## Vertex assigned CP's

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Brimstone
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### Vertex assigned CP's

This topic arose in another thread of this same board but that topic went astray so there might be people that haven't read about it. I got excited about this topic and told it to some guys I have a CP studying group with (in Spanish) and I think it has even developed so I am starting this thread to see if people think that it might be a good alternative for the plain and usually complex CP's.

The original idea was from Wolf and it consisted of assigning a direction to vertices (convex or concave) and note it so in the CP using V (for concave) and ^ (for convex) signs near the vertices. These signs don't tell the folder what directions the creases have and I agree that sometimes you can't tell if a vertex is concave or convex, but still noting the direction of most of them helps a lot.

When dealing with things that can only have one out of two options no information is also information, so you can note only one of the conditions and assume that when no information is provided, is because the condition is the opposite of the one noted. This way I decided that instead of the two signs I would only place the convex (^) sign but soon after, I realyzed that this can be confusing when vertices are close. After working with the idea for a while, I decided that for major clarity and easiness for the drawer, instead I would put dots on the convex vertices.

As a sample please review the CP for Miyajima's rabbit.
CP: http://www.angelfire.com/co/cubo/rabbitcpc.html
Original CP and Picture: http://www.h5.dion.ne.jp/~origami/rabbit.html

Also review this CP done by these two Spanish guys of Tanaka's butterfly:
CP: http://www.angelfire.com/co/cubo/tbutterfly.pdf This CP is done "white side up" so be careful if you like to fold CP's as I do keeping the lines on the outside, you have to "reverse" the convention, ^ for concave and V for convex.

Original CP and picture: http://www.h5.dion.ne.jp/%7Eorigami/shi ... erfly.html

Other Pictures:
Photo 1
Photo 2

Some URL's are quite long so make sure you got the whole thing or you'll get an error message.

Go ahead, try to fold the CP's, cheat using the vertex assignation and discuss if you think this method could be of some help. I personally find it quite useful.
wolf
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Cool - it's a CP I've never done before, so I'll give it a shot, both with and without the vertex assignment (as soon as I have time - HA!)

I prefer the dot method of marking vertices; I think the ^s clutter up the CP too much. Actually, using a coloured hollow dot might be better than a filled dot because you can then see the vertex itself.

Anyway, I'll post my thoughts once I've completed the CP.
wolf
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Ok - I've never finished a CP so fast before.

For this CP, it's particularly useful when collapsing the region where the wings meet the body. Once this part is in place, everything else can pretty much be done with standard folding methods.

Although using ^s and Vs are more visual, they are somewhat confusing - especially when you turn the CP upside down! Having dots would eliminate this problem; also it's easier to distinguish dots of different colours than thinking about ^s and Vs.
Brimstone
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wolf wrote:Ok - I've never finished a CP so fast before.
Exactly. thats the idea. Even though this is not a mega complex CP, the vertex direction indications help a lot.

I agree that the dots are more helpful than the ^and V signs. Those get confused when you turn the paper and you forget from what side they were supposed to be read.

So Wolf I think that your idea was GREAT. Where did you get it from?

And people it would be nice to hear your opinions on the method.
wolf
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Brimstone wrote:Even though this is not a mega complex CP, the vertex direction indications help a lot.
Again, especially for the central region. I've always found that CPs with lots of mini bird bases in the centre are really annoying to collapse, because it's often not obvious how the vertices should point (ie whether they lie on the outside of the model, or are hidden within the model). I usually end up tearing the paper when flipping the orientation of these vertices.
Brimstone wrote: So Wolf I think that your idea was GREAT. Where did you get it from?
From the original CP thread, of course. Plus my own observations after countless failed CP attempts...
steyen
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vertex assigned cp is a very new and brilliant idea!

this will be a new revolution of origami crease pattern and i think it will ease the folding of CP's in the future!

count me in this cp analysing group!im interested to contribute
Brimstone
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steyen wrote:vertex assigned cp is a very new and brilliant idea!

this will be a new revolution of origami crease pattern and i think it will ease the folding of CP's in the future!

count me in this cp analysing group!im interested to contribute
It is good to hear this. Some people have told me they don't see much advantage on them but I am possitive they can be of great help.

By the way do you think this would be the correct name for this method?
wolf
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I think that this will benefit CP beginners more than anyone else. If you already know how to do complex CPs, then probably it doesn't matter whether the vertices are marked or not. It'll be interesting to see how VACPs compare to PCPs (I don't see too many of those either). It's also less work to do a VACP than a PCP.

Anyhow, I might whip up some VACPs when I find the time to do so.
titan101
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I foled those 2 models by new CPs, but the rabbit is too easy and the butterfly is too confuse to use the mark itself. but I like the point(using at rabbit) idea, it can be a clue at the first challange.
Korean origaminist, jassu
My blog http://blog.naver.com/titan101.do
steyen
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hello i am very interested in this vertex assigned cp!
what is the vertex assignment for this cp?

http://www.asahi-net.or.jp/~qr7s-kmy/g/ ... enryu.html
OrigamiMagiro
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Here is the vertex assignment for Satoshi's Ancient Dragon type2 as I have read it:

The notation I have used is blue for "mountain-like" or "convex" vertices (vertices that stick out towards the observer) while red indicates "valley-like" of "concave" vertices (vertices that point away from the observer). There are two 22.5 deg central lines that I have added to the crease pattern to make it flat-foldable which are marked in green. Hopefully this may help you. Good Luck!

Also, I would like to put forth that there might be certain theorems embedded in such a vertex assignment as analogous to Kawasaki and Maekawa's theorems are to crease gender assignments. As of yet, I have found no general pattern for the distribution of the gender of the vertices as I believe vertex gender may not be helpful in understanding structure, but may indeed help some people fold crease patterns.

Thoughts?
wolf
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I agree that the vertex assignments are currently more folding aids than design ones. It does make it easier to visualise how the base collapses together though, without actually having to try it out on a sheet of paper.

Mathematically, I think the vertex assignments can be treated as node networks (like in graph theory), where the creases emerging from a vertex are the lines which connect various nodes together. With this, I can make the following observations, assuming that the eventual base is flat-foldable:

1) For every node, the genders of the nearest neighbour nodes cannot all be the same as the gender of the node itself. So a mountain node cannot be connected to all mountain nodes only, etc. This is really a trivial case.

2) For every node, count the number of mountain nodes and then count the number of valley nodes - their difference is either zero, or an even number. I suspect the reason for this lies in Maekawa's theorem.

The nodes mentioned above are those within the paper, not nodes at the edges of the paper (which are probably special cases).

Proof is left as an exercise for the reader.
steyen
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i will try it!

p/s: have anyone been ordering "Works of Satoshi Kamiya"? when will it be out?how to order?
OrigamiMagiro
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wolf wrote:I agree that the vertex assignments are currently more folding aids than design ones. It does make it easier to visualise how the base collapses together though, without actually having to try it out on a sheet of paper.
I actually didn't fold this model, but I pretty much know it works (except the cp leaves out a few lines that would otherwise make the base really flat-foldable). Please inform me of any error I might have made so I may change it (if it exists).
wolf wrote:1) For every node, the genders of the nearest neighbour nodes cannot all be the same as the gender of the node itself. So a mountain node cannot be connected to all mountain nodes only, etc. This is really a trivial case.
Trivial, eh? I thought about such a theory, but I think the nodes in the dragon are just a common exception. Take such a cp, valley/mountain on right, vertex assignment on left:

[img]http://www.bluegoo.net/~jason/demo.jpg[/img]

Here, the center node and all nodes connected to it are all pointing out (in the same topographical direction to the paper). This seems to be a counter example to your theory.
wolf wrote:2) For every node, count the number of mountain nodes and then count the number of valley nodes - their difference is either zero, or an even number. I suspect the reason for this lies in Maekawa's theorem.
Again, I think this happens as coincidence. It seems as if the proof left to the reader has its flaws in possibility, but I do enjoy the discussion.
bshuval
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First, I think that Jason's example is waaaaaay too complicated. I don't see how it adds anything more than an example of a simple bird base.

Second, I think that before we begin discussing theorems and such, we should first firmly define what a "mountain" vertex and what a "valley" vertex are. (I use these terms and not "convex" and "concave", because many people define convex and concave differently. A convex function looks like a 'U'...). A vertex is a mountain one with respect to what? Obviously, an assignment of m-vertices and v-vertices is relative to something. But what? We need a firm definition... And I am not sure that there is one...

Third, not all vertices can be designated m or v... Think of a simple twist. The vertices are as much m as they are v. An even simpler example is a kite fold...

Alone, a vertex cannot be designated as m or v. Kawasaki's theorem and Maekawa's theorem resulted from analyzing CPs from a vertex perspective. We are trying to analyze CPs now from a crease perspective, something that I fear will prove useless.