How I Precrease Paper for Tessellations

The image in this post shows a precreased sheet of paper, which upon collapsing folds to Triangles Tessellation. Usually, models like this are folded starting from a square grid (it would be 16×16 in this case) which fills the whole sheet, but I prefer to fold my tessellations in a neater way which leaves as few unnecessary creases visible in the final model as possible (preferably none). This requires extra effort, both for designing the clean pre-crease pattern and for rendering the model in actual paper, but I think it is worth it.

My design process usually follows these steps:

• Designing a molecule and folding a draft version from a small sheet with a complete grid;
• Unfolding the molecule (often, I fold a second copy for this purpose) and tracing an unoriented CP (i.e. without mountain-valley distinction);
• Marking the flat areas of paper visible in the finished model so that I can easily avoid creating creases through them;
• Trying to construct the creases therein using simple origami constructions such as dividing a segment in half or into thirds, bisecting angles, etc. This is a multi-step process, in which results of previous steps (already constructed creases) create reference points for subsequent steps.
• Adding new creases (as far as possible without going through the areas visible in the finished model) if I need them as new reference points or if they make precreasing simpler (for example, it is easier to fold a single continuous segment than two co-linear segments with a break in between).

Once the pre-crease pattern design is ready, I start working on a new sheet, and pre-crease all creases by hand by using a teflon bone folder. Sometimes, during this phase I spot mistakes in the design and refine the pre-crease pattern. After collapsing the model, no creases (or very few) are visible, and I end up with a neatly folded pattern, all folded from a single sheet of paper without any cuts or glue.

I gave a presentation about this topic at CfC2 in 2020.

Many pre-creased sheets are beautiful on their own and since pre-creasing is usually the most time-consuming part of folding my tessellations, sometimes I even regret folding those sheets rather than keeping them in unfolded state. However, I often take pictures of the sheets before collapsing, and you can find models with such pictures in the pre-creased sheet album.