Introduction: Articulated M.C. Escher "Curl-up" Model
I've been a fan of M.C. Escher's drawings since I was very young, and have always particularly loved his invented creature Curl-up (or Wentelteefje in Dutch). I recently had the opportunity, as an Artist in Residence at Instructables, to build a model of it. I have seen one or two previous attempts at the standing pose, but none that curled up, and I determined to make the attempt.
Step 1: Analysis
The first thing I did was to make some diagrams, expanding on Escher's creature as well as I could. Escher's published work containing the Curl-up constists of three drawings, Curl-up, House of Stairs, and House of Stairs II. These drawings are of course art, not a drafting diagram, so I had to make up some parts that were never shown clearly. I discovered that Escher cheated, too: the Curl-up's head is, in the standing posture, clearly the widest part of its body, but in the curled-up posture the smaller body segments encompass it, without any apparent stretching or bending. Although, with Escher's theme of realistically drawn but physically impossible objects, this isn't really surprising.
The Curl-up (in most cases I could observe) has a carapace with 18 body segments plus its head, with the last one, the tail, finishing in a rounded point instead of attaching to another segment. There is some inner flesh that protrudes a little from the head shell, but the underside of the body is never shown clearly. There are three pairs of legs, with a quite human knee and foot attached. I decided not to worry about the soft, squishy interior until I had the carapace working. I numbered the segments starting from the one just behind the head. The legs extend from segments 4, 9, and 14. The join between segments 1-2, 2-3, and 3-4 have a significant reverse bend in them when the Curl-up is standing normally, to keep its head raised. Some of the later segments also have a slighter reverse bend but this depends on the Curl-up's movement and activity.
Step 2: First Body Segment Design
I drew cross sections of the body segments and cut them out from heavy paper by laser, to get a sense of the proportions and how much bending would be required. Each segment fits over the one behind, and under the one in front, and flexes at the join points. Based on the paper model, I drew a segment in 3D, scaled it up and down by 3%. Using the Objet Connex 3D printers at the Instructables lab, I printed out the results to see what they'd be like. My first idea was that there would be small flanges at the front and back to keep the segments from coming apart at the top horizontally while bending. However I'd forgotten a way to keep the pieces together vertically; the segments could easily bypass their flanges by just falling out. Also, there was yet no clear axis along which to bend.
I considered using the Objet's multi-material capabilities to connect the segments, and printed some tests of the stiff "Vero" material, from which I made the shells, with some connective "tissue" of the flexible "Tango" material. The flex I got from this didn't seem quite sufficient, nor did it seem the connection would be strong enough... and finally, the finished Curl-up was too big to print all at once. So I abandoned the idea of a flexible connection.
Step 3: Curling Up Mechanism
Another major design choice was what I termed the spine. To keep the creature curled up, I planned to use a bit of nylon monofilament as a kind of "ventral tendon", which meant I needed somewhere for this to run. I first tried a cross-brace along the bottom of each segment, with a hole in the middle for the tendon, but this made the segment sides too stiff. They needed to flex a bit when the next smaller segment was shoved inside during the curling up process, and the cross-brace prevented this. So here I ended up with a vertical drop, like a stalactite, descending from the middle of the segment's top.
I tried a "dorsal tendon" at the top of the spine, as well, as a way to keep the segments together, but without a more detailed fit between the pieces where they met, this didn't keep them in line at all.
I tried several complex attachment options but none worked well. I ended up with a simple hole-and-peg hinge, for most of the segments. For the neck area, though, where the segments needed to bend backwards more than a little, I made a hinge in the middle of the side, with a flexible arm to hold the peg. To these segments I also added some extra length where they would gap open when curled up.
Step 4: Head
The head I probably had the most trouble with. I tried several different drawing techniques and four distinct files to make it. It took a number of tries to get something that was close to the original Escher drawing (and I think it is too big, although I tried to gauge the proportions right). The first drawing I was happy with enough even to print out a test of, I forgot to add the hinge pegs to attach the first segment, so I could only hold it in place to test the fit. I did add them to the next version though!
Step 5: Legs
The legs were really fun to draw, especially the little toes! I had originally thought I would let each pair of legs attach across the middle of the segment, with a join in the middle to let each rotate separately. However the segments' movement when curling up had to cross into the area the leg would take up (and it would stiffen those segments, like the cross-brace idea). So I made the legs end just a little beyond the inner shell edge. After initial test prints, I also made a flange to keep them from falling out, and made a press-fit plug to allow them to come apart and be set into place through the shell.
The hole for the legs I added to segments 4, 9, and 14 after they were all drawn (see the next step). It consists of a small round flange filleted smooth.
Step 6: Designing the Rest of the Segments
Once I had what I thought was a reasonable attachment system and shape, I needed to generate the rest of the segments, down to the tail. I could not simply scale them down for a couple reasons. One, the wall thickness had to be more or less the same all the way down, or the segments would be too thin and brittle at the small end, or too thick and inflexible at the big end. Two, the hinge assembly similarly needed to be the same size, not scaled down, and it had to fit correctly from one segment to the next. And the hole in the spine should remain the same size too. Here's how I drew each of the other 17 segments based on the first.
I named the satisfactory drawing 'segment01.3dm'. I had used a number of construction objects: a set of wireframes to loft the segment with, a "spine" object, a hole for the spine, a height placer for the hole in the spine, and a pair each of hinge pegs and holes, each with a placement plane I could resize without affecting the hinge size. After completing each segment, retaining these construction objects (pic 1), I copied the segmentXX.3dm file to segment[XX-1].3dm. I opened the copied file and hid the existing shell object (pic 2). I shrunk the wire frames, spine, spine hole placer, and hinge placement planes by 1% around the origin (pic 3). I moved the unshrunk hinge peg and hole objects to the planes' new position (pics 4, 5). I expanded the wire frames back by 1.01% in the wall thickness direction only, towards the inside of the shell, retaining the outside dimensions (pics 6, 7). (One of the frames is diagonal and I used a construction plane for that one.) This re-expansion was only barely visible on each segment, but over them all it added up. Then I did the loft (pic 8) and mirrored it to create the shell, boolean unioning the two pieces (pic 9). I copied the spine and hinge peg objects to retain them for the next smaller segment (hiding the copies), then unioned to the shell (pic 10). I repositioned the spine hole to fit the new size of the hole placer, keeping it the same size (pic 11). Finally I cut the spine hole and hinge holes, retaining the cutting objects (pics 12,13). I could then export an stl for printing, copy the file to the next smaller segment, and repeat.
Step 7: All the Pieces, First Try
I made the following changes:
- made all the segments a little shorter to allow the curling to be tighter
- redrew the head completely
- flanges on legs
- redrew tail segment
Step 8: All the Pieces, Second Try
I reprinted the redesigned head and body pieces in the translucent grey "Vero" resin, and the flanged legs in the white "Vero", plus black eyeballs. There was improvement, but I now noticed the neck area problem, where the Curl-up could not hold its head up hight enough to match its portrait. There were also some issues with the hinges, where a few of the segments' hinge holes broke and did not hold the pegs. Oppositely, the pegs on segment 17 were too long and the tail segment wouldn't fit on at all.
I modified and reprinted the neck segments and a couple of the interior segments which had failed hinge holes. It is now very close, I think, to a Curl-up that Escher might recognize!
Step 9: Done!
Wentelteefje FTW! I took some pics in House of Stairs type attitudes just for fun...
I may push this project a bit farther as there are more improvements I can see, such as adding a squishy "brain", improving the neck reverse bends, maybe just maybe if I get really ambitious, adding a motor to make it curl up automatically... Either way, I think M.C. Escher would approve.