Ridge crease


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

Ridge crease is a fold that could be easily recognized at least in the case of a box pleating technique. But at the same time, ridge crease’s definition is quite esoteric at least if it is defined in more general mathematical terms.

So, the general mathematical definition of the ridge crease is as follows: a Ridge crease is a fold or a crease that extends inward from a polygon corner, at the same time acting as an angle bisector of that corner.

For those who do not know, an angle bisector is a line that cuts an angle into two equal parts.

In folded form, a ridge crease could be easily recognised because it is a central fold of the flap that goes up and down along the flap’s whole length.

Ridge crease illustration.
Figure 1

Simplest form of a ridge crease

But let’s go back to the main definition, one that defines ridge crease as an angle bisector of a polygon corner.

Polygons

Ridge crease on a crease pattern.
Figure 2

If a polygon is a simple square, as most often is, defining the positions of its ridge creases is simple because all ridge creases are square’s diagonals (Look at figure 2 – Polygon A). In other words, ridge creases are lines that run at a 45-degree angle. Even if the square polygon is not fully visible, meaning part of the polygon is out of the paper, the principle is still the same. Ridge crease is always an angle bisector of a polygon’s corner (Look at figure 2 – Polygon B). But be cautious. If you look at Polygon B, it might seem that points (a), (b), and (d) are polygon’s corners. But they are not. Points (b) and (d) are points on the polygon’s edges. At the same time point (a) is a polygon’s centre, not its corner.

Be aware of that, especially when a large portion of the polygon is out of the paper. In this case, the only visible corner from which a ridge crease could be drawn is point (c).

So, to avoid mistakes, in case of polygons that are on the paper’s edges (figure 3 – polygons B or C), I would highly recommend the following procedure. First, draw a complete polygon even if it means drawing lines out of paper. Then, draw all four angle bisectors. Finally, erase everything that is outside of the paper. This way, you are going to avoid all possible mistakes.

Ridge crease definition on a crease pattern.
Figure 3

Rivers

Defining rivers’ ridge creases is even simpler. At every river’s bend, we just have to add a ridge crease at a 45-degree angle. And that’s all.
Look at figure 4. The river is highlighted in green. As you can see the river has four bends, therefore, there must be four ridge creases running at a 45-degree angle. There is nothing more to it.

Ridge crease and origami river
Figure 4

Ridge creases form a continuous line

Now that we know how to define ridge creases, at least in its basic form I would like to point out a few interesting features of ridge creases. If you look at figure 4, it is easy to notice that all ridge creases of various polygons add to one another. In other words, ridge creases have to form a continuous line from one edge of a paper to another. Simply, continuous ridge creases (one that emerges as a result of adding one ridge crease to another) cannot appear or disappear in the middle of the paper. Look at figure 4. You can see one such continuous line (marked in red) formed by adding together various ridge creases. Occasionally, such continuous ridge creases can go in a circle which is fine too. One such continuous ridge crease, marked in blue, you can see in figure 4.

Ridge creases that do not run at 45 degree angle.

Since we are familiar with basic rules, let’s talk about so-called special cases.
You see, there are few cases when ridge creases do not necessarily run at a 45-degree angle. But even in these cases ridge crease is an angle bisector. Remember that. Basic rules are always respected.

Case 1

Let me show you one such example (figure 5).

Horizontal ridge crease (case 1)
Figure 5

Polygon A in figure 5 (marked by a pink circle) is not a square but a rectangle. The fact that the polygon is not a square is a prime reason why not all its ridge creases run at a 45-degree angle. To better understand this case, let’s try to draw all its ridge creases (ridge creases of all other polygons are already drawn).

Horizontal ridge crease development procedure.
Figure 6

Let start with corner A. According to the rule, a ridge crease at this corner is a bisector of an angle between edges (a) and (b). See the dashed line.

The same applies to corner B. Again, a ridge crease is also a bisector of an angle between edges (b) and (c). These two bisectors (ridge creases) cannot run forever since they will collide sooner or later. What then? What will happen when they collide? Well, they will end there. But, as you can see in figure 6, from the collision point another bisector (a ridge crease) will continue horizontally. No matter how strange this seems, this horizontal line is also an angle bisector.
A bisector of an angle between edges (a) and (c). And since these edges are parallel its bisector is parallel with them too. I hope this is clear.

The same procedure can be applied to corners C and D. From these corners, two angle bisectors run at a 45-degree angle. These two ridge creases will also end at a collision point. I hope this is clear.

Case 2

Before we move on let me show you one even more complex example (figure 7) As you can see this polygon is a bit more complex, but do not worry all the rules still apply. In figure 7 you can see all its ridge creases. Again, all of them are angle bisectors of various corners. Even the horizontal and vertical ones. I hope you understand how these horizontal or vertical ridge creases originate.

Ridge crease (case 2)
Figure 7

But, what I would like to talk about is this small ridge crease in the middle (marked in red).
Is it a bisector of a corner? Yes it is, it must be. But, the question is, which one?

Can you guess?

Well, it is a bisector of the corner between edges (c) and (f) (look at figure 8). I hope you understand why. I hope you understand the logic behind this.

Ridge crease (case 2) analysis
Figure 8

Even more complex ridge creases

Sometimes the shape of a polygon is highly irregular, meaning not all of its edges are horizontal or vertical. As a consequence, not all of its ridge creases will run at a 45-degree angle nor will they be horizontal or vertical.

Ridge crease and highly irregular polygons
Figure 9

If you look at figure 9, you will see one such example. Something like this is not uncommon in modern-day origami since it is most often the result of the implementation of a technique called Pythagorean stretch. Explanation of what is a Pythagorean stretch and how it is implemented is way beyond the scope of this blog post. So, just take note, that this, a bit odd polygon is a result of implementing a Pythagorean stretch. That’s all.

So, let’s analyse one of these polygons (the polygon marked in red). As you can see this polygon has five ridge creases (marked in purple) that run at some strange angles. Except for the ridge crease number 5, none of the ridge creases run at a 45-degree angle. But, again angles are not chosen by chance. They are angle bisectors of various corners as they are supposed to be.
Specifically, …

  • ridge crease (1) is the angle bisector of a corner between edges (a) and (f),
  • ridge crease (2) is the angle bisector of a corner between edges (f) and (d),
  • ridge crease (3) is the angle bisector of a corner between edges (d) and (e),
  • ridge crease (4) is the angle bisector of a corner between edges (e) and (f) and
  • ridge crease (5) is the angle bisector of a corner between edges (b) and (c).

And that’s it. There is nothing more to it.

Final word

As a part of a final word, I would like to mention one interesting feature of ridge creases in the context of a crease pattern. You see, the crease pattern is defined by three types of creases: hinge, ridge, and axial creases. So if we want to analyse a crease pattern in order to define where individual polygons are, or if we want to define their shapes, only what we need to know is the positions of all ridge creases.

If you do not believe me, just take one more complex crease pattern and analyse it. You will see that the ridge creases are literally all you need to find positions and shapes of all polygons.