Crease Pattern Discussion

[steyen]

hi, i would like to start my discussion of crease pattern with two questions
:
1) Given a crease pattern of a model, will there be only one way of collapsing it?
2) is a crease pattern unique or degenerate? (one crease pattern can result in many many bases)

Thanks

[stuckie27]

1. No a crease pattern can be colapsed more that one way but usually there is only one right way
2. a simple example of proving this wrong would be to look at the crease pattern for the the square base and the triangle base. If you arrange the mountain and valley folds in opposite order you will get the other base.

[TheRealChris]

1) Given a crease pattern of a model, will there be only one way of collapsing it?

I think, every CP can easily collapsed into a nice an habile boulder
you may probably recognize, that crease patterns are a eternal riddle to me :/

2) is a crease pattern unique or degenerate? (one crease pattern can result in many many bases)

though origami is math, I can't imagine that one CP can produce two different models, as long as you follow the CP including mountain and valley folds exactly. If you tell someone the way to your home, he also can't come to a wrong place, as long as he follows the instructions and the instructions are clear.


Christian

[steyen]

i see---
thanks very much--i was so eager to know because i am on my semester holiday now,and i am studying crease pattern by folding models with diagrams. Then i will unfold them to study the projection of creases.
I realised one thing, that is : i find it hard to recognize creases that run through a few layers, means that i could hardly differentiate them. For example, a crease pattern of a bird base can be folded as the usual bird base, or when you fold the paper in half diagonally before starting to fold, all you will get is double rabbit ear.

I would like to ask for some tips about this--thanks!

[bshuval]

TheRealChris said:

though origami is math, I can't imagine that one CP can produce two different models, as long as you follow the CP including mountain and valley folds exactly

I disagree. First of all, origami IS NOT math. Yes, it has math-related aspects; but, it also has many other aspects that are not math.

Second, the SAME crease pattern can produce different models. I know of an example of a 3D tension folded model that can be formed in two distinct ways, to have two distinct appreances, and have the SAME crease-pattern (including gender of creases). I have other example I can think of, but I need to check that their creases have the same gender.

[TheRealChris]

First of all, origami IS NOT math.

B L A S P H E M Y Y Y Y Y Y

...but, it also has many other aspects that are not math.

so tell me just three of those many aspects.

Second, the SAME crease pattern can produce different models. I know of an example of a 3D tension folded model that can be formed in two distinct ways, to have two distinct appreances, and have the SAME crease-pattern (including gender of creases). I have other example I can think of, but I need to check that their creases have the same gender.

that's absolutely wrong.
to transform a model, you either need to change the type of a crease (mountain to valley... what you called the "gender") or add new creases. even if you just bend a paper, it's nothing more than a lot of very very soft creases.

it's impossible to change a model without changing something. and in the moment you change ANYTHING, it will produce a different crease pattern (even if it would only be a very little crease somewhere in another crease.

so why don't you just give a concrete expample instead of a loose indication?


Christian[/quote]

[wolf]

Origami is origami. Math is just used to describe it, just like it can be described in English or Klingon or whatever.

Once you get to 3d, CPs aren't unique anymore. The way they're drawn, there is no way of telling at what angle you fold up the paper along the axis of the crease. A valley fold is just a fold with an angle of between 0 and 180 degrees around the crease axis, but turns into a mountain fold from 180 to 360. And all this can be done without bending the paper either.
So unless you describe the angle at which a particular crease occurs around the crease axis, CPs are not unique.

Simplest trivial example is a sheet of paper folded in half - you can fold it completely in half so it's 2d, or open it up so it's a V-shape.

[TheRealChris]

Origami is origami. Math is just used to describe it, just like it can be described in English or Klingon or whatever.

I had this discussion over and over again. origami is "what happens, if I fold a piece of paper in a special angle into a special direction and then fold another flap..." just a complex way of Flächenlehre (I don't know the english term for that).... dividing paper into angles and so on. this discussion always stoppes at my disability to describe it in english :/

Once you get to 3d, CPs aren't unique anymore. The way they're drawn, there is no way of telling at what angle you fold up the paper along the axis of the crease.

that's half true, because you need to combine ALL THE CREASES, and not only a single one. of course a crease can be done in many different angles, but if you're doing it in the wrong angle, you won't be able to do one of the following creases without producing new creases that are not in the original crease pattern. the paper has to collapse into a model. at a special point you may have many tiers (?) of paper above each other, that have to be folded into one direction. If you may have done one former fold in a wrong direction or angle, you may not have enough paper, or the creases aren't at the right place.

Simplest trivial example is a sheet of paper folded in half - you can fold it completely in half so it's 2d, or open it up so it's a V-shape.

wow, you just tried to describe math with prime numbers
but a single crease is not a real crease pattern... and please don't start nitpicking. we all know CPs, and we're talking about more or less complex models, and even the smalles model will produce a multiple crease pattern.
try it: make two creases and try to collapse it into different models. IT's IMPOSSIBLE. try it, and you will see, that a CP is unique. make a couple of more creases and you will see, that it's impossible to create different models without producing new creases or without bending the paper into a direction. try it ! oh wait: in the moment, when two creases CROSS EACH other, it's impossible to create different models (just to stop you from creasing one crease beneath the next!).


Christian

[wolf]

just a complex way of Flächenlehre (I don't know the english term for that) .... dividing paper into angles and so on. this discussion always stoppes at my disability to describe it in english :/

Planar/Euclidean geometry. Go ahead and describe it in german, I can follow.

wow, you just tried to describe math with prime numbers
but a single crease is not a real crease pattern...

No, all I need is one counterexample to disprove a statement.

oh wait: in the moment, when two creases CROSS EACH other, it's impossible to create different models (just to stop you from creasing one crease beneath the next!).

Counterexample: The preliminary base CP - in 3d, there's two distinct ways of folding it. The CP has 4 creases, all of which intersect at the centre of the square. The two main diagonals are valley folds while the vertical and horizontal book folds are mountain folds. You can fold this one way to get a waterbomb base, where all the flaps are at right angles to each other. If you now "turn the paper inside out" but still fold with the same sense of creases, you get a preliminary base where the flaps again are at right angles to each other.

[TheRealChris]

Go ahead and describe it in german, I can follow

it's very unpolite to those, who don't speak german, so I will just give a small answer:
es geht beim Origami primär darum, Flächen (eben das Papier) zu unterteilen, denn jeder Knick ist nichts anderes als eine Unterteilung des vorhandenen Papiers an einem bestimmten Punkt in einem bestimmten Winkel. Reine Mathematik... gesteuert von einem kreativen Grundgedanken.

No, all I need is one counterexample to disprove a statement.

yes, but until you don't find a good one, you don't disaprove anything, don't you?


your counterexample is completely wrong, because you obvously learned to fold those bases with each one extra unneeded crease (for comfort?). look at this, I have drawn it with all the needed creases, and they are different CPs.
http://img78.exs.cx/img78/6898/bases.jpg
try to make a preliminary base from a waterbomb base without adding a crease. impossible, you see?

go ahead, another example.


- Christian

[wolf]

No, my example still holds because I'm describing a three dimensional object, not a flat folded one. What you've shown there is just the flat folded bases - the CPs for those are unique, because one crease is unnecessary.

However, the 3d waterbomb and preliminary bases need that extra crease. And once you allow the crease angles (the dihedral angles) to take on any value (not just 0 for flat folded ones), the CP is no longer unique. In the 3d preliminary base, the diagonal folds have 90 degree dihedral angles while the book folds are at 0 degrees. For the 3d waterbomb base, it's the other way around - the diagonal folds have 0 degree dihedral angles and the book folds have 90 degrees. So, the sense of the creases are the same in both 3d bases, just not their dihedral angles.

[OrigamiMagiro]

I think we've established plurality for 3 dimensional models.

So, the questions now arises, are flat foldable cps unique? The definition of unique must be called into question. I believe I have a counter example to this question which consists of a model that has two or more different possible final (flat) structures, but it depends on how one defines "unique."

I will be drawing up my counter example on freehand as I wait for a reply.

[wolf]

...but it depends on how one defines "unique."

Right - there's always the trivial example of overlapping flaps.

I think a starting definition for uniqueness for flat folded models would be whether the silhoutte is unique; ie the uniqueness of the polygon outline of the final flat folded base. Naturally, reflections and rotations don't count!

[bshuval]

Jason, you took the words out of my mouth. I thought about this most day at work today. However, I do not think that we have a problem with the definition of unique; this is easy: a crease pattern is uniquely flat-foldable if there is one, and only one, distinct model it flat-folds into.
Our problem is, in fact, in "distinct".

(NOTE: for the rest of the discussion, any reference to an origami models references a flat-foldable model)

Continuing with our logic, two origami models are distinct if they are not identical. So, what makes two origami models identical? First, we must realize that two models can be identical even if folded from different types/sizes of paper. So, our definitions need to take that into consideration. In the sequel we will assume that the models are folded from identical pieces of paper (and this time, there is no problem in defining what identical is)

One definition could be that two origami models are identical if they have the same crease-pattern. This, of course, really defeats our entire question, but is a valid definition.

Another definition would be that two origami models are identical if they "look the same". This, as Jason remarked, does not take into consideration the inner arrangement of flaps in the model. A trivial example would be making two overlapping folds on a square. This results in two edges of the paper, one lying on top of the other. Without changing or adding any creases, one can change the order of the edges. This does not result in a different model according to this definition. However, our intuitive understanding of 'identical' says that these models are NOT identical.

A third definition would be that two origami models are identical if they are indistinguishable by a third party. Imagine I had an oracle. I would give that oracle two origami models, and he would find their differences. According to this definition, the above layer-ordering example is sufficient to disprove Christian's conjecture.

Here are some related questions to ponder about
(1) Under what conditions does a crease-pattern fold uniquely?
(2) What is the maximum number of distinct models a crease pattern can fold into?

Our discussion above also answers the following question: Can two identical models have distinct crease-patterns?
The third definition of identical answers this question trivially. The second definition leads to the answer "yes". Fold any model, the crane, say. Now, blintz the square and fold the same model. Presto! Two models that "look the same", but with distinct crease-patterns. The first definition requires no discussion.

[TheRealChris]

@Wolf
nice, let's go on nitpicking. your 3D-waterbomb-preliminary-example is still wrong, because you change the gender of the tip. if we would draw a really exact CP, we would realize, that the tip of the base is a accumulation of very little creases; and if you change a 3D preliminary to a 3D waterbomb, you change the gender of the tip (turn it inside out). that produces (in a nitpickers definition) a different crease pattern. if you would cut the edge off, we wouldn't have any crossing creases.

now this discussion get a very weird touch. just to disaprove something you're coming with funny definitions like inside and outside identity. I don't like to follow this discussion in that way, because that's a waste of my time.
just another few words:
if you bend a paper, it's a accummulation of very soft creases. so you'll never get the same CP twice, even if you will start to hide paper to force a disapprovement.
nobody but Wolf gave a example. you only talk and talk and talk. so why don't you give me another example.
"exceptions prove the rules", so don't give me special cases to tell me I'm wrong. I know that there will always be special cases (that's what I ment with "to describe math with prime numbers"). and please stop those a- couple-of-creases-examples and start with real CPs, because that's what we are talking about.


Christian