Introduction: Fold Infinite Recursion
Anyone who starts playing around with origami pleats and tessellations will probably discover this fold themselves, sooner or later. I did, and later learned that it is commonly referred to as Fujimoto's Lotus after being described in the 1960s by origami artist Shuzo Fujimoto.
The basic idea is to take a square sheet of paper and collapse its corners so that they frame a smaller square, and collapse its corners so that they frame a smaller square, and so on and so on. It's a pretty simple model to fold, but playing around with it will help cultivate an intuitive understanding of more complex recursive origami, such as Chris Palmer's famous Flower Tower and Eric Gjerde's amazing origami tessellations.
Step 1: Mark the Major Divisions
The folds for this pattern are not even subdivisions of the full sheet of paper, so we'll need to do a little work with pencil and ruler just to get things rolling.
We'll start with a blank sheet of paper. I'll be using one that measures just under ten inches, but it doesn't matter exactly what size the paper is. Whatever size paper you have, you'll need to subdivide it along each edge into six equal parts with a little bit left over. I'll be dividing my 9-3/4" piece of paper into six 1-1/2" parts so I have 3/8" left over all around. This 3/8" strip will form the border for the finished piece.
Fold the paper in half and make a short crease in the center of that edge, but do not make a fold all the way across; unfold the paper. Measure out and mark 1-1/2" and 3" on either side of the center line or, if you have a piece of paper that is a different size, use whatever measurement makes sense so that you end up with six even subdivisions with a little left over on either end.
Repeat along the adjacent edge.
Step 2: Fold the Major Divisions
At each of the four marks that you made along one edge in the previous step, fold the paper over and make a crease all the way across, perpendicular to that edge. Unfold and smooth the paper back out after making each crease. Repeat along the adjacent edge.
Step 3: Fold Outer Diagonal Creases
Next we'll need to make diagonal creases at each of the four outer intersections. First locate the intersection between two of the existing outer creases created in the previous step. Fold the paper over along a line that extends away from this intersection at 45 degrees from each of the two existing creases and pinch the paper at the intersection. Holding the pinch down with the tip of one finger, line up the existing creases on the folded over flap of paper so that they line up perfectly with the ones below. Once you have the flap properly aligned, set the fold in place by smoothing it down with your finger, stopping when you reach an existing crease, close to the edge of the paper. Give the fold a strong crease and unfold. Repeat at the other three outer intersections.
Step 4: Fold Inner Diagonal Creases
We're going to do the same operation as in the previous step, but now at the four inner intersections.
Beginning at an intersection between two of the inner creases, fold the paper over at 45 degrees and make a pinch at the intersection. Line up the flap with the paper below and smooth along the fold with your finger, stopping where this line intersects an existing crease close to the edge of the paper. Unfold and repeat at each of the four inner intersections.
From here, you have the option to either continue pre-creasing for smaller and smaller recursive folds or to dive straight into folding the model. The advantage of doing some more pre-creasing up front is that the model will come out a little more clean and precise and you'll likely be able to fold a couple of layers further down, but it will also increase the time it takes to complete the model. If this is your first time following this Instructable, my suggestion is to jump ahead to step seven and start folding!
Step 5: Make Pre-Creases for Recursive Folds
At this point, pre-creasing for each successively smaller layer of recursion requires just four simple folds. It might be difficult to see where they go at first, but there's a clear pattern which you'll pick up after a little practice.
Looking at the center of the paper, find the smallest square formed by the existing pre-creases. It should be centered, oriented orthogonally to the edges of the paper, and about one third the dimensions of the sheet of paper on each side. I'm going to refer to this smallest square as the 'base square.'
Now, imagine an even smaller square placed inside of the base square, rotated at 45 degrees and with its four vertices touching the four center points of the base square. Now extend each of the four lines that form this new square so that they continue for the same distance in both directions outside of the base square that they cover inside of the base square. Each of the new lines will terminate at an existing intersection, which you can use to start and finish the fold. It sounds complicated when explained in writing, but if you look at the paper, there's some straight forward geometry that will tell you where these new folds need to go
Once you've made a crease for each of these four lines, you should have a new smallest square. This is now the base square
Step 6: Repeat Step Five and a Couple of Tips
Repeat the previous step as many times as you can, rotating at 45 degrees each time and creating smaller and smaller base squares. I'd recommend getting down to a square that is about 1/4" on a side. You can go a little smaller, but not much before the thickness of the paper will prevent you from making such tiny folds.
Tip #1: Once you've found the intersection that will terminate one of the new lines, if you take the two existing lines that project from that intersection and fold them over so that they meet all along their length, and then smooth the paper along the new fold this creates, that will automatically give you the new line that you wanted to fold. Just make sure to stop when you reach the intersection at the other end. If you continue these lines straight across, the paper will become an unmanageable thicket of creases.
Tip #2: When you're pre-creasing tiny lines close to the center of the paper, you might find it difficult to get the paper to do what you want. If you place the paper on the edge of a desk or workbench so that the edge lines up with the new crease you'd like to fold in the paper, you can bend it over the edge to get things started and should then be able to pick the paper up and finish the crease.
Step 7: Start Folding!
The pre-creases are finished and it's tine to start folding. Decide which face of the paper you want to be up in the finished model and flip the paper over so that this side faces down. Using the four major divisions that you created all the way back in step one, fold valley folds at the inner divisions and mountain folds at the outer so that the two outer divisions come together and meet in the center. Repeat in the other direction. Flip the model over and give it a good press to set the folds in place. You should see that you have a square with a border projecting around it. You're not quite done, though, because the four corners are trapped and you'll need to free them before folding any further.
Step 8: Free the Corners
Open the paper back up, but don't unfold completely. There should be a pre-crease that extends away from each of the four corners at 45 degrees; turn this into a valley fold and start bringing everything back together so that the model looks the same as it did at the end of the previous step. This is some real paper wrangling. Check out the video to see just how it's done.
Step 9: Check That All Four Corners Are Free
Each of the four corners should now be a flap which you can fold back so that the tip touches the center of the paper. Pulling back this flap should reveal two identical right isosceles triangles sitting adjacent to each other.
If you can peel back the flap but don't see the two triangles, that's no big deal. It just means that one has been flipped over the other and you need to rearrange the layers of paper. Play around with it and you'll see what I mean.
If you're not able to fold the corner all the way back to the center, that means that it isn't completely free. Head back to the previous step and work out the snags.
Step 10: Fold Each Flap Inside of Itself
Choose one of the four flaps, it doesn't matter which. If you completed steps five and six, you'll just need to fold the flap back a little to activate the pre-creases. If not, fold the tip all the way back to the center of the paper. Give it a good strong crease along the edge and unfold.
Work the flap open a little so that the four layers of paper that make it up are about equally spaced from each other. Once you've done that, gently press on the tip of the flap and it should start to puff out. Keep working it out until the flap has been entirely puffed out and makes a flat square.
You should see a crease running horizontally across the square you just created. Turn it into a mountain fold, bringing either end in toward each other. This should cause the whole thing to close up, bringing the top of the model back over so that it returns to its original position. Smooth everything down.
The flap should now be entirely reversed back inside of itself. Repeat for all four flaps.
Step 11: Repeat As Many Times As Possible
Repeat the previous step as many times as possible, each time folding in the four flaps and revealing a new, smaller base square in the center of the piece. Try to get down to a square that is about a quarter of an inch across.
It can get dicey as you're trying to make smaller and smaller folds. At a certain point, I start using tweezers. Just be careful not to pull too hard and tear the paper.
Step 12: Finished!
When you're folding infinite recursion, there's no real beginning or end. Just go as small as your fingers and the paper will let you, and call it a day. The thickness of the paper is the only real limiting factor to how small you can get. As far as how big, the sky is the limit! Start out with a piece of paper 20' on a side, and you'll end up with a model much larger but with many more folds and just as much detail. The trick is that each step reconfigures the inner geometry of the model, making it possible to fold one step smaller, reconfiguring the internal geometry of the model, making it possible to fold one step smaller .... etc etc. Have fun!