Circle Packing

[Jonnycakes]

Drawing circles inaccurately can lead to circle packings that look right but don't actually work. It is good practice to draw your circles as accurately as possible (graph paper helps), because it gives you a better idea of what the ~actual design would look like. For example, orislater, your circle packing won't work the way you have drawn it due to the inaccuracy of some of your circles. You have the right idea with axial creases, though! (It looks like you missed just one between the 2 big circles)

[orislater]

yes i know it would not work because of my circles. and i drew that line after i took the picture. also can someone explain how to make a river? and what does a river do?

[garrasdecaiman]

a river is a separation between two parts of the diagram which is not a real point.
for example if you have a tree for a human figure you will want a torso separation between the legs and arms, so the torso if it goes around the leg points will be described as a river, sometimes the rivers will be just there theoretically since the points will be too long and you will have a river which is half absorbed by the points, meaning that the points will be used shorter than their full lenght.
So you can think about rivers as a middle graft on a tree, to make them clearer you can think about this, take a square and make a waterbomb base, now take a longer rectangle and make a waterbomb base, as you can clearly see the structure of the points is the same but the distance between them is different, the rectangle has obtained a large space between the two pairs of tips, that can be viewed as a river in your crease pattern, and if you add for example two triangular molecules on the long edges to get four tips on one side and the original two on the other, since the short sides have not changed thus their tips may not stretch further inside the construction, the space that remains in the middle is the river, in this case you could see it as a big point splitted tip, but the most important part of using rivers is that they can meander around multiple circles and points mantaining the distance between all of them the same and separating bunches of points.

I would post a simple picture but this inkscape keeps crashing on me and I don´t have a camera handy, but I hope this helps at least a little
X

[FlareglooM]

*Revives thread*

Currently I am trying to calculate circle packings. I can calculate most of it up to four circles with simple geometry. (so my max basically is the fish base base). The problem though is that whenever I try to do things like rivers, a couple of circles with different radii or more circles I fail.

So basically my question: Can someone point me in the direction of incorporating math in designing (circle packing and tree theory). Because today during my Thermodynamics class, Lagrange Multipliers came by and they seem like a wonderful tool to use in designing. Feel free to post relatively difficult math (second year of my physics bachelor so I should be able to handle things like differential equations/vector calculus/matrices etc)

Thanks if someone could help me out ! (If there is need for an example where I get stuck I can work it out)

[Falcifer]

I'd be interested in seeing an example, Stacy.
Even after reading the thread, I'm still a little confused about rivers, but when it comes to maths, I've only ever had to use basic trig and algebra.

An example of what you're trying to do and why you can't seem to do it would be good.

[FlareglooM]

Okay here is a problem which I managed to solve:
Removed Image

This problem however I cannot solve:
Removed Image

[Falcifer]

Thanks, Stacy.
I'll take a look (although I already took a look earlier, and used TreeMaker to help) and see if I can offer some help.

I will say that I never heard of Lagrange Multipliers until you mentioned them, so I'm sure they aren't necessary.
And I'm not much of a teacher, but I'll do my best to put something together which helps...

I did manage to solve the same problem with all the flaps and the river of the same length. Having different lengths might make things slightly more awkward (with extra terms, etc.) but it should be doable.

[Falcifer]

I've only had a quick look at the maths so far, but I noticed that my initial assumptions regarding the circle packing were very wrong.
As Jon said, inaccurate circle packing causes problems.
I fed the numbers into TreeMaker, which shows that flap G ends up on its own. It should be touching the other two upper circles.

As far as I understand it, you either need to increase the size of flap G so that it fills the space between the flaps F and H, or you need to add some more circles, to fill in the gaps.
I think that increasing the flap size is easier. It certainly keeps the maths simpler, as well as keeping the number of flaps the same.

Increasing the size to 7/6 works out. But then it's more than double the value that you want.

If you want to see some working and explanation, I can try to draw up something tidier than what I have. But I won't be until the morning.

[FlareglooM]

Thank you Falcifer for the reply.
Lagrange multipliers can be used to maximize/minimize function with more than 1 variable under conditions which are called constraints. I though it might be possible, but as I hear not necessary .

Increasing the flap length is a good idea. Even though it may be too long, I at least got a circle packing to start out with.

Furthermore some explanation would be nice, like how you approach these kind of problems (if possible with some math). Take your time though .

[Falcifer]

Here's my working; Page 1 | Page 2 | Page 3 | Page 4

I will say again that I don't have much experience with designing models, from crease patterns or otherwise, so I tend to use TreeMaker to get a better idea of how the circles will look when packed. Then I use that as the basis for my working.

I hope this helps somewhat. Feel free to ask is anything's not clear.
I assume that you have some understanding of the maths, so while I have probably expanded things more than I usually would, I haven't included much actual explanation for what I've done, but hopefully it's clear enough.

And finally, I'm not actually sure if the maths is completely proper. It's all correct and the numbers work out, but it's possible that I'm applying it in a way that I shouldn't. For example, I have made the assumption that the circles along the left and right edges are both touching the river. If this isn't the case, then the maths doesn't really apply (it does, but there should be en extra variable). But TreeMaker agrees.
However, I've done enough maths to know that even the wrong maths can sometimes lead to the right answer...

[FlareglooM]

Wow you did a lot of work there. Thank you for taking your time to explain!

Though I'm gonna read through page 3 when my mind is a bit more focused to it

[Falcifer]

I just realised that my Flickr Pro account runs out today, so you won't be able to see the original/full size images after today.
I have uploaded the images to my webspace, though, and edited to post above to link to them instead of to Flickr.

I also simplified the CP a little and even folded it. I don't have a photo of the collapsed base, but I did draw it up in Oripa. You can see Oripa's fold, as well as the CP and how to find the reference (the point where all the crease meet near the top along the vertical center) from TreeMaker. The method of finding the reference uses more creases than others, but it is the most accurate. Though, the extra creases probably cancel out the precision to some degree.

Measuring the length of the flaps in the real collapsed base gives flaps of 0.95, 1.52, 0.52 and 1.14, which would be about 1, 1.6, 0.55 and 1.2. This is compared to the desired lengths of 1, 1.5, 0.5 and 0.5.

[Jonnycakes]

Wow, that looks like some heavy math you're getting into...again, I am not familiar with Lagrange multipliers, or complex math for that matter (no math since high school calculus). Treemaker has options in place to optimize flap sizes as well as locations (for example, you can enlarge a flap until it bumps into other flaps)-this sounds similar. Looks like you are retracing Lang's steps-that sounds like a useful thing to understand.