Origami Leek

Made for the PCOC 2025 design challenge, whose theme is "Farm to Fork", because apparently Sacramento is proud of its cuisine from local farms.

Leeks are member of allium family, together with onions, scallions, chives, garlics, and shallots. They are big and tasty. Easily recognizable by their long and white edible bottom half and leafy dark-green top whose base form an overlapping pattern. The outer top parts are usually too fibrous to be eaten, but they can be simmered in soup for flavoring.

Design

Let's gather the characteristics of a leek:

1. Base of the green part forms an overlapping pattern. This is the most important characteristic so the leek doesn't look like broom.
2. The green part sprout out alternately, slowly becoming shorter while their top reach the same height (this happened because of the way they are harvested).
3. The leaves should fan out in a single axis when seen from the top.
4. The bottom half is cylinder.

Leek with the overlapping pattern highlighted

Leek from top, the leaves fan out in a single axis, alternating between left and right.

The naive design would be to somehow have a rectangular paper with color changed bottom half, and slits on top half that progressively get shorter. This rectangle can be rolled, and the slits can be crimped down to form the green part.

Sounds straightforward but making slits are expensive. Basically each part between slits is a flap and with this structure, we'll need a long rectangle instead of square.

The next idea is to use tree-based design. Imagine a skinny rectangular strip that is crimped to form a simple overlapping pattern. The basic building block is as follows:

If the starred flap at step 4 was longer, then we could repeat step 2 to 4 on it:

Slowly the overlapping pattern emerges. This can be repeated on the starred flap to have more individual leaves. Every repetition with reduced 1 unit length will cause the top parts to reach the same level.

The leaves fanned out too widely. They can be easily made narrower by making the crimp angle smaller.

This structure works in theory but eats paper like crazy. Also there will be unholy thickness for each leaf. But why does it have to be a single strip? Can we achieve the same result when the part that is folded to itself is represented as a branched flap? Turns out the answer is yes. Upon closer look we can reduce the tree structure formed by the strip into a classic, efficient tree structure.

Made from hastily packed tree

I guess I was too preoccupied with the overlapping pattern that a simple approach like this was missed.

Anyway this strange tree just needs adjustment on the river length to make a recognizable leek. Each river should be half unit length, the leaf on the left follows 1, 2, 3, ... arithmetic progression, and the leaf on the right follows 1.5, 2.5, 3.5, .... This rule allows the length of the river to be equal to the width of the leaf, forming the overlapping pattern.

That tree forms an interesting pattern when packed. It gave me idea of using recursive structure like what Brian Chan did on his romanesco. The hope was recursive structure allows scalability to arbitrary amount of leaves, giving freedom to the folder.

First I came up with this scalable layout that starts from corner. Each flap is later made fatter to simplify the crease pattern. This has terrible efficiency and the leaves progressively get thicker.

It doesn't have to be that inefficient. The basic packing building block looks like below. 

Start with a single leaf, then progressively add leaf on the left and right with river and increased leaf size.

This linear structure can be bent so it efficiently fits in a square.

See highlighted square for reference.

There is a hole on the left side of highlighted square. We can fill it with extra leaf. What's nice about this structure is all the leaves are efficiently packed without extra fattening, while leaving precisely square space on the lower right corner. This space (marked dark blue) can be used for leek's bottom half. But can we bend it further and just use all edges for the green top? Yes, but the crease pattern will not be easy to fold.

Folding up the flap in half for color change is not efficient, but I went with it because it yields a clean looking color change.

I settled with 6 leaves that fit in a boxpleated grid of 10. There are partials and transition to figure out but I know how to solve it as it is similar to Kamiya's dragonfly. Finally I added 1 unit of graft along the border so each leaf is rectangular instead of the usual pointy boxpleated flap. This made the crease pattern to be based on grid of 12 as shown on the final crease pattern.

Final test fold

Fold

I thought of making a life sized leek. This model's efficiency is about 46%. I'll need an 85 cm square to create 40 cm leek. That size seems overkill and wasteful so I opted to just create small leek. I still want it to look big so people don't confuse it with puny herbs or spices. The idea is to fold several of them, and present them in a cardboard box just like how they are shipped to grocery stores.

As usual I prefer to use standard origami paper for simple model. However this one is special because it needs the layers to be tightly packed. This is better done on absorbent paper that takes MC well. Fortunately I had a lot of wenzhou and green tissue scraps, left from cutting a larger square.

Folded base

The cardboard box was modeled after a random produce cardboard from Costco. It is scaled down, and I cut a thick kraft paper to match its net. What's cool is the net is just a rectangle, with slits and holes here and there, then folded entirely with mountain folds (or valley folds, if seen from other side). Most likely so that they can be easily mass produced; where all the cutting and folding are made by machine.