## geometry of modular origami and extrapolating to more units

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### geometry of modular origami and extrapolating to more units

hey all,

this might be a dumb question but ive been making modular origami for about 9 years and I’m sort of at the point where I’m bored of making the same models over and over again. There are a bunch of 30-unit spheres that I really like and I want to extend them to 90-units or higher. I understand the geometry of the 90 unit models I have made, but I’m just wondering if most pieces can be extended to higher numbers of modules. For example, I’ve folded Tomoko Fuse’s Seastar model with 30 units, and i see how the units come together in vertices of 5, so I would assume I could create a 90 unit module by creating vertices of 6 in the buckyball configuration just like sonobe/phizz/etc. and similarly, ive loved making all sorts of phizz creations because they can combine to make toruses & spheres & almost any shape you can imagine. And I’ve always been curious why this isn’t true of lots of other modules. Because I know that the pentagon/hexagon/heptagon formations create the curvature, but that seems true of any modular piece. is there something that makes phizz units especially flexible when it comes to the variety of shapes that can be made?

and you’re probably all thinking “why don’t you just try it out?!” and my answer is that I am really limited in my supply of paper so I get really stressed out about “wasting” paper trying new modules. which I suppose is not exactly ideal for someone who wants to start experimenting. anyways, if anyone is particularly keen to explain any of these concepts that would be wonderful (and if anyone could recommend any resources— if they exist— about how units combine & the number of edges/vertices/etc.)
Baltorigamist
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### Re: geometry of modular origami and extrapolating to more un

I believe the thing that makes Phizz units so compatible with tori (to use your example) is that its ratio of height to length is so low (i.e., it's not as tall vertically as, say, Sonobe units). In addition, models made from Phizz units have a lot more open space in them than Sonobe models. This all makes models with negative curvature (e.g., tori) very difficult to construct on a small scale with Sonobe modules. (The reason for this is that Sonobes necessarily need polygonal faces which can be constructed by a pyramid of equilateral triangles, meaning nothing with more sides than a hexagon is possible. It involves a lot of math that goes slightly above my head, but it has to do with the dihedral angles between faces of the polyhedron or torus in question.)
You can, of course, extend the Buckyball configuration by adding more hexagons between the pentagon, but then the friction between the units becomes an issue as the structural integrity of the entire model becomes weaker.

Regarding how units combine, here is a list of polyhedra that can be made. Each unit would represent an edge, as you may already know. But, as I said, the angles between the planes make some constructions impossible in practice.
There are a few skeletal units (see Hull's Five Intersecting Tetrahedra for an example) that could probably be used to make a torus, but the negative curvature could be a problem there as well.
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