagorean stretch explained
Figure 6

As in the first example, we have to shorten the Pythagorean stretch element since the ridge creases of various elements cannot intersect one another. Again, the procedure is almost the same. The only difference is that before we shorten the Pythagorean stretch element, we have to deal with the collision problem between two ridge creases on the left side, here marked in green (figure 6).

As I said, two ridge creases of various elements cannot intersect. They have to end at the collision point (in this case, at point A). That is logical. But from that point, and this is essential to remember, another ridge crease has to continue in a horizontal direction until it collides with another ridge crease (figure 7, point B).

Origami Pythagorean stretch explained
Figure 7

A complete, detailed explanation of why this had happened is out of the scope of this blog post. After all, I believe that you, who are regular readers of this blog, understand the theory behind the ridge crease construction. For those who are not familiar with the theory at hand, please, read the blog post on Ridge crease design. There, I have tried to explain what a ridge crease is and how it is supposed to be designed. 

Having said that, we can go back to our problem. As you can see, now we have both intersection points on the Pythagorean stretch polygon (points B and C), so we can add an additional ridge crease,  effectively splitting our Pythagorean stretch element in half (figure 8). 

Origami Pythagorean stretch explained
Figure 8

Now what is left is to add all axial and hinge creases (figure 9).

Origami Pythagorean stretch explained
Figure 9

Example No. 3

Now, the interesting question. What would happen if the non-rectangular polygon is even wider (figure 10)?

Origami Pythagorean stretch explained
Figure 10

Well, if it is wider (figure 10), we would be able to move the Pythagorean stretch element even further into the non-square polygon. That would be possible since the wider polygon will allow for the movement of the inscribed circle even further to the right. That way, the circle centre would not be on the Pythagorean stretch moving path anymore (figure 11).

Origami Pythagorean stretch explained
Figure 11

Again, the Pythagorean stretch element has to be shortened, the same way it was shortened in the previous example (figure 12).

Origami Pythagorean stretch explained
Figure 12

But here, I would like to pause for a moment, so we could take a good look at a lower-left polygon (figure 12). As you can see, it is perfectly rectangular, meaning the complete overlapping area between previously overlapping polygons is subtracted from the upper-right polygon. If you think about it, this is quite logical since the complete Pythagorean stretch element is pushed inside the upper-right polygon. In other words, the fact that the circle can move significantly to the right allows us to cut a complete corner out of the upper-right polygon and still manage to inscribe a circle in it.

But let’s think about what this really means. You see, if something like this is possible, then it is fair to ask: Why did we use a Pythagorean stretch technique in the first place? Isn’t it possible to rearrange ridge creases in the upper right polygon to achieve the same result without the Pythagorean stretch element?  Of course, it is possible. This way, we will get a simpler and more understandable ridge crease configuration. In other words, we will get a classical ridge crease configuration without complications caused by the implementation of a Pythagorean stretch technique (figure 15).

But before I end this blog post, I would like to show you that those two approaches, though different, are in no collision with one another. If you take a good look at figure 11, it is obvious that the Pythagorean stretch element could be moved even further. It could be even moved out of the upper-right polygon if you like. But there is no need for such extreme moves. It is enough to move it above the horizontal ridge crease of the upper-right polygon, and as a result of ridge crease rearrangement, the Pythagorean stretch element will disappear (figures 13 and 14). 

Origami Pythagorean stretch explained
Figure 13
Origami Pythagorean stretch explained
Figure 14

Again, a complete, detailed explanation of how to perform a ridge crease rearrangement is out of the scope of this course. But, if you feel the need to deepen your knowledge on the subject, please read the blog post on Ridge crease design

Now, what is left is to add all axial and hinge creases (figure 15), only to see that even if we implement a Pythagorean stretch technique, the final result can and will be the same. The only precondition is that a circle can be moved sufficiently to the right.

Origami Pythagorean stretch explained
Figure 15