Lines on a paper and the shapes they represent.
A crease pattern is a drawing of an unfolded base of an origami model where the positions of the creases are marked with lines. They can be both a blessing and a curse, as they are simple and fast to make, but somewhat of a challenge to decipher.
To go about explaining this I will first attempt to impart some information on the different kinds there are, and how they compare to other ways of spreading origami models, then to show you a practical example of how this stuff works. For the latter part you will need one folded bird base1, which is one of the traditional origami bases, and a pencil.
Before I begin, I ought to extend a word of warning, as this write-up is intended to serve as a fairly brief introduction to the subject. A full exploration would be beyond the scope of a single write-up2 .
On the documentation of origami models.
There are two ways you can teach someone to fold an origami model without meeting them in person, one is to send them a diagram of the model, the other is to send them a crease pattern. And while they both seek to accomplish the same task, this is also where the similarity ends. To sum it up in an oddly unrelated metaphor: A diagram is to a crease pattern as being lead carefully down a mountain trail, stopping to take in the sights and sounds at each turn, is to being kicked of said mountain with nothing but a parachute.
A diagram will present you with a drawing of each individual step needed fold the model from start to finish. Crease patterns will on the other hand indicate that: "this part of the paper will end up as a pointy thing, probably a leg, and that part may be an arm." In its simplest form it will present you with an image of the designer's thoughts as to how the paper will be distributed in the base of the model.
The base of the model? In order to not cross over the edge and into complete information overload, a crease pattern will only tell you how to fold the base of the model, which would be the point in the folding process where you have just isolated each of the flaps needed to produce the desired number of appendages in the model. Quite like a drawn stick figure is to a drawn human.
In an optimal situation these will be the features to look for:
It should be fairly large, so that if one were so inclined it could be printed on a sheet of paper.
It should clearly differentiate mountain and valley folds. Creases that can be either should also be indicated.
A separate crease pattern indicating how to locate any obscure and unintuitive reference points, if needed.
A picture of the base produced from the pattern.
A picture of the finished model.
However reality tends to frown upon our conception of the optimal situation, so you are actually unlikely to find crease patterns which satisfy all these conditions. The picture of the finished model will, for obvious reasons, nearly always be present. The other points, apart from one, are fairly trivial and included mostly for the sake of convenience. Which means that the deciding factor when determining the difficulty of any crease pattern is how (or if at all) the creases are marked.
For a crease pattern where all the creases are appropriately marked, folding it will usually be a relatively simple matter of folding each crease and collapsing the entire thing into the base, followed by further shaping creases transforming the base into the fully realised subject. If on the other hand your crease pattern is a collection of identical black lines the experience takes on a different shape altogether.
To illustrate this last point a bit closer, imagine you have a 1 centimetre wide strip of paper 30 centimetres long, divided in 30 squares. In one scenario you possess exact instructions as to the direction of each fold; there is only one way the strip can be folded together. If, on the other hand, these instructions merely tell you to make 30 creases and fold the thing up, the number of possible ways this can be done 3 may prove to be something of a challenge.
Fortunately it will not be quite as complicated as that when you are dealing with an origami model since most of the creases are connected, thus meaning that there are only a few different configurations which will produce a base which folds flat. At this point the major deficiency of crease patterns should be apparent; they are inherently unfriendly to inexperienced folders, as the only way to make sense of them is through experience.
Why then is such a cumbersome and impractical thing allowed to exist? As opposed to drawing a diagram for a model, which not only takes time, but also is rather tricky to do well, a crease pattern can be made by anyone, and with minimal effort. The main benefit of this is that a large number of complex models, created by people who normally would not be inclined to diagram them are still available to interested folders. Crease patterns also give you a very clear image of how the paper in the model is utilised, something a diagram seldom reveals. Some folders also consider them highly entertaining challenges.
And in reality?
This would be a good time to get the bird base and pencil I mentioned earlier. Lying on whatever surface you have in front of you will be an approximately kite shaped object. You should pick it up and examine it, you will probably notice that there are four large triangular flaps extending from a small triangular flap in the centre of the paper. In relation to the flattened kite shape the large flap will be lower triangle and the small one will be the top. When lying flat two of the long flaps will be trapped inside the model, while the two others will respectively occupy the near and far side. Choose one of the accessible long flaps; notice how it is connected to the rest of the model. The connecting creases should act as a hinge; mark them with your pencil. Do this with the remaining three flaps too.
Unfold the model.
Each corner of the sheet before you will be covered with very rough quarter-circle, although it really looks like a quarter of an octagon right now. If you were to fold the model up again and narrow each of the flaps, and then unfold it again the line made by the hinge creases would look more like a quarter-circle. Actually the narrower the flap, the closer it would resemble a quarter-circle. There are two things worth noticing about these quarter-circles, none of them overlap, and the radius of each equal the length of the flap they indicate.
But what about the smaller triangular flap, the one at the top of the model? Well, its point is located at the centre of the sheet, so draw as large a circle as possible without overlapping any of the others, the centre of this circle will correspond to the point of the smaller flap, the centre of the sheet. The short flap requires a full circle on the sheet, while the longer ones only require a quarter-circle, which tell you that only the centre of a circle is required to be within the sheet in order to create a flap, and also that the closer the centre of said circle is to a corner the less paper will be used to create the flap. A flap made from the edge of the sheet will need twice the amount of paper as one made from a corner, and a flap from the interior of the paper will need four times the amount.
Armed with this knowledge you should be able to look at a crease pattern and be able to determine how many flaps the base will have, and from what parts of the paper they are made. Looking for circles is a good way to start analysing a crease pattern as it helps you to build a mental image of how the paper must fold in order to reach the correct shape.
By going the other way, first creating a set of circles that properly describe a model you wish to fold, one for each arm, leg, etc. And then by finding an efficient way of packing them into a square4 you can create a base with any number of flaps of arbitrary length.
This page contains illustrated instructions. If you feel that instructions should be included in the write-up tell me and I will add some.
If you still want to know more after reading this I recommend that you look for the book "Origami Design Secrets" by Robert J Lang.
Due to my atrocious lack of math skills I have elected to leave the task of determining this number as an exercise for the reader.
Useless trivia: The bird base is incidentally the most efficient packing of 4 equal circles in a square.