A crease pattern [almost] defines an origami design so this is the closest to a formal description you can get. It's actually good enough for quite an advanced mathematical apparatus to have been developed which allows you to prove lots of things about origami in a strict, mathematical way. However, the crease pattern is still a 2D drawing, so while much simpler to describe than the final 3D shape, it's also too complex to be neatly described in words. You can, of course, spell out all the folds, their angles and lengths, but usually it will be too long to be called a name. For simplified cases with additional bounds, it is, however possible - in the post I list one such approach (by Goran Konjevod) which works for a certain type of tessellations.