Circle Packing

[FlareglooM]

Title says it all: I’m trying to learn how to make circle packings. I get the idea, a flap = circle, so draw up some circles with the correct radii as the length of the flap, non overlapping, connect centers of circles that hit each other (the axial creases), fill with tiles and voila a cp that folds into a base in the direction of what you want.

Drawing up some circles isn’t hard, but for some reason drawing up the axial creases gives problems.. (probably the definition confuses me or not). I’ll explain my thought and hope someone can help out (I’m sure you can!)

(Note: I left out all the axial creases on the edge of the paper, which connect the center of the circles)

This is a random circle packing I drew up with a small river on the top right corner. (I know the river isn’t that smooth but it is about the idea)

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Well my first question, do I need to fill up the center space (for example with another circle) or can I just leave it empty and draw up the axial creases.

If I leave it empty I would say that I could connect the center of the following circles (paired by color).

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But they aren’t considered axial creases since the circles aren’t touching each other or are they? But now looking at this there aren't axial creases in the square only on the edge, since none touch inside the square, only at the edge.

So, if I need to fill it up with another circle I can draw up something that looks like this. The red dotted lines are what I think are the axial creases.


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I do not need to make a line between the centers of the red circles since they do not touch, right? (same thought with the pic before) Though when I tile I noticed I make that line, but it is not an axial crease as far as I understand.
Furthermore the center circle must touch the river if I want to connect it’s center with the red circles and call it an axial crease?

If someone could bump me in the right direction it would be much appreciated.

[Sushy]

I don't know much of circle packing since I haven't tried myself, but I have read the section in Robert Langs Design Secrets.
As far as I know, it doesn't matter whether filling the middle with a circle or not. But in most times it is easier to do so, since it gets easier to fold. As you can see, you will get triangles, which you can fill with rabbit ear like folds. And the unneeded flap can be easily tucked in.
If you leave out the middle flap you'll have a pentagon, which needs to be divided into a triangle and a quadrangle to find a nice crease pattern. For quadrangles there are 4 different types to fill in these, but I cannot remember them, need to look them up.
Whether the middle circle needs to touch the river, well I don't know, but I think it might work as you draw it.

I know, I was not much of a help, but I can learn from this as well.

I hope you'll work it out.

[DavidW]

Well I think that the center circle is needed, the nodes for those big circles on the tree diagram would cross creating the node that corresponds to the center circles. It's just like a bird base, you can't have the four corner flaps without the center one.

Anyway I think that the packing is a no-go right now since the space isn't filled. Either two small circles need to go in the gap (between the river and the center circle) to complete the packing or the two big circles on top left and bottom right need to shrink to accommodate the river (which needs to be larger).

[Jonnycakes]

You guys are on the right track. Axial creases absolutely cannot run through empty space. The circles/rivers need to be touching where an axial crease runs through them. I would add the stub circle in the middle. Otherwise, as you said, the axial creases will just border the paper and you will need to use a nasty universal molecule to fill in the creases.

With the stub flap added in, you can slightly alter your circle packing to get something that works out rather nicely:



Here is the circle packing I came up with. Try to find the crease assignment yourself. If you absolutely can't figure it out, click on the picture to see a version with the creases filled in. Good luck!

[Sushy]

Hey nice work out jonnycakes. But what if he needs the river to have a neck/body to come up with?
Maybe you can draw the river down, so it touches the middle flap, than add another circle above the river, which touches the smaller ones and the river.
I would try it myself, but I dont know a good program for that right know. Need to read more the forum.

[DavidW]

It would be interesting to know what the aim is. It seems as if a standard splitting for that top corner would be easier on the whole design and just use pleats to slim the middle flap if it needs to be smaller. That way it just becomes a bird base + splitting not unlike Lang's shells in Origami Sea Life, which is simple and elegant to fold.

Anyway I like Johnny Cakes packing, those interior flaps if unusued could be stretched or sunk to put the extra space back in.

[FlareglooM]

Thank you all

Jonnycakes, one small question left regarding the pattern. I tried to fill it in myself, this was my result:

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My steps:
1. First I drew the axial creases (the red lines)
2. I folded all the angled bisectors between the axial creases (this ofcourse includes the ones on the border of the paper)
3. Then I tried to make it flatfoldable, by using the rules that the creases around a point should be even and that the odd numbered angles should number up to 180 degrees and the even numbered angles should add up to 180 degrees.

That got me up to what is shown in the pic. The only difference are the blue lines. Without it, it is also flatfoldable if I have seen right. I haven't folded it yet, but currently I'm thinking it has something to do with the distinction of the small circle in between the blue lines.

A clarification would be nice, or maybe a reference in ODS if that is necessary.

[origamimasterjared]

The bird base strikes again!

It is flat-foldable without those creases. That is a flap though, and it's good to see that. Also, in that CP, that section is folded up, away from the long flaps.

[Jonnycakes]

Oops, I forgot the river

I was thinking of a 3 point split, which provides a river as well. It shouldn't be too much harder to get that river in there, though.

As Jared said, I was just lazy and filled in every possible crease. Some hinge creases won't be folded.

[garrasdecaiman]

those blue creases are tangential creases A.K.A. hinge folds
some will be needed and some not depending on your actual orientation of flaps.
although yes a pont splitted bird base is a simple option it is not the only way to make a base out of the circle packing, you can use an arrowhead molecule or a gusset quad.
also before crystallizing the crease pattern with the central circle in the birdbase position you can push the central point up into the river and crystallize there later stretching it to get the river back.
X

[FlareglooM]

Well the river in the example was just so I would understand that a bit more. I get the idea of a river, but I thought, might as well add that, since I never applied/used it.

I finally understand the concepts better the main problem was that my circles didn't always touch, but I did draw the axial creases.

I will first try to get comfortable with circle packing without rivers in it. If questions arise when I do want them added, you'll hear from me again .

Thanks again!

Small question arised since a new post appeared before I was finished with my post

Crystallizing = Making the circle packing "fixed"?

[Jonnycakes]

One last quick note: if you want to add in that river, you can alter that little square area in the top right. Instead of a fish base, replace it with a quadrilateral molecule (probably a gusset molecule) where the small circles are a tad bigger.

[garrasdecaiman]

yes crystaliizing means expanding the circles until they touch each other and cannot move further and are thus pinned in their places.
And rivers are just spacers to expand the area between the points, so you use them to add a chest area between arms and legs, or a neck division or something.
Lastly the cirles must touch except if you have added a river. If they do not touch they will shift around while you fold and you will have extra paper in the space between them.
hope it helps.

[orislater]

im trying to understand and learn circle packing. im not making anything specific right now but i wanted to try it out.
here is my packing

i know it's messy but have i done the axial creases right?

[garrasdecaiman]

Pretty much except the lost paper should be somewhat larger, it doesn´t matter if it cuts into a circle since the paper will be lost, so the triangles should run from the centers of the circles and not outside.
Everything else is well done!
keep at it, and collapse the pattern and try to make something of it eaven if it´s not succesful it will help you with the practice and the ability to read,write,build crease patterns faster
X
not to criticize you but your circles are terrible! jaja