Origami Frigatebird v3

My frigatebird's color is fading, so I thought about refolding it with something better than double tissue. While doing so, I got some ideas to update it. Now it has distinction of lower/upper beak, and individual toes. Also used this opportunity to fold the female version.

Design

The previous version looks like this:

The bird has no toes in that version, only stubby flap made by splitting the already tiny flap in the bottom corner. So my first intention was to add toes, by making the bottom corner larger, and somehow split it into 2 feet, each with 4 toes.

We can use the good old point splitting to first split a flap into two for feet, thin them to 22.5° flaps, and do another point split there to get toes.

From top to bottom, left to right:
1) Start with a corner flap
2) Point split the corner into 2 flaps
3) Thin the split flaps to 22.5° flaps
4) Point split one of the split flap (now an edge flap) into 3 flaps

Frigatebird has 4 toes on each foot, where the 4th toe is much shorter than the rest. We don't need to split each foot into 4 smaller flaps like what was done on my condor. Instead, split it into three flaps and use the excess paper in the bottom to make the 4th flap. This way we get three long flaps and a short flap. 

Left: Point split the edge flap into 4 smaller flaps.
Right: Point split the edge flap into 3 smaller flaps, and create the 4th flap from the excess paper.

The result, with extra thinning

But how can we enlarge the bottom flap? Looking closer to the previous version's tail & leg part, the molecule in the bottom corner is actually a fish base, born out of a half bird base flap which is split into two.

The construction of the frigatebird v2 lower half by point splitting a half bird base.
The highlighted region shows a fish base.

To use the flap enlargement plan, we should make the fish base's small flap larger, which means instead of a fish base, we want a bird base. Of course, the proportion with other flaps (wing & tail) must not be altered. Using common 22.5° molecules, I thought about reconfiguring the layout into this:

Wing : tail ratio remains \(1 + \sqrt{2} : 1\)

However this makes the head part strange. We will have really long flaps there. It's a good problem to have though, since having a long flap means we can split into few smaller flaps. So the idea of adding upper/lower beak appeared. We need to do something like this:

It is a natural extension from the previous layout. However, notice that there is a small flap that runs across the pleats, and exits on the top edges. This adds thickness to the already thick wing to body connection. I tried to find another solution.

This one eliminates that small flap and makes the wings longer. However, the pleats to make feather along the bottom edge of the wings are reduced, and we don't need that wing's extra length.

Another solution eliminates the small flap by altering the proportion. The bird base part is smaller, which means the bird's tail and legs will be smaller. This is not acceptable, even though the structure is quite beautiful.

While playing in Oriedita, I found a solution that keeps the small flap around, but it is shifted to the middle of the wing. This way it won't get in the way of the wing and body connection, so the thickness problem is gone.

That small flap in the form of pleat is still unused, so there is a wasted paper. Can we use it for something?

The easiest application is to use it as level shifter. For the wing feather tiny flaps, instead of pivoting on hinge like the previous version, we can let it terminate this small pleat. This reduces the thickness on the wing to body connection by 4 layers. During the fold, we can also shape this as additional feather along the bottom edge of the wing.

I made some test folds and it worked pretty nicely. The upper/lower jaw distinction and toes have good proportion.

At the time of this design process, I read John Montroll's single sheet polyhedra book. So I had an idea, what if the frigatebird's air sac is folded as polyhedron? Something like icosahedron or bipyramid would look rad.

Random polyhedron shape for air sac

I spent some time experimenting with it. Folding polyhedron out of the air sac part is possible, but it uses a lot of paper. The zigzag color change in the air sac sides will have to be sacrificed. At last I decided to give up the polyhedron idea. Maybe that would be exercise for another day.

Lastly, I want to fold this model in pair. Female frigatebird doesn't have air sac, but its chest has white feather. It was easy to modify the air sac part to be a color changed chest.

Fold

I got some black, white, and red unryu from Paper Tree in San Francisco back in 2023, and finally managed to get them used after almost a year. The design efficiency is decent, with wingspan of 0.86 from the paper's side length. It is reduced to around 0.77 with some shaping to bend the body and wings. 

Both the male and female version are made from 45 cm double unryu. Knowing how absorbent unryus are, I furiously spray them with water before treatment. They have to be completely drenched so they take MC well. The resulting double unryu has the following characteristics:

1. Soft: there was no paper crinkling sound when I folded them. They still have decent crease memory and reversal, but the crease won't be sharp.
2. Thick: around the thickness of triple tissue, but easily compressible with MC. Once wet shaping is done and the model dries, it is guaranteed to have a stiff model that holds its shape.
3. Strong: resistant to tears. This model's head area has some stressed points that tore on my test folds with foil, but this duo unryu handled it just fine.

Obligatory initial square papers

Midway collapse

Almost finished collapsing

Wet shaping starts here

The shaping spanned over a week, where every day I wet certain area, clamp them with clothespins, and let it dry overnight. With some improvisation, the air sac was shaped to be polyhedron-like, with some curves on top. Hopefully I was able to express it well.

In the end the legs are kind of silly. It took a great deal of pain to fold them out, but they are barely visible in the final fold. Can't complain though, the actual frigatebird also has its tiny feet barely visible.

I promised to send the folks in Paper Tree of what I folded out of their paper, so I emailed them the picture of this models.

Flight

Making this model's tail forked was quite bothersome. Instead of a flap, we need two. Actually why is it even forked? I knew that birds with aerial agility tend to have forked tail, but why is that? I got hooked and read a comprehensive publication about bird's tail morphology.

To start, we need to know about the concept of lift in flight. Lift is an upward force relative to the entity produced by airfoil. When a bird is gliding, its wings remain outstretched and generate lift which counter its weight. This makes the bird lose altitude slowly, or even gain it if the lift is higher than weight. To turn left, bird tilt their body so that the left wingtip is lowered and right wingtip is raised. This "banked" position causes the generated lift to point slightly to the left.

Left: upright flight
Right: flight banked to the left, where the lift vector has a sideway force indicated by "Lift (x)"

Now that the lift force has horizontal component towards left, the bird will turn left. To turn right, bank to the other direction. If any of you played video games with flying like Acecombat or GTA, I'm sure this feels familiar.

To turn more sharply, the bank angle can be increased. There is a limit, however, as the more banked the position, the smaller the vertical lift is generated. If the vertical lift is much smaller than the bird's weight, it will quickly lose altitude. So how can it turn even sharper without stepping over this limit? By generating more lift!

Tail feather that can be spread into narrow triangle effectively makes the tail acting like additional wings. This extra lift generated by tail helps the bird to make a sharp turn. According to [1], the optimal shape is an isosceles triangle with height of 1 unit and its sloped side has length of 2 unit. In other words, the outermost tail feather should be twice longer than the innermost tail feather. This is how frigatebird's tail feathers are structured.

When such tail feathers are not fully spread, they form a forked shape. So this is why forked tail is associated with birds with aerial agility. Additionally, birds can independently adjust how their tail is banked, so they can precisely control the amount of lift produced.

From left to right: the optimal tail shape when fully, moderately, and lightly spread.

Why do frigatebirds need aerial agility anyway? It's because like to bully other seabirds and steal their food mid air. Being proficient in flight helps them survive. Check out this amazing shot of frigatebird dogfight. You can see how their tail feathers are used during tight turns, slow flight, or quickly gaining altitude. Being able to move like that in the air despite having large body is incredible. 

Well, that was interesting. Anyway I got deeper into the rabbit hole the moment I learned that frigatebirds are not dynamic soarer. They are thermal soarer, which also includes vultures and condors! So why does their wing shape differ so much?

According to [2], there is a hypothesis that soarer like vultures are optimized to generate lift even at low speed, while soarer like frigatebirds are optimized to travel horizontally as far as possible for each altitude lost. However the conclusion is unclear, so more research is required. Perhaps it has something to do with their habitat, take off/landing requirement, feeding habit, or flight speed. Could also be that there are more than one optimal wing shape for thermal soarer.

[1] A. Thomas. "On the Tails of Birds," in BioScience, vol. 47, no. 4, pp. 215-225, 1997.

[2] Rader, J., et al. "Functional Morphology of Gliding Flight II. Morphology Follows Predictions of Gliding Performance," in Integrative and Comparative Biology, vol. 60, no. 5, pp. 1297-1308, 2020.