Axial creases


Post is a part of a larger series (Creases):

Axial crease is one of the three basic origami creases. By definition Axial creases are creases that run in the direction that is perpendicular to the polygon’s hinge creases.

Simplest form of a axial creases

In most basic form, meaning in the models that do not use more advanced origami techniques such as level shifters or Pythagorean stretch, axial creases are always horizontal or vertical, fully coinciding with lines that make up the grid.

Let me show how this looks in a simple example. In figure 2, we have one such example.

Crease pattern example
Figure 1

As you can see, all ridge creases are red while all hinge creases (boundaries of a polygon) are orange.
Now, let’s analyse, for example, polygon A. If we want to add axial creases to this polygon, we must draw lines that are perpendicular to the hinge creases. At the same time, these axial creases must coincide with lines that make up the grid.

Axial creases are always perpendicular to the ridge crease.
Figure 2

Interaction with a ridge crease

In figure 2, you can see something interesting. All vertical and horizontal axial creases will collide at the polygon’s ridge crease location. That brings us to one of the most important rules concerning axial creases. Axial crease must change direction whenever it intersects ridge crease. And it must do that respecting the following rule: axial crease must reflect across the ridge crease. In other words, the angle at which the axial crease hits the ridge crease is the same angle at which it will leave the ridge crease. If you are a bit confused, look at figure 4.

Axial crease reflecting over the ridge crease.
Figure 3

s you can see an angle a is the same as an angle b. In this particular case, since this model is a simple one and all ridge creases run at a 45-degree angle, all axial creases upon hitting a ridge crease will turn at a 90-degree angle. But do not be mistaken, this rule is universal, meaning it stands for all angles. Remember that.

An example shown in figure 3 is a classical one, and in most cases, the axial crease will reflect across a ridge crease as shown in figure 4. But, there are three exceptions. Again, do not be mistaken, the fact that we are talking about special cases does not mean that the above-mentioned rule does not apply. It does, believe me.

First of these special cases is shown in figure 4.

Interaction between ridge and axial crease (special case no. 1)
Figure 4

As you can see, the axial crease hits the intersection point between two ridge creases. In this particular case, the axial crease should bounce back into the direction it came from.

How do we know that? Well, let’s think about it. If we move the axial crease a few centimetres to the left it will hit ridge crease B. Then it would reflect over it turning right at a 90-degree angle hitting another ridge crease (ridge crease A). Again, the axial crease would reflect over that ridge crease too, turning downward into the direction it came from.

Now, if we start to move the axial crease incrementally to the right, it will always bounce back into the direction it came from. Only the distance between the incoming and the returning axial crease is going to be smaller and smaller. In the extreme case, when the axial crease hits the ridge creases’ intersection point, the distance between incoming and returning axial crease will disappear. Meaning, the axial crease will bounce back as I have told you.

In figure 5, we can see another such case.

Interaction between ridge and axial crease (special case no. 2)
ure 5

As you can see the axial crease again hits the intersection point between two ridge creases but, this time it does reflect across both of them effectively splitting itself into two axial creases. Please, pay attention to the fact that the axial crease hits the ridge crease C at 90-degree angle. The result is an axial crease that keeps going forward as if there is no ridge crease in front of it.

All of this brings us to yet another rule. All axial creases are, in essence, continuous lines that march across the crease pattern until they run out of the paper, come back to their point of origin or end in the intersection point between two ridge creases (case no. 1).

Defining all axial creases on a crease pattern

Now, that we are familiar with the basic rules, let’s try to define all axial creases in a simple example. For the beginning, let’s start with a crease pattern in figure 2. This one is interesting since it does not implement advanced origami technique (there are no Pythagorean stretch nor level shifters).

In a simple example, like this one, on one side of a ridge crease, an axial crease is always horizontal, and on the other side, it is always vertical. There cannot be both types of axial creases on the same side of a ridge crease.
A direct consequence of this rule is that on the crease pattern there are distinct sectors of only horizontal or only vertical axial lines. The lines that unambiguously mark the transition from one sector to another are none but ridge creases.

So let’s divide the sectors bounded by the ridge crease in two groups. To make the whole process easier, we will colour these sectors in two colours: orange and blue. Bear in mind that two sectors cannot be coloured the same if they are next to each other. That’s why we need to colour our crease pattern so that it resembles a chessboard.

Crease pattern divided into the 
sectors bounded by ridge crease.
Figure 6

Look at our crease pattern in figure 6. There are no sectors of the same colour touch one another. If we somehow found two sectors that do, then there is something wrong with our design. Most likely with ridge creases. There is definitively a mistake somewhere.

This procedure allows us to almost automatically divide the crease pattern into sectors of horizontal and sectors of vertical axial creases. But again, there is one small problem. We still don’t know which colour, blue or orange, represents horizontal and which colour represents vertical axial creases. Be very careful with that. It is not all the same.

To solve this problem, we only need to define the direction of axial creases in just one of these sectors. To do so, we have to pick one of these sectors and find a hinge crease in it. Then, all we have to do is to draw all axial creases that are perpendicular to that hinge crease and at the same time that coincide with the grid. And that’s it.

Crease pattern.
Axial creases in the first sector.
Figure 7

At this moment we know the direction of all axial creases. Blue sectors contain horizontal while orange sectors contain vertical axial creases. Now, we can define all axial creases, by drawing horizontal lines in the blue sectors and vertical lines in orange sectors.

Crease pattern and all axial creases.
Figure 8

That was easy, would you agree?

But, please be aware that this simplified approach is possible since this example is simple. It does not contain any of the more advanced elements. Meaning, all hinge creases are either horizontal or vertical.

Before we move on to the more complex cases, I would like to underline yet another future of the axial creases. All axial creases are, in essence, continuous lines that march across the crease pattern until they run out of the paper, come back to their point of origin or end in the intersection point between two ridge creases (case no. 1).

Axial creases as a continuous line.
Figure 9

More complex cases

There are, of course, crease patterns with more asymmetrical polygons, the polygons that are bounded by hinge creases that run at various angles. But even in such cases basic rules still apply.

Crease pattern example
Crease pattern exampleFigure 10

Look at figure 10. Two polygons (A and B) are highly asymmetrical since their edges (or hinge creases) do not necessarily run horizontally or vertically (just for your information, this kind of polygon shape is a result of the implementation of a technique called Pythagorean stretch).

Even in such cases, it is possible to divide the whole crease pattern into the distinct sectors bounded by ridge creases.

Crease pattern divided into the 
sectors bounded by ridge crease.
Figure 11

gain, as the basic rule implies, each of these sectors should hold only a set of axial creases of the unique direction.
So, let’s start.
It would be wise to start with sectors that include the hinge crease that coincides with the grid. So, let us start with the sector (1). In that case, everything is the same as in our previous example. We just have to draw all lines (or axial creases) that are perpendicular to the hinge crease (orange line) and at the same time coincide with the grid. The same applies to sectors (2), (4) and (5). They all have a hinge crease that coincides with the grid.

Crease pattern with sector 3 market in green.
Figure 12

With axial creases in the sector (3) (one in the middle), you should be careful. You see these axial creases should be perpendicular to the hinge crease H. That is obvious. But, since these axial creases do not coincide with the grid, we have to be careful while positioning them. The difficulty is that they should be in line with the axial creases of adjacent sectors. They should be positioned as if they are just extensions of axial creases in adjacent sectors. They should look as if they are the same axial creases.

If it is so, then the solution is quite simple. We have to extend the axial crease from adjacent sectors into the sector C, having in mind that axial crease upon hitting ridge crease should reflect across that ridge crease effectively changing direction. Their new direction must be if we did everything right, perpendicular to the hinge crease H.

Crease pattern and all axial creases.

That would be all for this blog post. I hope the explanations were clear and understandable.