Category: Origami

It’s been a long time since I bogged about my ongoing origami book. Well I recently finished diagramming two more models. One is my Medieval Dragon, clocking an just about 100 steps. That’s twelve pages as I currently have it laid out. The other is my U.F.O., which is ten pages and a shade above 60 steps, but from step 40 or so onward it’s all fully three dimensional, so each step is a major work of drawing. These are by far two most involved models I’ve done so far. The only thing that comes close is my Adirondack Moose, which is also about 60 steps, but it’s only 3-d for the last few steps. Both the Moose and the Dragon are fairly traditional designs, using variations on classic bases and inspired by models in John Montroll’s first book. The U.F.O. however is completely modern and without an antecedent, and well over half the folding sequence is precreasing. So while it may not seem like a lot in terms of number of models, it’s a lot in terms of page count, and I knocked of two major models to boot. This brings me up to 16 models done, about 90 pages worth. I’m definitely more than halfway done with the book, so I think its time to start looking at the publishing process.

I signed up to teach both of these at the upcoming OUSA convention – only a week away. So I’m glad I got the diagrams finished but as usual I have lots of ideas for things I’d like to fold but haven’t had the time to fully develop. So I think this weekend will be a big folding jam.

I’m posting pics of the final step of each model as a way to motivate myself to continue by showing the result without giving away the whole diagram. I may go back and post pictures of the final steps of my other completed diagrams to create a sort of evolving table of contents. I need to update my origami site anyway. I have a number of models, going back more than a year, that I need to photograph and post. Well July will be a good time for that. I always get a big boots of energy and creativity from the convention, and that will fuel me into the summer to continue developing things.

My first origami tessellations was a grid of pentagons. Now I’ve made a tessellation of triangles and squares to go with it. And mean that in a very specific way. This mesh is the dual of the other. If you draw a dot in the center of each cell of the pentagon grid and connect them all, you’d get this pattern. Five cells converge at each vertex. In the other pattern all the cells were the same shape, but either three or four cells converged at a vertex. I’ve never seen anyone fold this pattern before and the two of them make a nice set.

Here is a crease pattern for an origami Great Dodecahedron. This fascinating shape is something like a sunken icosahedron, and can also be seen as twelve intersecting pentagons with a raised star on each face. I tried several iterations of the layout because the details of forming tabs and pockets to close the model took some trial and error to get right. The basic idea is fairly straightforward. I use fivefold polar symmetry, and the whole pattern embedded in a single pentagon that takes up pretty much the entire square sheet. I was able to divide it into a grid of parallelograms using simple ratios. Each parallelogram then gets subdivided into the triangles that form the faces of the shape.

Since I just fold back the corners of the square to form the base pentagon, I tried a version folded from a pentagonal sheet, but this turned out not have enough extra paper around the edges to from the tabs and pockets. The pentagon’s height is slightly less than its width, which results in a then strip of unused paper at the bottom edge of a square sheet. I decided to try folding the strip around all five sides (except where it gets truncated at the corners), and that turned out be just the trick.

I’ve successfully folded a couple of these now that stay together well. Pictures as soon as I make one out of nice paper.

Here’s an idea I’ve been working on for a while. I’ve seen this kind of thing done as modular and thought it was doable as a single sheet, and I figured I’d go for it. Single-sheet stellated polyhedra are pretty advanced, but the color change brings it to a whole ‘nuther level of complexity. It turns out to be a very rewarding shape to fold, and the design is replete with all kinds of interesting symmetries.

The first challenge was to work out how to achieve the arrangement of alternating colors. Once I’d worked that out the resulting (flat) shape would serve as the base for the 3-d phase. I needed two corners to come to the center like a blintz, but offset. Working out the amount of offset for the grid to be the right size was the key problem. It turns out the key angle is 67.5 degrees, which is 3/4 of 90 degrees and easily derived. It also turns out the angle has a slope of 3/2, which is also easily derived from a square grid. From this I was able to work out the arrangement of the squares in the inner rotated grid and the outer triangular grid areas, which correspond to the blintzed flaps. The 3/2 slope made was convenient because the grid is has an integer relation of the unit the whole. Each square of the grid has a length of 2/13 the edge of the paper, as you can see in the crease pattern. Neat, huh?

Another interesting property of the model is that once you’ve made the base and put in the color change squares, the easiest way to deal with the leftover paper on the two remaining corners is to flip the model over and do the same thing. The result is the pattern is the same on both sides, although made with opposite colors, and either side can be used for the outside.

The finished model fold together well and looks really good. I went ahead and made a few.

As an added bonus Lizzy and Michelle were folding bowls and picture frames, so here’s one.

It’s been a rough week. I had a bit of a pain in my leg which turned out be caused by a swollen or pinched nerve. I tend to have a few aches in my legs this time of year due to the cold, but this got bad enough I needed to go see a doctor. Maybe it was a blessing in disguise, since I’ve been feeling down about my work situation lately and I had a few days off, a break from that unpleasantness and drama and from commuting in the cold, and I took a few days to rest and reset. Now that I’m on the mend physically, I’m much less burned out mentally too.

I got a chance to work on some more origami ideas this week. I have a handful of polyhedra I’ve been trying to perfect for a while now. One of them is the Stellated Rhombic Dodecahedron, and I had a half-fold one I started ages ago and I thought I’d finish it off. But as I got into, I realized the layout was kind of weird and I could do better. So here is the new improved layout. As you can see, the layout is nice and symmetrical, and that makes the precreasing relatively straightforward. The forty-right triangles that make up the faces are based on the geometry of the silver ratio. There is a border that runs the whole way ‘round the paper to provide flaps at the corners to tuck in and finish the model. As luck would have it, I picked a good size for the flap to provide a reference to set up the silver ratio without having to find any other landmarks. I’ve folded a couple of these now, working out a clean way to close the model. Look for completed model in the near future.

Now that some things with the weather, work and the recording project have been sorted out, it’s time to get back to origami. My turkey has been sitting half-folded on my desk fro weeks and I finally had the chance to finish it off. Good thing too, I’m teaching it this spring, and need to get all the details worked out.

Last week Lizzy took a class trip to the American Museum of Natural History and while she was there had a chance to check out the origami holiday tree. I had hoped to cruise up there to meet them for lunch and tell the kids a bit about the tree and Origami USA, but alas it was not in the cards. I was stuck in an all day project planning meeting. It got pretty contentious and by the end my boss was openly calling for the bad product manager (you may recall him from a post a few months back) to be sacked. Today was had part two of the meeting, in which some differences were resolved and workable project plans were agreed upon. At one point I had to call out the bad manager and asked him to articulate the difference between his new plan and my existing plan, and he admitted he hadn’t read my plan therefore couldn’t say. So my plan prevailed, much to the amusement of some of my colleagues. Frankly I’d rather avoid all this unpleasantness and stupidity, and I’m beginning to loose my sense of bemused detachment. Although I must say all the developers on my team are really excellent, so I don’t see any deep worries, just management B.S.

In any event, Lizzy and her classmates enjoyed the tree, and she had been the museum before for special folding sessions as well as to see the museum itself. She was able to tell the other kids something about the tree, and even spotted some of my models and took some pictures. In addition to the models I donated this year I was happily surprised to see a turtle which I’d donated a previous year.