Origami Lamprey

Lampreys are fish with primitive features. Together with hagfish, they are the last surviving jawless vertebrate. These creatures don't exist in my home country, so I've only known their existence from encyclopedia back in elementary school. Their somewhat scary circular mouth is full of teeth and will surely etch into memory. 

Design

After the design challenges on PCOC 2025 2nd night, my brain was charged with origami ideas. One being the sperm whale, and the other was this lamprey. The motivation also came from seeing hexagonal tessellation. I thought about incorporating that in figurative origami, and lamprey with radially arranged teeth in its circular mouth came to mind.

Besides their mouth, let's gather the other features of lamprey:

1. There are only 3 fins located in its rear part: 2 dorsal and 1 caudal.
2. On each side, there are 7 gill holes located in a row behind their eye, slanted towards bottom as it reaches the last gill hole. They make lampreys look like having 8 eyes.

Upon further thought I ended up not using tessellation. We could create row of pleats, form teeth, and roll them.

Three rows of pleats, with central flap reconfiguration.

Teeth made from swiveling the pleats.

Rolled to make lamprey's mouth.

Crease pattern for the 3 rows of pleats.

There are several ways of producing pleats. I started with standard boxpleating that is akin to Kamiya's Ryujin scale:

It produced pleats but eats a lot of paper. The other solution is to distribute a one unit pleat into numerous uniform pleats separated by rivers. I've explained them before (original and construction) so this post won't go into the details. As my understanding of this technique has increased, it can be upgraded with larger squash to make it slightly more space efficient.

Left: old
Right: more compact version, occupying less vertical space.

This ginkgo-leaf-like structure can be put in pair, connected with long pleats to form its body. This long pleat can terminate with simple forked structure to make the fins. We'll do that later.

Now I need to make a row of 8 color changed spots on each side: 1 for eye, and 7 for gill holes. The answer is obviously to use the paper edge, somehow pleated to expose tiny flaps that can be color changed. I tried to use standard pleat but it ended up producing flaps that are spaced too far away. 

This feels like deja vu of ibex moment. Should I use teardrop structure like what was used on ibex? I experimented on actual paper and can't get it to work. There was not enough space. So how about blintzing it? It indeed provides a lot of paper to work with, but it came with the cost of doubling the thickness of teeth part. That part was a corner flap and turned into edge flap. I rejected this idea.

The design stopped for months. One day I was listening to a boring part of an audiobook while staring at my test fold. An idea came to use standard 22.5 angled pleats, and pleat it with 11.25 to make them parallel with the body.

This alignment solved the problem of producing gill holes. The first pleat can be deliberately made longer to produce longer flap, which produces larger color change and can be made into eye. 

There was a problem with where the row of 8 tiny flaps start. They need to be slightly pushed to the back or the eyes would be located immediately behind the mouth. I thought of some solutions:

1. Move the teeth part towards the top. This reduces the paper for the teeth.
2. Start the angled pleats slightly to the back. This wastes paper.
3. Use irregular angle. This is ugly to draw.

Among those, the 3rd solution has the least drawback. So it's chosen. There was a pleasant surprise that the gill holes are now slanted towards bottom, which is how real lamprey's gill holes are arranged.

Aligning the flaps with the target drawing.

With this color change done, I realized that the teeth are not color changed. So if the model was folded with duo color paper, the teeth would use the colored side. I can think of 2 ways to color change the teeth:

Brown line shows colored side, black line shows underside color.
A) The current state, with colored teeth.
B) and C) are solutions with color changed teeth.

Currently the pleats become narrower as it is closer to the tip. Option B doesn't fit this layout because the outermost teeth row has to be made with the pleats near the tip -- there won't be enough pleats to produce many teeth. Option C doesn't have this issue, but eats more paper. Moreover when I imagined this cross section to be rotated to form full tube, there will be self intersection as the top layer has to travel down, and at the same time the bottom layer has to go up. In reality it's not that complicated, we could use rabbit-ear like maneuver and let the pleats form loose tube.

Rabbit ear through the pleats, keeping the pleated part loose and dented at the bottom.

The top view. Arrows show the curve to make as the model is rolled.

Roll completed, with teeth pleats are tightly placed inside the roll.

Next I started to work on the rear part of the test fold. With some improvisation and standard 22.5 squashes, the 2 dorsal fins and caudal fin are produced.

I compared this with the reference image and realized they are located too far to the back and the fins are too short. Actually it was an easy fix, we could change the forked structure with the 11.25 version instead of 22.5 version, producing skinnier 1st dorsal fin that is pulled to the front.

Left: 22.5 version.
Right: 11.25 version.

The fix on the 2nd dorsal fin was also easy. I played around with the test fold and ended up with different structure:

The design was basically done. All that's left was to practice folding the mouth part before doing final fold. I also used this opportunity to sacrifice some mathematical accuracy for ease of folding. For example, the gill pleats' width are actually not uniform. The first 2 are slightly larger than the rest, all because I want the pleat starting point line up with 22.5 reference.

With uniform pleat width, there will be ugly misalignment.

Altered pleat width to line up the reference and simplify folding.

Also thanks to this practice fold, I realized that the middle row's teeth has a gap. This is because the pleats are not facing the same direction; they are mirrored. So the symmetry line that has the pleats facing away from each other produced gap. The fix is to arrange the pleats to face the same direction. 

Top: the symmetric part produces wide gap, interrupting the teeth production.
Bottom: Fully aligned pleat direction without gap.

It is not hard to solve, but a shame that the structure is no longer symmetric. Basically the right ginkgo-leaf-like structure has to use the sunken version of the left one.

Fold

This model turned out to be highly efficient, with efficiency of around 1.0. My test folds used 30 cm paper which produces nice little model. Unfortunately the teeth pleats are around 2.2 mm, too small to absorb the thickness as the teeth are technically middle flap. Going with this size carry the risk of drowning the teeth in the mess of thick layers. If I go big with 50 cm paper, the finished model will be 50 cm long, kind of too big to transport and display. So this high efficiency is both blessing and curse.

In the end I settled with 50 cm paper. I have brought most of my fancy papers back to home in Indonesia, so what's left are just tissues and Wenzhou, which are used for this model. I posed the model like letter 'S' to allow people to simultaneously see the front and rear part; and to keep the model footprint relatively small.

Folding this is surprisingly not difficult. Every test fold can be completed within one sitting.

Obligatory starting square.

Precreasing fin pleats, before they became hard to access once collapsed.

Begin transitioning the pleat into 8ths.

More in progress.

Three rows of teeth to be made out of these pleats.

Finished 3 rows of teeth.

The teeth are seamless when seen from outer side.

Rolled teeth.

Remarks

I am not satisfied with how the face part looks. The eyes might need proper color changed sclera and pupil. Also the second and third row of teeth can't fully cover the circular mouth. I might have to optimize the structure to surface more paper for the teeth. Those would be exercise for another day.