## Origami and circles

How many times you have heard that origami is based on circles? Is it true?  and if so, what does it mean?

Well, in this blog post, I will try to answer these questions. So, let’s start. Right at the beginning, I will give you the answer to the first question. The answer is definitely YES. An origami is indeed based on circles. But if we want to know the real meaning of this, maybe it would be the best to start with the basic rule. So, the rule is as follows: Number of circles we can pack on a square piece of paper defines the number of flaps our model (or its base) will have. This is a pure definition but what does really mean asks for a little bit more of an explanation. So, let’s start from the beginning.

## A Simple example

So, if you look in Figure 1 you will see one really simple origami base. This base, as well as every other example in this blog post is made by using so called “box pleating” technique. This technique assumes implementation of a grid (in this particular case 16 by 16 grid was used), meaning, all basic elements are in line with the grid. Specifically, centres of circles are always located on the intersections of horizontal and vertical lines that define the grid. Also, the circle’s sizes are always defined as a multiple of basic grid size.

 Another name for this technique is Polygon packing.

## Basic rules

In Figure 1, it is clearly visible that this very simple base consists of six flaps of various sizes. Being aware of that let’s find circles we mention before, since according to the origami theory every flap must be defined by its corresponding circle. But, in order to utilise this theory some addition rules are needed.

1. circles that define flaps cannot overlap. This is quite logical since one and the same piece of paper cannot be a part of more than one flap.
2. a part of a circle can be outside of a paper, but its centre must always be on the paper or at least on the paper’s edge. Or more precisely, the circle centre can be anywhere on the paper but must be on the paper.
3. a circle’s radius defines a flap’s length.
4. a tip of a flap is always in the centre of a circle.

## Crease pattern

Now that the ground rules have been established, let’s analyse our simple base in more detail. At first glance, it is obvious that the base consists of six flaps of various lengths. One flap is distinctly longer while five are shorter. Now, if we unfold the base, we will get a square paper with more than a few distinctive creases on it (look in Figure 2). This kind of representation of an origami model on a fully flattened paper with all creases being visible sometimes is called a “crease pattern”. Here I would like to stress clearly that this is not a “crease pattern” by definition. This is something that only looks like a “crease pattern” in a sense that all the creases are shown. That’s all. Real “crease pattern” is much more informative. From real “crease pattern” one can clearly see where axial and hinge creases are. In this case, since axial and hinge creases fully coincide with creases that define a grid, there are not clearly visible. Also, real “crease pattern” shows crease orientation which is not the case here.

## Where are the circles?

Now, by looking in Figure 2 you must ask yourself where are the circles we were talking about. Where are the circles that define flaps?

In Figure 3 circle are clearly visible. Circles are in all four corners of a paper with two additional circles on the paper edges. I hope that from Figure 3 you can clearly see that all rules are obeyed. Circles do not overlap (Rule 1). Circles are partly outside the paper, but their centers are on the paper (rule 2). They are in the paper’s edges. Also, large flap is defined by larger circle while smaller ones are defined by smaller circles (rule 3). And finally, tips of the flaps are at circle’s centers (rule 4).

To make this theory as clear as possible I would like to show you yet another example (look in Figure 4).

Again, the main idea is to pack a needed number of circles on a square piece of paper. From Figure 5 it is clearly visible that all the basic rules were obeyed. The number of circles corresponds to the number of flaps. Circles are partly out of the paper, but their centres are definitely on the paper. Circles sizes are proportional to the flap’s length and finally, all flap’s tips are located at circles centres.

## River

I hope a have managed to explain the importance of circles in origami design. Nevertheless, before I finish this blog post I would like to show you something else. You see, origami models hardly ever, not to say never, consist exclusively of flaps. In most cases, the model will include something we call a “river”. A “river” is part of a model, but it is not a flap.

To make this more understandable let me show you a “river” on our last model (look in Figure 6).

As you can see, a river is not a flap and as such, please remember this, is not defined by a circle. Look in Figure 6. A hope you can see the river. It is marked in green.