Recursive Models

malachi · Post by **malachi** » October 26th, 2005, 5:20 am

I was wondering if anyone had thoughts on recursive models, by which I mean a model that has one or more areas folded in a particular way then a subset, or maybe sometimes a superset, of that area is folded in the same way, and then a subset of that area, etc. Usually this goes on until the folder gets bored or a limitation of the paper and/or folder is reached.

The best examples I can think of are the Fujimoto Hydrangea, the Fujimoto Lotus, and Andrea's Rose. There are also some designs in Tomoko Fuse's Spirals II that would qualify, although some of them have an optimal number of levels. I don't think something like the Fujimoto Tessellation would qualify because it has a number of levels that are determined ahead of time and are all the same size, although it could be debated.

What is the finest level of detail that you've managed to fold?

I've gotten a Fujimoto Hydrangea to 10 levels with a 12" square and Andrea's Rose to 8 levels with a 6" square (which should be around the same as 10 with a 12" square). I didn't count the number of levels I got to with the Fujimoto Lotus, although I think it was probably around 8 with an 11" square.

What other recursive models are out there?

wolf · Post by **wolf** » October 26th, 2005, 5:40 am

There's a 3d recursive pyramid structure (Endless folds - square) by Ushio Ikegami in one of the Tanteidan convention books. Model database gives it as 5th and 6th, I'll have to check this for sure since I don't have access to the books now. Ravi Apte's website has an image of this:

http://www.vishakha.org/gallery/OUSA2005/101_0119_IMG

Also, do a Google search for Ushio Ikegami, there's quite a few links relating to his folded fractal models.

malachi · Post by **malachi** » October 26th, 2005, 5:49 am

Is this the model?
http://www.origami.gr.jp/New/Meeting/9905/endless.jpg

bshuval · Post by **bshuval** » October 26th, 2005, 6:29 am

malachi said:

Is this the model?

No, it isn't.

As to the original question. Yes, there are many people who have considered "recursive" models. They are usually called "infinite folds." In 1994 Jun Maekawa published a paper about these in the proceedings of the 2nd OSME. It's an interesting article, and well-worth seeking out.

Peter Budai has also dabbled with these folds. Check out his webpage for a listing of infinite folds.

Dave Mitchell once published an infinite fold in a BOS magazine. Nick Robinson (I believe) later published inifinite flapping birds in the BOS magazine, based on a similar principle (I may be wrong as I am working from memory here.) This is not to be confused with Robert Lang's Generations, which also produces an infinite progression of flapping birds.

As for the so-called Fujimoto Tessellation, I have found a folding sequence that allows you to continue it infinitely without precreasing at all, so does that turn it into an infinite fold?

malachi · Post by **malachi** » October 26th, 2005, 6:39 am

http://members.tripod.com/~PeterBudai/O ... ite_en.htm

That is interesting, although most of those, as he says, are not models as much as they are mathmatical experiments. It does remind me of a couple of similar shell models by Lang or Montroll in Origami Sea Life.

I guess I would need to see a picture to understand what you mean about the Fujimoto Tessellation.

bshuval · Post by **bshuval** » October 26th, 2005, 7:06 am

malachi, what I am saying is that I don't agree with your claim:

I don't think something like the Fujimoto Tessellation would qualify because it has a number of levels that are determined ahead of time and are all the same size

The reason being that the way I fold the tessellation is without precreasing and without determining apriori how many levels I want to do.

Now that I think about it, Yoshihide Momotani also has a nice infinite fold -- his fractal flower (also called by other names, but I'm not surewhat they are right now) in "Arte Dos Mesteres De Origami" and other places (it appears in many of Momotani's books). This model is, in a sense, the inverse of the Hydrangea (which was also independently discovered by Momotani, probably before Fujimoto). In fact, I have a folding sequence that transforms this fractal flower into a Hydrangea.

JMcK · Post by **JMcK** » October 26th, 2005, 5:49 pm

Chris Palmer's flower tower would be another example, I suppose.

Post by **origami_8** » October 26th, 2005, 7:20 pm

Larisa PerelekhovaÂ´s "Cube Transformer" and "Box with a bow" do have a part that can be folded endless and also KawasakiÂ´s Seashell.

mleonard · Post by **mleonard** » October 26th, 2005, 8:33 pm

What about endless modulars? The only one I can think of at the moment is the Stacking Pyramid Box by (ahem) yours truly - it was in the BOS mag a while ago.

JMcK · Post by **JMcK** » October 26th, 2005, 9:51 pm

mleonard wrote:What about endless modulars? The only one I can think of at the moment is the Stacking Pyramid Box by (ahem) yours truly - it was in the BOS mag a while ago.

Oh - there's also Jeannine Mosely's recently completed level 3 Menger sponge (from a mere 66,000-odd business cards.) Not endless, but it does have that sort of structure that repeats on different scales.
Sponge photo from Boaz's Yahoo! album

Then a scottish guy (Kenny I think - Dennis Walker knows him) did an origami version of the Sierpinski tetrahedron. You can see a photo at the bottom of this page:
Link

And Daniel Kwan has done a complement of the Sierpinski tetrahedron - a sort of negative of the original shape.
Link

JMcK · Post by **JMcK** » October 26th, 2005, 10:08 pm

wolf wrote:There's a 3d recursive pyramid structure (Endless folds - square) by Ushio Ikegami in one of the Tanteidan convention books. Model database gives it as 5th and 6th, I'll have to check this for sure since I don't have access to the books now. Ravi Apte's website has an image of this:

http://www.vishakha.org/gallery/OUSA2005/101_0119_IMG

Also, do a Google search for Ushio Ikegami, there's quite a few links relating to his folded fractal models.

Sorry, nitpick mode:

The Ushio Ikegami models in Tanteidan 5 and 6 are flat folds with squares of different sizes. The photo Malachi links to is of the model from Tanteidan 5 (just checked my copy).

Maekawa's 3D recursive pyramid in Tanteidan 4 that Dani mentions (link, scroll down a bit) is v. similar to the Ikegami one in the photo that Wolf links to, but after looking at the images carefully I think they are slightly different.

wolf · Post by **wolf** » October 26th, 2005, 10:34 pm

Yep, John's right; the Ikegami models are just the flat squares, and it was Jun Maekawa's pyramid I was thinking of.

The crease patterns are different, as far as I can tell. Maekawa's pyramid is more akin to what Budai has done on his website, just extended to four sides of the square. Ikegami's pyramid CP extends that idea further by having each arm of the pyramid sprout a new level of arms.

malachi · Post by **malachi** » October 27th, 2005, 1:33 am

bshuval wrote:malachi, what I am saying is that I don't agree with your claim:

I don't think something like the Fujimoto Tessellation would qualify because it has a number of levels that are determined ahead of time and are all the same size
The reason being that the way I fold the tessellation is without precreasing and without determining apriori how many levels I want to do.

And I am just saying that without seeing it, I can't understand what you mean well enough to know if I would agree with you or not.

Do the tessellated squares get smaller and/or are they folded inside of part of an already existing square, or did you just find a folding method that starts in one corner and folds until you decide on a center?

Post by **Daydreamer** » April 4th, 2006, 7:26 pm

I came across one of those infinite folds by accident in one of my own designs.

If you fold the diagrams for my Four-Leaf Clover up to step 11 and then sink all of the corners in step 12 (instead of only three of them) you can repeat this sinking process on the center square again and again to get this after four iterations:

I was wondering if this is a known folding sequence, because it somehow looks familiar to me.

Post by **origami_8** » April 4th, 2006, 10:35 pm

ItÂ´s the same folding sequence as in Larisa PerelekhovaÂ´s Cube Transformer.