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If you want to do it without mindstorms then this is a solution (skip to 0:53 to see the control, and 2:14 for the internal mechanism - sorry for awful video quality). I think I may still have instructions for this...

Here is by far my most ambitious project. It is a purely mechanical device which draws a fractal curve, driven by only one motor. So far as I know, this is the first time such a thing has ever been done in any medium, Lego or otherwise! Specifically, it draws a Heighway Dragon Curve, a space-filling curve that never crosses itself. It in fact draws a "twin dragon" which would in principle join up to form a loop, given enough time, floor space, and paper. (In practice, slight inaccuracies mean that it would not quite join up). I actually made this over a year ago (and it was the result of about a year of hard work), and showed it at BrickCon 2012. The long delay is partly due to the difficulty of filming it! The "brain" on the top layer consists of a series of Geneva mechanisms. These required many designs to get the motion smooth and precise enough - they also require a 45 degree offset. Each Geneva wheel is connected to two cams, one on the left side, and one on the right. The cams lift the two levers running along the sides in a particular sequence, related to Gray code (a variant of binary counting). When translated to left and right turns, this gives the Dragon Curve. The bottom layer contains two identical mechanisms controlling the two wheels, to enable precise turns. When one side is triggered by a cam, the wheel on that side needs to make exactly one turn. After many failed attempts, I was able to do achieve this using a variant of the "Tomy Armatron" mechanism (which was discussed here before), combined with a Scotch Yoke linkage. A worm drive ensures that the wheels are locked and cannot turn when not triggered. Enjoy! Here is what it would theoretically draw, after 256 turns, given enough space:

Thank you very much for all the nice comments! Here are a few answers to questions. JGW3000: I will indeed try to produce instructions at some point soon. The paper is actually an annoying issue. In the video I am using two rolls of width 3ft taped together. This is not really big enough to complete the entire 256-step curve, and it is already a nuisance to handle and to get flat enough. Really I should perhaps redesign it to make the drawing smaller, although that would require putting the wheels "inside", near the middle of the vehicle. Gilbert Despathens, captainmib: The basic idea of a mechanical fractal-drawing machine definitely came first, and a dragon curve seemed a particularly natural choice because of the simple description, and because it looks interesting even without too many iterations. The lego implementation was by far the hardest and most time-consuming part. The Geneva mechanisms came first. There are several designs out there, but none of them quite worked well enough here - they need to be very precise and reliable, and yet have little enough friction to be able to have 6 turn each other in sequence and still have power to lift the side-arm. I'm happy with the final design for these. The motion is silky-smooth, and the 45-degree offset was achieved in a neat way. The other very hard part was the mechanism for turning a wheel by precisely one turn when triggered. I tried many many things here, including many which looked good in theory but simply didn't work, and others which worked fine "on the bench" but not under load. One just has to keep on trying different approaches. The "Armatron" design does work, and is just about reliable enough, but this in one area that could probably be improved further. legolijnte, Mestari: As will be apparent from my youtube channel, I have a strange obsession with doing mechanically what other more more sane individuals would do with Mindstorms.... Philo: Yes, Wikipedia is your friend here - I have now added some links. You are right about the way it works: when the k-th bit of Gray code changes, exactly the first k Geneva wheels make a turn, and only the k-th one triggers the arm mechanism. Since each axle has two cams at opposite ends pointing in opposite directions, at the times when this happens it is alternately the left and right sides that get triggered. So we get a left turn or a right turn according to whether the Gray code bit is changing to a 0 or a 1, exactly as in the wikipedia description. The last axle is different - it has 3 cams on one end (at 90 degree offsets) and 1 on the other, in order to get the "twin dragon". TinkerBrick: Yes, I think a few other curves might be possible with minimal changes - haven't thought about this properly yet. Definitely with some more extensive redesigns... D3K: the motor is indeed working hard. It is better than the many previous designs where it would stall completely! Victor Kojenov: it was definitely BrickCon 2012 (I didn't make it to 2013)

Genius! I love it. Reminds me a lot of some of the creations of the legendary akiyuky, especially these:

Thanks for the (re-)post! This is by far my most popular MOC on youtube. It worked nicely at Brickcon - sometimes people don't see what it is at first, and then they suddenly realise! Of course, making it into a multi-digit counter would be nice, but requires some kind of of powered carry mechanism, which I don't really know how to do. (Or NXT, but that would be cheating in my view).

That's a great idea, and nicely executed. However, I suspect this construction would not be strong enough. The axles running through the special tires would have alot of bending force on them. And I suspect the radial axles might have a tendency to pop out as well. Can you find a way (within the demanding constraints that you point out) to support the tires at both ends?

I would just move the top connection point one hole to the right and the bottom one hole to the left. Then two 4L axles will exactly fit!

Amazingly, it turns out that the "impossible cube" CAN be made using 3L push pins, even with all of the bushed ends internal to the 5x7 frames, despite the fact that two such pins may not be inserted directly opposite each other (as mentioned earlier). All the relevant pin holes are filled, and no bending was necessary to assemble this! The method is left as an exercise...!

Wow, that's beautiful - I don't remember seeing it. These remind me of this , which the maker claims is impossible to open or take apart without the combination.I think a new kind of puzzle might be possible among these lines - models that CAN be taken apart (without pliers etc), but only if the steps are done in exactly the right order. If some of the steps involve some parts pivoting or sliding, it might be very difficult to discover the right sequence.

How about this? I have actually used this - it was essential for the rotating knot. I found that a crown gear gives a tighter fit (which I needed). I can confirm that the turntables can be taken apart easily without damage to lego or the user.

Incidentally, this can also be modified to give an "auto-reverse" mechanism (e.g. to make a LA continually expand and contract): connect the two outputs of the gearbox so that they turn in opposite directions, and put the differential _after_ the gearbox in the drivetrain.

I love these challenges! Here is a very reliable solution that does not rely on friction. (Bracing and compactness could certainly be improved). The basic mechanism is: motor -> one side of differential other side of differential -> selector gearbox with changeover catch differential housing -> spring-loaded axle with two stable positions -> "clutch" with large amount of slack -> changeover catch

Edit: scroll down for the improved version. They said it couldn't be done! Well, actually, no-one said that, but I had serious doubts whether it could work. After the rotating ring, I'm sure everyone has been thinking "that's very nice, but can it be tied in a knot?" I'm happy to report that the answer is yes! Enjoy!

Here is version 2! Found a way to make the anchor points much more rigid, greatly reducing the wobble that was present in the previous version.

Wow, that's amazing performance! The stripped-down chassis looks simple, but getting something to work this well is anything but.

Just a few things I've been playing around with...

That's awesome! This trike setup (2 steered wheels in front, 1 powered behind) is great for stability. Maybe you could use 2 wedge wheels + 3 pins in place of those crown gears to keep the wheels on better?

That is interesting. It's amazing what can be done with 3D printing!

Wow! Awesome work, Jennifer and Blakbird! Wonderful to see this incredible model and its builder reappear! :thumbup: Is there any chance we might see the legendary Compact Excavator resurrected at some point?

Wow, this looks amazing, looking forward to further progress! I imagine controlling it will be quite challenging!

Wonderful!

I would like to see a technic dolphin! You might take a look at mahj's Da Vinci flyer - the wings achieve an amazingly fluid-looking motion with only one internal joint:

Yes, the chain one was inspired by exactly this. (Arthur Ganson's other sculptures are also well worth checking out). I tried for a while to send the chain upward, like this, but found it too tricky. That's a great idea! I believe this could be made to work. There's actually very little friction. The more delicate point is that the internal ring of axles needs to be held rigidly in place, and to hold up under the weight. So adding too much more weight is not a good idea. But I think your "scissors" could be done! A few GBC ideas are indeed in the pipeline...

Awesome work, Burf! As a child it was my dream to own enough Lego to do this... Have you examined your axles after this? Is there any significant wear or bending?

Thanks all for the encouragement! Lost_In_Noise: Thanks! Actually there's no differential, just two back-to-back turntables. lego muppet9: A smaller one definitely should possible, but tricky. The technic frames seemed perfect for the purpose, which is why I chose this scale. Fitting everything in to the driving section at the bottom was also challenging, so making that part smaller would certainly require some thought...