aeh5040

Pen-Dragon Mechanical Fractal-Drawing Device, version 2

Recommended Posts

Hi all,

Just over a year ago I presented version 1 of this MOC. It is a purely mechanical device that draws a fractal curve - I believe it is the first time such a thing has ever been done, in Lego or otherwise. It uses a single L-motor, a series of Geneva mechanisms, and a stepped transmission mechanism to draw a Heighway twin-dragon, a space-filling curve that never intersects itself. (No Mindstorms here!)

One problem with version 1 was that the final curve was huge - about 2m by 3m. Consequently I never found a big enough piece of paper and enough space to run it to completion - it just wasn't practical.

In the new version I have managed to solve this problem. I filled the one remaining space with an add-subtract mechanism comprising two differentials. The previous version made a 90 degree turn by turning one wheel by a full turn while holding the other fixed. In the new version, one wheel makes 2/3 of a turn while the other makes 1/3 of a turn in the opposite direction. Consequently, the final drawing is 3 times smaller, about 0.7m by 1m. In addition, I redesigned almost every part of the mechanism, and made it far more robust and reliable. I think it is now at a stage where it would be reasonable of other people to make it - I am working on an LDraw file...

img_2429.jpg

The full drawing takes 256 turns and just over an hour to complete. It is accurate enough that the curve never intersects itself, and is within a few cm of joining up to itself at the end. Not bad for pure mechanical dead-reckoning! Here is the actual drawing, compared with a mathematically perfect computer generated version.

img_2426.jpg

dragon.png

img_2456.jpg

img_2441.jpg

More pictures and LDraw file on brickshelf and bricksafe

Edited by aeh5040

Share this post


Link to post
Share on other sites

Now that is one cool machine. I remember seeing your original version at BrickCon a couple of years ago, and this one looks even better. How many individual Geneva mechanisms are there? How sensitive is it to initial conditions? Can you start it at any point in the plot? I imagine that the relative position (clock angle) of the various Geneva mechanisms has to be just right when the model is assembled in order for the sequencing to work.

I look forward to studying this in much more detail.

Share this post


Link to post
Share on other sites

That's an amazing piece of work. Can I ask if you worked it all out first and then built it, or did you get building in stages and just keep resolving issues

Share this post


Link to post
Share on other sites

Now that is one cool machine. I remember seeing your original version at BrickCon a couple of years ago, and this one looks even better. How many individual Geneva mechanisms are there? How sensitive is it to initial conditions? Can you start it at any point in the plot? I imagine that the relative position (clock angle) of the various Geneva mechanisms has to be just right when the model is assembled in order for the sequencing to work.

I look forward to studying this in much more detail.

Great questions. There are 5 Geneva mechanisms in sequence, on 6 axles. Each axle has a cam at each end.

It is not sensitive to initial conditions - it can be started at any point, and it will just start at the appropriate point in the drawing (it is one closed loop). Since it is essentially doing binary counting (more precisely, Gray code), it cycles through all possible states during the loop.

Of course, if you want the drawing in the center of the paper and with the desired orientation, you do need to figure out where it is starting on the loop! But it is easy to advance the Geneva mechanisms to the desire position by hand as in the video.

The set-up of the Geneva mechanisms is also quite straightforward. It requires two things: (1) Each axle has the two cams at the opposite ends pointing in opposite directions (180 degrees from each other). (Except that the last axle has a "triple cam" on one end, with actuators in 3 out of the 4 positions - the cam at the other end is in the 4th position). (2) The cams are parallel with the two red pegs that activate the next Geneva mechanism. This ensures that only one cam at a time actuates the side bar.

Edit: what I said here was not correct - the setup of the cams requires more than this. See the later posts in this thread for corrected information.

The design of the Geneva mechanisms does involve a sneaky 45 degree offset in order to get the blocking "macaroni" pieces in the right place relative to the red pegs. This is achieved by two 1/2 pins slotting into the stud holes on the underside of a round plate (I was pleased with how that worked out).

That's an amazing piece of work. Can I ask if you worked it all out first and then built it, or did you get building in stages and just keep resolving issues

I did start with the basic idea of what it would do ahead of time, but the actual build involved dozens of iterations for every mechanism, and took several years! Every detail has to be right for something like this to work reliably...

Edited by aeh5040

Share this post


Link to post
Share on other sites

Excellent work! I'm looking forward to the ldraw file, it will be quite educational. I'm amazed how close the start and finish points are.

Do you have any other fractals planned? Can this machine be adapted to draw them, or is it a completely different build?

Share this post


Link to post
Share on other sites

Amazing!

I likem, that You just didn't thought about functionality, mechanisms etc., but it also looks very good as object.

Share this post


Link to post
Share on other sites

A mechanical marvel!

Now we just have to wait for an in depth review by Blakbird!

Share this post


Link to post
Share on other sites

This is awesome on so many levels, thanks for showing us such a unique creation. Technic rules!

Share this post


Link to post
Share on other sites

Fantastic! The gap at the end of the sequence really is tiny in the scheme of things. Great to see something so different.

Share this post


Link to post
Share on other sites

And one more time someone proved that nearly anything is possible with LEGO Technic!!!! Very well done! I am like super-impressed by all the mechanics and the result is very good when you think about how "imprecise" Technic is.

Share this post


Link to post
Share on other sites

Wow. Masterpiece of mechanical engineering. I love the precision it has I don't know much about these machines but I wonder, youldnt it be possible to modify it on order to get it drawing infinite fractal instead of closed loop?

Share this post


Link to post
Share on other sites

I am really impressed by the accuracy of the machine. Some mechanisms still baffle me. Can't wait to study this more.

Share this post


Link to post
Share on other sites

As remarked by many others, it is very precise. Maybe due to the fact that it goes left then right then left then right etc., the imprecisions cancel out? I.e. since there is only one circle more in counter-clockwise direction, there is no cumulative effect?

Share this post


Link to post
Share on other sites

That is one great bunch of mechanism included, I would definitely like to build one someday, and study.

Share this post


Link to post
Share on other sites

Simply mind boggling. This ranks very high up there, along with some of Akiyuki's GBCs and PG52's spirographe.

Now, without going much into the math behind fractals, the drawing is made up of a sequence of left/right/left/right turns of 90/180/270 degrees. The sequence is such that the curve never intersects itself ...

Edited by DrJB

Share this post


Link to post
Share on other sites

Another vote for an incredibly well executed machine. It's something i'd really love to build and is kind of the reverse (or upside down) version of the 8094 plotter that I still really love.

Share this post


Link to post
Share on other sites

Btw. I hereby vote this as an entry for the Hall of Fame!!!

Oh do you now.....why don't you use the official topic for suggestions?

Share this post


Link to post
Share on other sites

Oh do you now.....why don't you use the official topic for suggestions?

I knew I forgot something!

Share this post


Link to post
Share on other sites

Simply mind boggling. This ranks very high up there, along with some of Akiyuki's GBCs and PG52's spirographe.

Thank you very much for the kind words, everyone. This in particular is the highest praise I could hope for!

As remarked by many others, it is very precise. Maybe due to the fact that it goes left then right then left then right etc., the imprecisions cancel out? I.e. since there is only one circle more in counter-clockwise direction, there is no cumulative effect?

To be honest, I am pretty amazed by the accuracy myself (even though I put a lot of effort into precisely this). I have done a few computer experiments to try to understand the "cancelling out" a bit better. Basically, the "cumulative" angle of each turn is very very precisely 90 degrees. If it did 4x10=40 left turns in sequence then it would end up pointed in the same direction as before (after 10 complete rotations). If it did the same thing with turns of 89 degrees or 91 degrees, or random angles between 89 and 91, the result would looks quite poor (as confirmed by computer simulation). However, there is another source of "error" that miraculously cancels out - this comes from backlash in the gear trains. The result of this is that if it does a sequence of left turns followed by a right turn, this right turn will be less than 90 degrees (maybe 80 degrees - it is fixed a predictable amount). Then any subsequent right turns will be 90 degrees again. It turns out that this type of error does "cancel out" - the final drawing will still join up exactly and have the correct overall shape. (I haven't quite understood exactly why).

Wow. Masterpiece of mechanical engineering. I love the precision it has I don't know much about these machines but I wonder, youldnt it be possible to modify it on order to get it drawing infinite fractal instead of closed loop?

It's not possible to get the "real" infinite fractal with only a finite number of states. By adding one more Geneva mechanism, we would get the next approximation, twice as large, with 512 turns, and so on. (One or two more might indeed be possible in practice, but eventually friction would be too much, of course). By changing the arrangement of cams on the last Geneva axle, it would be possible to get an infinite pattern, but it would just be the same figure repeated over and over along a line.

Amazing!

I likem, that You just didn't thought about functionality, mechanisms etc., but it also looks very good as object.

Thanks! Every single piece is functional here - really the only aesthetic considerations were color choices, but I'm happy with how it ended up looking.

Excellent work! I'm looking forward to the ldraw file, it will be quite educational. I'm amazed how close the start and finish points are.

Do you have any other fractals planned? Can this machine be adapted to draw them, or is it a completely different build?

Ha ha! I'm not sure I want to devote another year or more of my life to a "completely different build"... I don't know whether this machine can easily be "reprogrammed" to do something else interesting - I'm open to suggestions...

BTW, the Ldraw file is nearing completion - it will definitely need some testers...

Edited by aeh5040

Share this post


Link to post
Share on other sites

BTW, the Ldraw file is nearing completion - it will definitely need some testers...

I'll test it!

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now

  • Recently Browsing   0 members

    No registered users viewing this page.