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Origami Challenge #13: Beyond Halves/Quarters

First and foremost, I’m not a high level mathematician. If I don’t need to be one to design origami models, neither do you. If I took the time to burrow down that rabbit hole, I’m certain it would open up a whole new world of design possibilities for me. Perhaps one day I will, and when I do, I’ll share what I’ve learned. But that day is not today. For today, I’ll work within the confines of what I know. 

If you work with the traditional bases, you’ll notice that there’s a lot of folding in half (as in a Book Fold) and bisecting angles (as in a Kite Fold). However, there are many ways to divide an area. Halves and quarters are always tempting because the reference points are obvious. Just bring the edge to the opposite edge or a crease. It’s more likely to be memorable and precise. You have to do more work to divide an area into threes, fives, sixes, etc, either eyeballing the proportions (which can be imprecise) or doing a few measurement steps beforehand to find the right reference point (which makes the sequence longer). But if you’re looking to expand your origami palette, it’s worthwhile to incorporate new divisions. 

Incorporating divisions beyond halves and quarters allows you to find an infinite array of new structures. Imagine working from a Kite Base and finding it too wide for what you need, while Kite Folding it again would make the model too thin. If you work with a 30 degree angled corner rather than the 45 degree angle of a Kite Base or 22.5 degree angle of a thinned Kite Base, it might hit the spot. Folding the paper into thirds (or another uncommon division) also arranges the rest of the paper in a new way, so you’ll have new options. This Elephant starts from a Kite Base-like shape that uses a 30 degree angled corner rather than the 45 degree angle traditional Kite Base. That structure gave me additional flaps that I could turn into legs and a longer, thinner point that I could manipulate into the ears, head, and trunk.

You can also play with unconventional grids to find new structures and references. Origami artists who use box pleating technique for designing basically create a grid to measure and work from. 4X4, 8X8, 16X16, 32X32 and so on grids are popular for box pleating because they’re easy to measure, and you can find uncommon reference points within those grids. Making a 3X3, 5X5, 6X6, etc. grid opens up even more possibilities. This Aardvark has major reference points from a 6X6 grid, which led me to a structure I would never have otherwise found. 

One way to find new structures for bases is to adapt traditional bases to new measurements. Think of a Square Base. The closed tip of the Square Base is the center of the original square. What if you created a structure where the closed tip was off center? Create a 3X3 grid, use one of the corners of the center square as the closed tip point, and collapse to see what you get. You’re still working with squares, so you can pull from your normal toolbox to adjust and shape them, but the composition, alignment, and size of those squares are different. Doing this with a 3X3 grid in particular is nifty because the resulting structure looks like a 2X2 grid with more flaps to play with. That’s what I’ve done with this Camel. After I created the 3X3 grid Square Base variant, the rest of the sequence involved mostly 45 degree and 22.5 degree folds. But I wouldn’t have been able to get the proportions and structure for the Camel without using the 3X3 grid. 

You can also use these kinds of references to practice spatial awareness. I’ve improved my ability to eyeball thirds immensely simply by incorporating thirds into the shaping portion of my models. When the measurement is needed to create the base of the model, I take the time to do the extra steps that will measure it out. But for shaping shapes, where the outcome isn’t integral to the overall structure, eyeballing can be great practice. It also helps connect the intuitive and mathematical brains by identifying your intuitive shaping choices. This Helmet Base Rabbit variant (similar to the Yoshizawa one from the last challenge, but with determined references for teaching purposes) and the ears of my Simple Skunk are eyeballed thirds (as I mention in the Challenge 11 post). 

There are many resources for ways to measure out different angles and grids. John Montroll’s book Origami and Math gives ways to do this and talks about using different measurements to adjust bases for specifically proportioned models. There are also many places online to learn how to divide paper into different angles and grids. In addition to the models I found through this design, I thought I’d share a couple examples I like from other artists that use references beyond halves and quarters:

  • Penguin – Jun Maekawa, Genuine Origami
  • Orca – John Montroll, Origami Fold-by-Fold
  • Kangaroo – Jo Nakashima, PDF available online
  • Rhinoceros – John Montroll, Origami and Math

Eventually, you may be able to calculate the measurements you need to get the features and proportions for a specific subject. Let me know when you get there!

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