I started with the inner 12 hexagons which was by far easiest, then all the heptagons, then the pentagons, and then finally all the remaining hexagons to close it out. This is the only picture I have from the final assembly - got on a bit of a roll! They slot together without any tape or glue which is nice. Two pictures for hours and hours of work spread out over several weeks. Now folding, first I did all the 'accordions'. The cutting actually took a long time because I wanted to be relatively accurate, so I only cut through one sheet at a time. I used a big straightedge and an xacto knife to cut the squares. It is nice and crisp, holds a fold really well, and doesn't crack. It is relatively thick for origami but is really good for wetfolding, tesselations, or this sort of modular origami. I used Zanders Elefantenhaut which is a book binding paper. Oh yeah, I also made a quick visualization of the graph: There are more efficient ways of solving three colorings, but this was fun. I used the fact that the graph has 12 fold rotational symmetry and I encoded 1 / 12 of the three coloring as a SAT problem and solved it with glucose. To find the 3 coloring I folded a test one (yeah, an entire test torus out of just regular printer paper.), labeled the vertices / edges, and worked out the graph on a sheet of paper. The graph which is formed by the individual PHiZZ units is a 3-regular graph, in that each vertex has exactly 3 edges leaving it, so we know that it can be edge colored in at most 4 colors. I wanted something a little bit bit different to go with the elefantenhaut paper I wanted to use, so I set out to find a three coloring. I've also seen a rainbow pattern which looks really cool. There are a couple really nice color schemes out there for this model, it certainly looks elegant in a single color. This might make it more tedious, but I think it helped make the final model more consistent. I first cut all the squares, then I folded all the units - even breaking that up into stages. I would not suggest cutting a square, folding the unit, and then cutting another square. Some splaying will happen, but any sloppiness translates into the final torus. If they are perfectly square it will make the rest of the process easier! There are a lot of squares to cut, but don't rush them. I used paper that was about 3 inches square, and ended with a torus that was about 16 inches in diameter.Īlso, if you want to make your own torus I have a few of my own recommendations: I was able to do it with their schematic - but I had a little trouble. Unfortunately this link has since stopped working.Īt the bottom of this page I included a schematic which would have helped me a bit more than the schematic at the above website. I'm not sure which torus I saw first or where I saw it, but when I was starting this project I searched for more information and found this website with some good instructions and recommendations: I did not come up with the idea in the first place - I saw a picture of another Torus and wanted one for myself. You can create loops from the pyramids To construct a dodecahedron use 30 units and make each loop consist of five edges. Three PHiZZ units are joined to create a pyramid, then additional pyramids can be formed from the other ends. He originally used them to construct dodecahedrons, and other shapes such as buckyballs.Ī single PHiZZ unit is only a couple of folds. The PHiZZ unit was originally created by Tom Hull. When you are folding 360 of something, it helps if they are relatively easy to do! The PHiZZ unit is a simple unit for modular origami, which makes it ideal for larger models such as the torus. Huge thanks to my wife for taking the above picture - she makes it look so much better than it actually looks. The Sonobe unit and the PHiZZ unit (pentagon-hexagon zig-zag unit) are two versatile units which can be used to make many different models. There are many different kind of units, some are custom to a particular model, and others can be used for many different models. The individual pieces of paper are often square, but not always. Modular origami is a branch of origami where the final model is made up of smaller units each folded from an individual piece of paper. However these kinds of restrictions are made to be broken - which can lead to better art! Normally origami is done with a single square of paper no cuts, no glue. It's nice to spend time on something and see it completed. Origami is one of my favorite hobbies, it's nice to create something in the physical world instead of writing some code which you can't pick up and hold.
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