Circle Flap Concept

In origami, models come from bases, and bases are made up of flaps. A root concept of designing is that all flaps on a base are just circles. This information may seem pretty confusing, so let's look at a simple example.

In the image shown above, the square represents a sheet of paper, the second image represents a base, and the third image represents the finished model. Notice how the base already contains all of the main flaps required for the finished model.

The numbered flap tips correspond to the flap tips in the finished model. Now, if we unfold the entire paper, it will produce the crease pattern. Notice how the flap tips on the paper are all made from circles.

After the paper is unfolded, we can clearly see that the five flaps used for the crane are just actually circles. Box Pleating follows a simple idea, however with the use of a grid.

Rules for Packing

When packing circles, it is important to remember the two main rules that apply to them. 1, circles cannot overlap each other. 2, circles can be packed anywhere on the paper as long as the center is touching the paper or touching the edge or corner of the paper. 3, the circle size must not be larger than the paper size. Let's see a few examples.

In the first image from the left, all the circles follow the three rules. Although the corner circle is not completely on the paper, its center is still touching the corner, which makes it a valid circle. None of the other circles are larger than the paper, and they are all touching the paper in some way. In the second example, it is not valid. The two circles' centers are touching the paper, but they overlap. In the third example, the two circles in the corner are still touching the paper, but their centers are off the paper, making it invalid.

In the future, you will learn about other things to take up space on a sheet of paper, such as rivers and level shifters, however, for now, just know that the entire paper must be completely filled with flaps (circles), or else it cannot be turned into a complete crease pattern.

Taking Up the Whole Paper

You may be confused on what it means for the flaps (circles) to take up the entire paper. However, there is an important detail that I left out that you should know. In box pleating, all circles should be packed as usual, but afater packing, each circle should be surrounded by a square just large enough to fit perfectly on the circle. Ideally, none of the squares should overlap, but if they do, it will require a pythagorean stretch. However, I won't get into pythagorean stretches yet, as they are tedious to implement, so just try not to have overlapping squares. All this information may sound like a lot, so let's see a few examples.

In the image above, you can see that there are many different ways you can pack circles with their squares into the paper. However, you will see that if two of the squares overlap, it will create a squiggly zig-zag shape in the overlapping segment, which is a stretch. Another concept you should understand is that larger flaps generally should be placed at the corners or edges of a paper. Although this may not work in all situations, it is highly recommended, since the circles and squares, or flaps take up less space when they are placed on the edge or corner of a paper.