The standard rules of origami dictate that you fold models from a single square sheet of paper, with no cuts. I’m going to start off this blog by breaking two of those rules. Recently, I’ve been watching the lectures from Erik Demaine’s MIT course on folding algorithms (which are great, by the way). Lecture 3 is about single vertex flat foldability, in which vertices are generally drawn at the center of circles, and I think it was while watching this that I started wondering what you could fold from circular paper. Some internet searching didn’t reveal much. There are things you can do with circular creases on curved paper, but any searches for modular origami models just gave models that are vaguely circular, but made from square sheets. So I decided to come up with something myself. (Luckily, I didn’t come across this rather discouraging post until later.) A while ago I discovered a few books by Meenakshi Mukerji at my public library, and I folded the QRSTUVWXYZ Stars model. It’s 10 intersecting 9-pointed stars, and mine looked like this: I decided to try to do something similar, but to take advantage of the circular paper to make intersecting circles. Here’s what I came up with.

Note: Paper

After cutting out a few circles that I used to come up with the units, I decided that cutting them all out myself was going to be pretty annoying. (I even tried putting an X-Acto knife in my compass, but that didn’t work as well as I hoped.) Turns out there aren’t a lot of options for buying circular paper online, but I got this pack from Amazon, and it’s been working pretty well. It’s colored on both sides, has lots of colors and folds nicely. It doesn’t have the same shiny finish as most origami paper, but otherwise it’s great.

I ended up doing seven intersecting circles, because the angles work out nicely when each unit is ⅙ of the circle. The crease pattern is pretty simple, just radial creases making alternating 15° and 30° angles. If that diagram makes you a little dizzy, here’s one with colors instead. This gets folded so that the edge of the paper makes this shape (not to scale): Then the units slide together like this: To keep them together, I folded down the corners of the pockets once the units were attached to each side. (This GIF shows only two units going together, but you’d really fold these corners down after adding another brown unit to the other side of the pink one.) The assembly is the same as in Mukerji’s models, and there are some pretty handy tables and diagrams here. (This model is analogous to the TUVWXYZ.) And when it’s done, you get this! This isn’t folded and assembled as cleanly as I’d like, but I’m pretty happy with it as a first attempt! Now I’m working on a version with nine circles.

Edit: Here’s a version with nine circles. There are diagrams here.

Note: Planarity

So it turns out these intersecting circles/stars/whatever aren’t actually planar. If you look at where two circles intersect, they meet at their centers, but not along diameters. This is especially clear in the photo below; see the red and brown circles. I think the variations with an even number of planes may not have this problem, but I’m not sure. Either way, using circles seems to make this much more obvious than something like the stars above, where it’s harder to follow each color individually. This fact is rather disappointing to realize halfway through assembly, but it was a little late to give up, and I haven’t figured out a way to fix it.

