How do you plot circles onto a square of paper?

[ftangdude55]

Well, as some of you may know, I have recently started to design original origami figures, and my most recent design, was created via circle packing. For this, I used a cardboard circle jig.

But, I was curious: Is there any other way to plot the circles on a square piece of paper with equal accuracy and effectiveness?

So that is my question to you: In your experience, what is the best way to plot circles on a piece of paper? If you do have a method other than a circle jig, please feel free to explain it.

I hope that this topic will end up being a good place for new designers to learn techniques on how to circle pack.
I also hope this topic doesn't end up locked because there is another topic out there that is similar, or has already covered this issue, but as far as I know, this is the only topic that focuses directly on how to put the circles to the paper.

Thank you in advance,
Ftangdude.

[Lux Aeterna]

You could use a compass... Or am I misunderstanding the question?

[Zoraz]

Well, the only other sane method I can think of is to use treemaker. Most of Lang's models that weren't designed with treemaker have most of the circles the same length, and have fairly simple point arrangements, or are boxpleated. You could try to learn all the algorithms and geometry/trig/higher level math to plot strange sized circles on your own, but personally I don't think it is worth it.
Another method I just thought up: You could pack squares first, draw circles inside them, then expand the ones that can be expanded. (Always easier to shorten a point than to lengthen it!). I have never tried it, nor do I know anyone who has, but it's worth a shot!

[origamimasterjared]

Yeah Lux, you are.

As much as I hate to say this, your question screams Treemaker. That is exactly what Treemaker does. It packs circles. It is a digital circle jig.

You can also draw it / estimate it free hand.

I'm not a fan of Treemaker'd designs, but I have used it once or twice successfully. What you can do it use Treemaker to find the "optimum efficiency base". Then try to approximate this arrangement using nicer folds (22.5 or 30 degree geometry). Robert says you can tell Treemaker to force creases to lie at a certain angle, but I never figured it out.

[ftangdude55]

Well, that answered my question (kind of), Jared. Thanks.

[origamimasterjared]

ftang, did you read my post?

Also, important!

DO NOT MAKE THE CIRCLES THE MEASURED LENGTHS. You NEED to make legs, and generally anything that will endure any detailing longer, as the flaps will become shorter in the process.

Edit: Looks like you deleted your long post.

[ftangdude55]

Okay, that makes sense. I'll try that.
But really, all I wanted to find was an alternative to a circle jig. I was hoping Treemaker wouldn't get involved.

[origamimasterjared]

I don't know many people who really use circle packing. I think Jon Tucker might. Derek McGann and Nguyen Hung Cuong use Treemaker when they do it I believe. Jason used to do a bit of circle packing, pencil and paper I think. I think Chad Killeen uses a form of circle-packing--maybe he can recommend something (though again I think he uses pen and paper).

You can also make a bunch of circles in your drawing program and try packing them. That might work.

I don't use circle-packing. I'm more interested in the shapes of the flaps and elegance of the design and structure, so I use a "holistic" style of design.

[mike352]

I was hoping Treemaker wouldn't get involved.

Treemaker is quite an easy program to use, so I wouldn't be afraid of trying it. Like Jared said, it is essentially the same thing as using a circle jig, but it's digital. Also, I imagine it's easier to keep things symmetric in Treemaker than using a jig.

Things that are tricky with working with the program are forcing it to have 22.5 degree angles, and selectively expanding certain circles. I would imagine, though, that these two things are beyond practicality with a jig. For example, expanding a circle is non-trivial. With some practice and reading through the manual, both these issues can be tackled - I've done it in the past.

What I've found useful for keeping things at 22.5 degrees is to work with well known structures from known bases, and use grafting techniques to assemble a model. For example, use part of a frog base for a head, part of a bird base for legs, etc. I've been able to make quite complex designs this way, keeping everything at 22.5 degrees so that also a linear sequence of folding can be found.

[WhisperPuffin]

Like Jared said, it is essentially the same thing as using a circle jig, but it's digital. Also, I imagine it's easier to keep things symmetric in Treemaker than using a jig.

Treemaker is actually a lot more than that. You can specify the relative lengths of flaps, and then optimise them so that the flaps are the most efficient possible. I think. But I always have problems with that, so I just use it as a circle jig. Dr. Lang, if you still browse this forum, could you clarify this confusion?

[mike352]

You can specify the relative lengths of flaps, and then optimise them so that the flaps are the most efficient possible. I think. But I always have problems with that, so I just use it as a circle jig.

When using a circle jig, you're also specifying relative lengths. Whatever the size of the smallest circle, you're making all the other circles ratios of that smallest size. Then, when you pack them in as tightly as possible using the rulers, you're finding the most efficient packing. So, the two methods are doing the same thing. It's much easier, as I mentioned, to impose symmetry conditions and make creases be at multiples of certain angles using Treemaker, though.

If you have particular questions about using Treemaker, perhaps I can answer them.

[Jonnycakes]

Mike is right about Treemaker-but Treemaker can do rivers, too, and (correct me if I'm wrong), a circle jig can't.

Jared, I do use circle-packing, but only as an initial step in a design. For me, circle packing is mostly a tool to decide the basic flap arrangement of a design. After that point I tile molecules or polygons (polygon packing) or use other techniques to fill in the actual creases. That way the design is generally foldable and I have control over what flaps and structures look like in the end.

[origami_8]

For rivers in your circle jig you could sew together some equally long strips of cardboard. But leave a gap between each part, it should look like this -I-I-I-I- (the horizontal lines being the string), so it remains flexible.

[Jonnycakes]

Is it really worth it though? I do just fine with pencil and paper or Treemaker (on rare occasions). Circle packing is a limited design tool, and I recommend against relying much on it.

[origamimasterjared]

I thought so Jonny. Something about your Spiny Orb Weaver told me, "yeah, this guy must use circle-packing. At least a little."

Also, Mike, I'd like to see some of your successful Treemaker'd stuff. Especially the 22.5˚ variety.

I've used Treemaker once or twice to get an idea of how relative arrangements of flaps might work--for instance, would it be better to have flaps of set A come from the interior or the edge, what about set B, etc.

Also, how to apply Treemaker to non-square polygons? Nguyen Hung Cuong used this on some rectangles for parts of his lucanus stag beetle.