Origami fractal waterbomb base corrugation

Designed and folded by Helena Verrill, March 2022.

Story and Crease Patterns

The above photo shows four levels of iteration folded of a fractal construction. This is folded from a single sheet of paper. This is a stretched version of a flat folding origami. The flat fold origami appeared in Figure 1 in my 2021 Bridges article. However, although a photo of the folded origami appeared in that paper, the crease pattern did not, because that paper (almost) only included details of patterns which are continuous variations of the waterbomb base fold (such as here), not fractal versions.
As in my other pages on the waterbomb base tessellation, following Ron Resch's original design, we start off with the waterbomb unit, generally put together as here.

Square units

My first unit in this fractal sequence is the following, where the pictures show the unit (left) and tessellation (right) the crease pattern formed by putting lots together. The crease lines are red for mountain, blue for valley. The boundary of the unit is either a valley fold or unfolded. I first fold a square grid and use this as reference for the other crease lines. The units are coloured in random colours to distinguish them from each other, so it's easier to see how they fit together.
Only four levels of the iteration are shown, since otherwise it gets slow to create both for the computer and for the person folding. The pattern is finished by pleats, rather than folding to infinity.

Here are some images of the folded version, either folded completely flat, except for pleated region, or somewhat opened. The second image in the second row is a model with less itterations, but the flat fold works better, since the more pleats are folded, the less accurate they get and the harder to lie on top of each other acurately. These are all folded entirely by hand, and the creases are not perfectly accurately folded.

A4 Waterbomb

Notice that when the unit is completely flat you can't really see what's going on properly, and when it's opened out a little, it doesn't really lie completely flat.
Contrast this with the waterbomb base unit, when stretched to have A4 paper side ratio, as the picture here appears on page load. When flat, we get a curved origami. However, whereas the tradition waterbomb base tessellation from square units is curved when opened out slightly, and makes a really nice curved lamp cover, for the A4 ratio unit, we get a very nice plane configuration, as in the following picture:

Here is a video version. It's fun to play with. Best option is to make it yourself!

Why A4?

I'm going to write A4, but this can be changed to any A series paper.
The A4 side ratio, √2 : 1, means that the diagonal of a sheet of A4 has length the diagonal of a cube side length 1, as in the first picture, which is just a folded along the diagonal sheet of A5, placed on a square.
The folded in half on diagonal is just a quarter of the waterbomb unit. So, several of the A4 stretch waterbomb units together fit together with parts of their sides flush against each other, like parts of little cubes lined up, as in the second and third pictures.

So, now we just apply the A4 stretch to the initial tessellation, to get the following crease pattern, with photo of folded origami at the top of the page. Actually, I had to use a sheet about twice as long as wide, and sloped down one edge. Also my folded version is mirrored left to right compared to this. Also the pleated region needs to extend to the right significantly.

Unlike the A4 stretch waterbomb base pattern, or the original fractal tessellation, this pattern is no longer a flat origami, because alternate angles no longer add to 180 degrees at vertices.

More A4 variations

The A4 ratio waterbomb fold seems worth investigating further. Here is another simple example, where the units are moved and the fold direction changed:

Non fractal multisize variations

We can also make variations on the fractal design to be less fractal, just alternating between different size units; many variations are possible, e.g., the following, given on square paper, but would probably look better on A4 series paper, and with more units:

More fractal variations

Different size ratio of iteration

Exactly the same trick used to decrease the unit size by half on each fractal iteration can be used to decrease by a third or a quarter, or other factors, as for example the following (also given in square version, you could experiment with stretching to A4 size):

LamiOri variation

Just as in the LamiOri method used to change waterbomb base tesselations, as applied in e.g., here, we can apply this to the first tessellation above to get the following, and similar patterns. Again, I'm just showing the square version crease pattern, but you can stretch to A4 size. In this example, the units are slid together by the maximal amount, but they could be moved further apart, or shifted not quite so close together. Note that when units are shifted together, if a mountain crease and valley crease end up on top of each other, they cancel out to give no crease line. Also the creases at the top of the pleated region need to be adjusted slightly, so the given crease pattern is not completely correct at this junction. The crease pattern is the mirror image compared with the photo.

Other fractal variations

There are many other fractal waterbomb tessellations. For example, here is another method. Just one fold here, but I've draw pictures from several views, and with extra gridlines folded or not. Adding the extra creases converts a mountain peak into a sequence of concentric squares. I have left out these additional lines from the crease pattern. I've also added a row of waterbomb units at the bottom, rather than adding rows of smaller and smaller units. Also in the folded version, the largest row, I have added extra creases for another waterbomb peak in the middle of the unit.