Pinwheel without cuts, 2016

I was deeply honored to receive an invitation to teach at the 4th Origami World Marathon (OWM4). Among the various origami models I proposed, the one they specifically requested me to teach is the one I'm about to share with you. I initially had some concerns because this particular model is rather intricate and time-consuming to fold. However, I was pleasantly surprised when they allocated a generous 2-hour session for me to teach this model. In a recent interview leading up to the event, Ilan even made a special mention of this particular design. I guess he must have liked it a lot ?.

The inspiration for this particular origami model came in a rather amusing manner. As many of us are familiar with the clichéd scenario where a specialist is humorously asked to perform stereotypical tasks related to their field by someone less knowledgeable, my experience echoed this theme. It all began when a layperson, upon learning of my origami expertise, inquired whether I could make a traditional pinwheel.

To be perfectly candid, I felt a twinge of offense at the question, as a pinwheel doesn't involve origami in the true sense. It doesn't require folding; instead, it involves cutting and bending the paper, which is fundamentally different from the art of origami. However, after I explained this distinction, a creative spark ignited within me. I wondered, why not challenge myself to design a pinwheel using only folding techniques, entirely devoid of cutting or the use of glue?

Fortuitously, I had already formulated the foundational concepts of my edge-river method (ERM), which adeptly addresses the challenges posed by such non-uniaxial designs. The crease pattern and the ERM map for this particular model are illustrated in the following figure.

In this map, the four lakes serve as the surfaces for the pinwheel. The gaps between each pair of lakes organically give rise to the river structures and the peninsulas, resulting in a spacious central area on the sheet that can be utilized to craft the spinning axle. Additionally, the entire layout is strategically tilted to facilitate the creation of locking components at the four corners, while strategically ensuring a straightforward and manageable initial folding sequence.

A commonly asked question regarding the initial folding sequence revolves around the significance of the 21/41 reference point. Let me clarify this intriguing detail. The core of the pinwheel structure necessitates an octagon with dimensions specified by the blue lines in the figure above. As previously mentioned, this octagon must be tilted to accommodate the locking flaps. While it's possible to adjust the tilt in a way that it occupies a larger area within the square (at the cost of shorter locking flaps), doing so would require locating all eight edges of the octagon, which would be rather cumbersome.

Instead, I opt to tilt the octagon in such a manner that its eight edges align precisely with the angle bisectors of the red triangle. By doing this, in conjunction with the tilted blintz fold, I only need to locate two reference points. It just so happens that the red triangle that fulfills these criteria corresponds to the 20,21,29 Pythagorean triangle, hence the magical ratio of 21/41.

However, one question lingers: why did I opt for this specific octagon rather than a regular one? The answer lies in the delicate balance of the spinning axle's width. My experimentation revealed that subdividing the center region into a 16\times16 grid made it too thick, while an 8\times8 grid wasn't narrow enough. As a compromise, I settled on a 12\times12 grid. Because of this choice, I decided that I should choose an octagon of integral proportion, so that the precreasing can be done simply by using the edges as references.

The entire folding process typically takes around an hour and a half, making it a somewhat tight timeframe to cover everything within the allocated two-hour session. Fortunately, we managed to achieve just that. I'm genuinely delighted to witness how this design of mine, which has endured over the years, has now found appreciation and enjoyment among others.