This article presents an origami tessellation technique that I developed to depict any pixelated design. It uses a grid of pixels that can be independently “sunk” or “unsunk,” corresponding to binary “off” and “on” states, or black and white pixels.
Design and development
The tessellation is made up of a system of barriers, where the junctions are created using waterbomb bases, forming a grid. This technique of using barriers in a model was also recently employed by Janet Hamilton in her Altoids-Tin Dividers.
My design began with making a large grid out of these barriers, but I was curious to see if there was a way to invert the squares formed. I found that each unit was indeed independently invertible, much like the individual pixels in a black-and-white digital image. This inspired the name “Pixel Paper Tessellation.”
There have previously been multiple approaches to this style of 2D representational origami, such as Jo Nakashima’s Origami Alphabet and Jason Ku’s typeface. My work shares similarities with the Origami Maze Font that Erik Demaine, Martin Demaine and Jason Ku developed at the Massachusetts Institute of Technology. The group used barriers in an algorithm that encodes characters or any orthogonal graph/maze as traced-out features running across a flat square grid.
While previous 2D styles have required separate design processes for each combination of characters, the Pixel Paper Tessellation uses a simple universal base. The tessellation is a breakthrough in representational origami because with it, any image can be simply encoded with each unit square able to be independently raised or lowered, corresponding to binary on/off states. One possible application is of course in depicting letters.
Variations
Since the tessellation boils down to just a grid of barriers, different sizes of pixels can be created by adjusting their placement. For example, a tessellation with pixels of size 2-by-2 can be created by increasing the spacing between barriers. Models of larger pixel sizes are slightly easier to fold, but 1-by-1 unit pixels have a more efficient design, enabling more pixels to be created using the same area of paper.
The pattern for each unit remains the same — a general rule is that a single a-by-b pixel is folded from an (a+2) by (b+2) square grid, so there is a scaling factor of approximately \( \frac{ab}{(a+2)(b+2)} \). In my model, I spelled out the words “pixel” and “paper” from a 64-by-40 grid of squares folded into a 22-by-14 grid of pixels. The grid I used is 64-by-40 rather than 66-by-42 because the border units use only two units instead of three. I discuss this later in the section “Adjustments.”
Inverting Pixels
Recall that the tessellation is made up of a lattice of waterbomb bases. The display side of the model is the one where the waterbomb folds are not visible, and the hidden side is the one where they are.
In this model, images are formed by the distance between the planes of the raised surfaces of inverted pixels and the lower planes of the surfaces that remain recessed. Full images are created by inverting all the desired pixels and then resolving the permutation of the waterbomb bases on the hidden side of the model.
To push a pixel that is raised on the hidden side so that it is instead raised on the display side, unfold the chosen pixel and surrounding waterbomb bases, then refold the model into the correct position. The structure of an inverted pixel differs from a noninverted one in the permutation of its waterbomb folds. For a single inverted pixel, one of the flaps of each of the waterbomb bases is reflected into the plane of the hidden side.
There are two main ways that pixels can meet: diagonally or horizontally. At meeting points, the permutation of waterbomb folds is resolved by rotating the flaps of the waterbombs and repositioning them so that they are not too concentrated in any single pixel. There is no single “correct” permutation for any design because the displayed side is independent of how the waterbomb bases are organized.
Folding Tips
As in many tessellations, one of the most time-consuming steps in this model is precreasing. Sharp precreasing is particularly important for this tessellation because it allows the waterbomb folds and inverted pixels to snap into place much more easily. Fortunately, much time can be saved because all the diagonal and horizontal folds align nicely. For collapsing the model, I find the following folding sequence to be easiest and most efficient:
Step 1: Begin folding row by row starting with the top row:
Step 2: Put in prefolds for the next row of pixels:
Step 3: Flip the paper over and fold waterbomb bases on each barrier junction:
Step 4: Repeat to complete the row, then repeat for each column, arriving at a result like this:
For inverting a row of pixels all at once, I find that the most efficient and precise way is to unfold the creases around the whole set of pixels that need to be inverted and then resolve each of the waterbomb permutations one by one.
Adjustments
As is visible in step 4 of the folding sequence, the difference of layers between the waterbomb folds and pixels in each row makes the tessellation resistant to lying flat. This can be resolved by adjusting the placement of the waterbomb bases into a permutation that allows the paper’s tendency to expand to push it into a more compact state rather than unfolding it in one direction.
The step of experimenting with waterbomb-base permutations is best performed after all the pixels needed to capture the image are inverted because having the waterbomb folds lined up in one direction initially also allows for easier inversion. Also, folding the crease pattern below will yield a few protruding waterbomb folds on the border in the areas highlighted in gray.
These can be hidden by being repositioned to point inward, toward the center of the tessellation, rather than outward, sticking out of the tessellation. However, the easiest way to deal with these extra flaps is to not have them at all, which can be achieved by removing the border of the design entirely. That is the approach I used for my Pixel Paper model. The 2-unit-wide outer pixels that result (also discussed in the section “Variations”) can be seen in the crease patterns below.
Paper Recommendation
Thicker papers with higher grams per square meter (GSM), like the Stardream paper (120 GSM) used in the display models, are more susceptible to bowing, and it is also more difficult to invert pixels. Because of this, paper with lower GSM, like double tissue or tissue foil (~40 GSM), is recommended.
Final Steps and Shaping
Once all the desired pixels are inverted, the tessellation will likely no longer be in a compact state.
There are a few remedies for this. One possible solution is of course glue, but a more traditional method is folding the waterbomb base flaps to lock them in place, which can yield a very nice result.
Only small bends or creases are needed, but for models with pixels of a very small size, the flaps may be difficult to manage, even with tweezers.
A final method is finding a container to compress the model into. It can be difficult to find a box with the correct dimensions, but this is an opportunity to revisit an idea that I mentioned earlier.
Recall that the tessellation is made up of a system of barriers. We can use the same technique to create a container of easily customizable size.
Once the model is placed inside, leaving it compressed for a few days will help the creases to set. The model can then be removed with sharper definition, or it can be left inside for display.
Finally, much more definition can be achieved by placing the raised surfaces in light and the sunken pixels in shadow.
References
Demaine, Erik D., Martin L. Demaine and Jason Ku, “Folding Any Orthogonal Maze.” In Origami5: Proceedings of the 5th International Conference on Origami in Science, Mathematics and Education (OSME 2010), 449–454. New York: A K Peters, 2011. The conference was held in Singapore on July 13-17, 2010.
Demaine, Erik D., Martin L. Demaine and Jason Ku, “Origami Maze Puzzle Font.” In Exchange Book of the 9th Gathering for Gardner (G4G9). Retrieved from https://erikdemaine.org/papers/MazeAlphabet_G4G9/paper.pdf. The conference was held in Atlanta on March 24-28, 2010.
Thanks to Kam Eng and Val Landwehr for their editorial help.