Category: Origami

You may recall that I started working on a new Penrose tessellation back in June at the convention. This turned out to be a really interesting challenge for a number of reasons. First, my initial attempts at a pattern for the tessellation were wrong because they didn’t go together according to the rules. There are lots of ways of joining the tiles the look correct at first but really aren’t once you start to grow the pattern. I learned alot about how to join Penrose tiles.

Second, I needed to systematically work out the widths of all the pleats. I discovered they came in at least four different widths. The ratios of the widths are all the golden mean, which I guess is not too surprising. More surprising is that the wide pleats tend to separate groups of tiles into clusters, like avenues, streets and alleys, or maybe branches on a tree. The groups are usually five tiles, sometimes ten, sometimes two or seven, and then these aggregate into larger groups.

In fact the whole pattern is fractal. I made a series of Penrose tessellations with 10, 35 and 70 tiles. For each one, you begin by precreasing the pattern of the previous tessellation and then subdivide it. This happens in an interesting way, since each tile subdivides into approximately 1.6 tiles, with the edge of parent tile forming the diagonal of a new tile.

I worked all this out when I was on vacation last month in Albany and Lake Placid. After that it took me a few weeks to actually fold the models. I’m satisfied that the approach can be taken to an arbitrarily large number of tiles. The next level out would be at least 120 tiles; the further out you go the harder it gets to find a logical stopping point. I might do that eventually, but for now I think I’m going to fold a 70’er out of good paper and declare victory. I suppose I ought to document the crease pattern too. But after that, I’ve been focused on pentagram-symmetry subjects long enough. It’s time to move onto other things.

Like compound polyhedra with a color change!

I’m participating in another origami exhibit. This one is eXtreme Origami and is part of the Origami Heaven convention in Stonybrook, Long Island, and runs thru early August. Go check it out. You can find out more about Origami Heaven here:

http://www.origamiheaven.org/

I’m not sure if I’ll be able to make to check out the exhibit, so if you want to take some pictures I’ll appreciate it.

The theme of my collection of models is single-sheet complex polyhedra. The models I’m contributing are my Stellated Dodecahedron, Great Dodecahedron, and
Tessellated Dodecahedron (a.k.a Penfractal Dodecahedron). All of these
models exhibit pentagonal symmetry, being based on the dodecahedron, which
is composed of twelve regular pentagons. Each of these models is folded from a single, pentagonal sheet of paper.

You’ve seen the Stellated Dodecahedron recently. I still want to fold a second one, but didn’t get done in time, so I sent the one I folded for OUSA. Did manage to make a nice version of the Dodecahedron Tessellation out of Wyndstone paper, and will show that in a future post.

But this post is about the Great Dodecahedron. It’s not exactly all-new; I folded one from a 12” square of Tant a few years back but was never quite satisfied with it. The new one has a refined CP. The main difference is that it’s from a pentagon, so the corners provide nice flaps and the model goes together well and holds its shape quite strongly. I didn’t even need to wetfold it.

The shape itself is a complement to the Stellated Dodecahedron. Both are composed of sixty triangles and form star shapes out of sets of coplanar faces. With the Great Dodecahedron, the coplanar faces form a pentagon with the star rising out of the middle in the negative space.

I did another class at the convention, teaching my mandala tessellations. Most tessellations out there are based on a grid, either of squares or triangles, but these are based on a circle. I originally started playing with tessellations because of models like my turtle and my U.F.O., which use pleating to form a pattern on the surface of the model. I actually came up with the mandala idea at a convention, at the Monday dinner, and have since refined it. Unlike grid-oriented tessellations, the pleats are arbitrary, by eye. This is not hard to get across to advanced folders who have a good eye, but people tend to like landmarks and it can take some getting used to.

People seemed really interested in these models back at MIT last fall, where I taught them informally, and my class at OUSA was in fact full. There were a few “legit” tessellation guys in the class, notably Jeff Rutzky, who has a book Shadowfolds with Chris Palmer that explores some of the same territory, but his ideas are more developed. So as I taught the class it turned into a conversation, which was great cuz I learned a lot as well as taught.

This also led me to begin work on a new model: a penrose tessellation. This is a new and larger pattern than the one on my web site. I folded a prototype using my arbitrary-pleat method, but on this model I reached the limit of that approach. I folded a second, more legit attempt, but it didn’t want to go together. So then folded the center portion only, working out the correct sizes for the pleats. I folded successively larger sections, until I got to the third ring, where there’s a vertex where seven angles come together, three wide and four narrow. So now I have to work out complex connection in order to complete the model. But that’s cool; when its’ done it’ll be really nice. I also feel like this will be another good CP to feed to the craftRobo.

This is the last of the current flurry of origami posts for a little while. I’m working on a exhibit for Origami Heaven that has to be done in three weeks. Five pentagon-based symmetry geometric models, including some of the ones described here. I’ll let you know how that goes. Meanwhile I hope to get back spending a bit of time on rock’n’roll and other pursuits.

So way back in early June I was invited to participate in an exhibit Origami as Fine Art being held at Kinokuniya Book Store in NYC just off of Bryant Park. Kinokuniya is a great place, BTW, full of all things Japanese, from manga, anime and robot kits to origami books and specialty papers, and all kinds of other stuff. The exhibit is really nicely put together by Sok Sung and Paul Frasco, and features a great variety of models from a good number of artists. I’m honored to be included among this elite set. The exhibit runs for another two weeks, until mid-July. You can learn all about the exhibit here:

http://origamifineart.com/

The curators asked me to contribute a Zeppelin. This was on the list of models I’d been meaning to refold out of nicer paper anyway. Last year at convention I picked up a couple sheets of paper specifically for this. The paper is blue, and strong and heavy like Wyndstone, but sparkly like it has flakes of mica embedded into it like Origamido paper sometimes does. I have no idea where it came from or what its called, but it turned out to be up to the task.

As I mentioned before, this is a very challenging and labor intensive model, and I only ever made one successful instance a few years back. I first made a study out of draft paper, and then commenced to fold two Zeppelins in parallel. The thinking was, if I messed one of them up I’d still have a good’n, and if they both came out good I’d have one for myself. The prefolding took a couple sessions and the collapsing took a couple more, since I wetfolded as I went, doing first the nose, and then collapsing the tail. This was all a very delicate operation. I also custom built two new stands for the models. As June progressed I went from staying up late once or twice a week to work on these to practically every night.

In the end I had a bit of a problem with the model staying puffed out at the transition from main body to the tail. I realize now that I should have wetfolded an internal flange of accumulated layers of paper to relieve the stress at the point. I may yet go back and fold yet another Zeppelin now that I know what I’m doing, and so I can be at peace about it. For one of my models I tried to wetfold the hull at that point. This turned out to be a mistake, and I ripped the paper pressing it from the inside with chopstick to shape it. At least I was able to patch it so it doesn’t show. The other one turned out better an the problem was largely solved by good ol’ fashioned wrestling the paper into submission. I had to re-open the tail after it had been wetfolded, but it worked out okay. So the better model I donated to the exhibit, and the other one is the one you see photographed here.

I folded new high-quality versions of my new models Timber the Dog and Sophie the Cat for convention, as well as few to give to my friend. As mentioned before, these models were inspired by the logo of the animal adoption service Timber’s Legacy (timberslegacy.org). The versions you see here are folded from a 12” square of Canson and a 10” or so square of Wyndstone respectively.

I taught both of these models at convention. I rated them as high intermediate level of complexity. I think that was pretty much on the mark. I taught them in a single 45 minute class for each. There were very few people in over their heads and everyone was able to get the model done in the time allotted. I found out that origami people are more cat people than dog people. The class for Sophie was filled to the limit of 25 and then some. Since it’s a new model and not yet diagrammed, I taught by demoing it with a large sheet of paper. With the larger class you spend more time walking around to be sure everyone’s got it right, but it was okay. In fact feedback was very positive, and I’d say I have two more models for a future book.

As mentioned previously, I had a great OUSA convention this year. If you’re interested there are lots of pics online here. One of the things that made it great for me is I had a lot of new models: Timber the Dog, Sophie the Cat, my Giant Squid, Cuttlefish, a new rendition of my Zeppelin, my Penfractal Dodecahedron Tessellation, and my Stellated Dodecahedron. I hope to say something about each of these, starting with the Stellated Dodecahedron.

Most people make origami polyhedra using the modular approach, putting together lots of triangles or whatever to create a shape. To me this is not very interesting since the assembly dominates the folding, and in my view it misses the essence of doing origami. Much more challenging is to fold a polyhedron from a single sheet. In fact it’s so challenging I only know three artists who work in this area: John Montroll, T.J. Norvell, and myself. John has written several books on the subject and has made it challenging indeed to find terra incognita in this space.

My Stellated Dodecahedron is the result of a long and arduous quest, one of a handful of advanced polyhedra I’ve been working on for the last few years. Now I can finally cross it off my list. In case you don’t know, the stellated dodecahedron is a shape based on the regular dodecahedron, which is a platonic solid composed of twelve (“dodec” means two and ten) regular pentagons. The stellated version replaces each face with a five-sided pyramid, resulting in a star-like shape (“stellated” means starlike) composed of sixty triangles.

I made several attempts at it a few years back, but it was beyond my skill at the time, and I only barely managed to make one out of a giant sheet of foil, and it didn’t hold together too well. This year I was determined to finally make an exhibit-quality one for the convention. I’d been working feverishly on Zeppelins (more on that in another post) but found the time to create this as well. In the process I think I leveled up as a folder.

The hard part of doing this model was closing it all off at the end. My previous version was made out of a square, and this created the problem of having all the flaps being different shapes and very difficult to deal with. For the new attempt I began with a pentagon sheet of paper rather than a square. The first step was to fold a pentagon out of a square anyway, so I just cut off the leftover bits rather than fold them under. Now when I got to the end I had a nice array of identical flaps that made perfect tabs, and I was able to use a twist-lock to finish the model.

I folded the Stellated Dodecahedron from a 50 cm regular pentagon of Marble Wyndstone (a.k.a. Elephant Hide) paper. It took me five sessions of several hours each, and this after I made a study out of draft paper. The first two were just to do the precreasing (you can see the CP below). The next one was to fold the bulk of the model, up to the close. At this point it became clear that I would not be able to close my study model because the paper was too soft, and I’d have to just go for it. Wyndstone is thicker and I’d precreased everything strongly with a bone folder, so it looked like it would have enough “pop”.

Once I got it got it collapsed there was some springiness that wanted to push the model apart from within. I applied a drop of water to every concave vertex, the ones where six triangles come together. Then I wrapped the whole thing tight with tape and string and let it dry overnight. Next morning – the day of convention – I unwrapped it, not knowing how the wetfolding had worked. To my delight it came out perfectly. The final model was tight and strong like a soccer ball. In fact I like it so much this model has now taken the spot on top of my piano previously occupied by the only modular I’ve ever liked well enough to actually fold – Tom Hull’s Five Intersecting Tetrahedra.

I want to fold another one of these soon, for an upcoming exhibit on Long Island. Between doing this and the Zeppelins, I’m now seriously considering getting a KNK Zing or a Craft Robo. Having a machine pre-score my paper from a .eps file would seriously reduce the amount of time I have to spend prefolding, plus it would guarantee everything to come out as precise as can be.

Here’s my prototype of a Penfractal Dodecahedron Tessellation. It’s a 3-D evolution my Penfractal Tessellation. Each face of the dodecahedron is subdivided into six smaller pentagons. The model also makes a reasonable representation of the dodecahedral analog of the tesseract, albeit in 3-d hyperbolic space rather than true 4-d space (trust me on this; I looked it up). Now that my proof of concept is done, I want make another version that’s more ball shaped by making the tessellated sub-faces bulge out. And you should know that although I almost always work with a square sheet, I’m considering using a pentagon for this one, so I don’t have to deal with all different ways to handle the unused paper in the corners. I think the subject may justifiy this.

I’ve been folding lots of other stuff too. I’ve perfected my dog and cat and made a whole bunch over the weekend. A few are to give away, and others are for my exhibit and the convention model menu. Soon as I get around to it I’ll take some pictures with the good camera. I’ve also been invited to contribute a model to an upcoming exhibit. More on that soon. The model they want is my Zeppelin. This is one of my hardest models. In fact I’ve only really successfully folded one. I’ve been meaning to fold one out of better paper for a long time. So now I’m making two, one for me and one for the exhibit. Luckily I diagrammed the first half of the model (the precreasing part) so I don’t have to work that out.

If I get that done in time, my next model will be a new attempt at a one-sheet Stellated Dodecahedron, since it uses the same base as the Dodecahedron Tessellation.

This weekend we’re having the first real hot weather of the year, in the mid to upper 80’s. It’s also the weekend of the carnival at the kids’ school. I helped out last night at the Fathers’ Club Food Tent, and will be helping out tonight and tomorrow in the games booths. These school events get more and more fun as I get to know more parents and families. There’s a great sense of community that I never felt in the old school. Plus it’s interesting, but I guess not surprising, how Establishment it is. The local cops and firemen are all out showing support. Heck some of them are parents of the school. I’m still conditioned from the days of my youth to see cops as a threat, so it feels weird to learn they’re normal, friendly guys.

I have alot of vacation time to use up this summer, and so I’m taking my first summer Friday to do some origami so I have some new models for convention. I have few things that I’m working on. First is a pair of new animals, a dog and a cat. I’d been thinking about doing a dog and cat for a while, but they’re pretty challenging subjects. There are lots of cats and dogs out there, ranging from simple and stylized to realistic and complex, capturing a wide range of moods and styles. Plus cats and dogs are very familiar to humans. It won’t do to simply have the right configuration of limbs; the model has to say something.

I found my way in to approach these subjects as a request from a friend. She had recently created a website for an animal adoption service in Queens – Timber’s Legacy (timberslegacy.org). The website logo features a dog and cat in silhouette and she asked me if I could fold them as gift for the lady who runs the service.

The idea that the animals are sitting allowed me to focus more on the overall shape and sculptural aspects rather than worrying much about the base and where the points for all the legs would come from. The dog in particular has pointy ears, and you don’t see that much in origami dogs. Once I had the dog worked out, I figured the cat would be a simple variation, but it turned out the pretty different. Not only is the head and face totally different, but the long tail required a whole new geometry and base. Finally, the body posture, neck and shoulders were different enough to have to come up with a new treatment.

I’ve been folding lots of cats and dogs the last few weeks. They’re not too complex and take less than a half hour to fold. I’m still refining the details, particularly on the cat, before I attempt to fold exhibit quality ones out of good paper.

The other thing I’ve been working on is an approach to combine polyhedra with tessellations. I came up with a successful prototype for a sixty-sided shape based on a dodecahedron. More on that soon.