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
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.
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!
I'm going to write
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
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):
Just as in the
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.