Thicket, not tree — On Origami Classification, Part I

Origami classification resembles a thicket of intertwining shrubs rather than a tree

This post is part of a series on origami classification: Part I, Part II

When you ask origami creators what types of origami they design, you hear terms such as traditional, modular, supercomplex, tessellations or dollar-bill origami. Most origami websites also group models into similar categories. On the face of it, it looks like there is a widely used and understood classification system for origami designs. Unfortunately, upon closer inspection, this “system” falls apart, showing it is not a system at all, but rather just a bunch of names used for a few commonly folded types of models. These types overlap in some places while leaving out some models without any classification in others. Some are very generic while others are overly specific. Categories are based on different criteria, and there are few, or no relationships between them.

About this post series

In this post I will try to point out some weaknesses of existing attempts at classifying origami models and suggest a different approach which I think can better solve some issues while staying mostly compatible with the system(s) origamists have gotten used to. Since the topic is huge, I decided to split the text into multiple parts. This should allow me to flesh out my vision in smaller, approachable portions, and to start the topic with just this introduction rather than waiting till I can write down everything in a single, probably very long, piece.

The topic of origami classification has been on my mind for about a year now, and was directly inspired by me creating a new version of my website in early 2021. I wanted to organize the several hundred featured models into a reasonable structure, which led me to more general investigation of how origami could be best classified. In early 2022, Der Falter, the German Origami Society’s magazine, brought up the issue of origami classification as well, which reminded me of the subject, and inspired me to finally start writing down my thoughts.

Why care about origami classification

Why do I think origami classification is of any importance (at least, for people interested in origami)?

The first reason is purely practical: origami websites and books need to organize models they present, and having a system of genres helps with that. Using a consistent system in different places should make things easier for everyone.

The second reason is more philosophical in nature: classification allows us to generalize observations of what we already know, and it may also allow us to find out things we don’t know yet. If a classification system predicts that a genre should exist, but no current model can be assigned to that genre, there is apparently room for creating such a model and possibly opening a new chapter in origami.

Issues with current approaches

Here are some issues I notice with current approaches to origami classification. Many of them are well illustrated by the list of over 80 genres at Origami Resource Center:

• lack of consistency: some genres appear in almost all systems, but they are often not understood in the same way; some genres appear in one system but not in another and the same model may be classified differently depending on the system used; different genres are (often only implicitly) based on different kinds of properties; thus, figurative origami belongs to a classification based on the subject, while modular origami is defined based on the number of sheets;
• lack of structure: dollar-bill origami may contain figurative and modular designs; should I look for Dollar Bill Hare in dollar-bill origami or among figurative designs?
• lack of precise definitions: authors often attach implicit assumptions to the genres; for example, in the case of modular origami, sometimes the subject (abstract 3D shapes) is also implicitly assumed, which leads to issues such as classifying a modular model which represents an animal; another common assumption is that units are identical or at least similar, or that their number is “large” which makes classifying models such as animals folded from two sheets of paper difficult; often, it’s not clear whether the genre classification applies to designs or works — if the Dollar Bill Hare is folded from regular paper, does it still count as dollar-bill origami or not?
• most importantly, usually there is no system at all, just a bunch of genre names which are not in any way connected to each other; they are just named islands in the sea of all possible models, without any structure or relationships between them.

Foundations of the system

Here are some properties that I expect a good origami classification system to have:

• it should be multi-dimensional: different kinds of properties (e.g. subject, number of sheets, techniques used, etc.) all need to be taken into consideration, but should also be clearly separated from each other;
• it needs to be hierarchic in order to accommodate the complexity needed to classify all types of origami; however, this hierarchy can’t be represented by a tree structure; multi-dimensionality of the scheme makes that impossible; a fitting representation seems to be a directed acyclic graph (DAG), a structure in which each node can have multiple parents; this representation becomes obvious once we try to use the categorization for the purpose of navigating a set of models by their genre on a web page: there are multiple ways to arrive at any given node/genre, e.g. modular polyhedra could be reached either by starting from modular origami or by starting from abstract subject; Origami classification resembles a thicket of intertwining shrubs rather than a single tree;
• it should be comprehensive, that is, it should have room for any possible shape made by folding paper, including those which no one has folded yet;
• it should be flexible: depending on individual needs, it should be possible to increase or decrease the level of detail in individual dimensions;
• it would be an advantage if popular genres that can be found in other approaches and that people are used to, could easily be placed within the improved system.

These properties lay the foundation for the new system. Given the provision for flexibility, this is actually a meta-system rather than a single, rigid system.

In practice

To put these very general rules into practice, let’s consider a few examples. For more, you can have a look at the list of models on my page, grouped by type. This practical implementation of the system is not quite finished yet, and you can easily spot inconsistencies within it, but it should give you an idea of what the system should be like, and I hope to improve the details over time. Note that on my page, a model means a specific piece of folded paper, so the classification applies to individual works rather than to designs.

In a tree-like hierarchy, a node’s position within the tree can be represented by a path leading from the root to the node. A common example is file paths used in computing. In a DAG, there are multiple roots, so multiple paths are needed to represent the node’s full position.

Now, for some examples.

A Simple Elephant would be classified as belonging to the single-sheet animal genre which has the following paths:

• Sheets: single
• Subject: figurative / living things / animals

Even in this simple example, you can see the flexibility of the system at work. Firstly, one dimension is responsible for representing the number of sheets used in a model. One can opt to only distinguish between single-sheet and multi-sheet models, but one could just as well create separate genres for each individual number of sheets (2, 3, 4, etc.) if it makes sense in some context. Secondly, figurative models are subdivided into subgroups such as living things which are then split into still finer subgroups. Again, the granularity of this division could be modified to fit the needs. My site focuses on abstract designs, so abstract subject genre is split into quite a complex hierarchy. On a website focused on models representing animals, abstract designs could be subdivided only roughly while animals could be subdivided into very fine subgroups (e.g. animals / vertebrates / mammals / elephants / african elephants).

A number of other dimensions might be relevant in some contexts, and could be added to the two mentioned above: a model using wet-folding or not (this property may be considered at design level or at work level), a dimension of design techniques used (box-pleating, 22.5°, etc.), perhaps a dimension of difficulty level, and so on. Note that sometimes placing a model within a group may not be obvious: if we introduced the dimension of difficulty level, there would certainly be controversies whether some models are still intermediate or if they should be classified as complex. Such doubts will always be there to some degree, and it is not a goal of my system to eliminate them. Many of these issues, such as what constitutes a separate design (or a separate genre), are very similar to those mentioned in my post on naming origami models.

In this example, I also created a mapping from a set of paths in my classification system onto a genre name similar to those used by other systems, single-sheet animal. With multiple dimensions and complex subdivisions within each, the number of combinations (and thus possible genres) is huge. Some will pop up only rarely, but it seems to be a good idea to give short, memorable names to those that come up a lot. They will often correspond to traditional genre descriptions as well since “types of models that come up a lot” is roughly what these traditional origami genres tried to represent.

In another example, a 90-edge buckyball from PHiZZ variant units could be classified as modular polyhedron (non-cube):

• Sheets: multiple / modular
• Subject: abstract / mathematical / balls and polyhedra / other (than cubes)

I have split Sheets: multiple into modular (with identical or similar units) and multi-sheet models (for models folded from multiple pieces of paper, each folded in a significantly different way). More about that in follow-up posts. You can also see that the path in subject dimension is rather long. It is adapted to my site’s needs thanks to the classification’s flexibility: balls and polyhedra are split into cubes and other since I have so many cubes on my site that I decided to give them a separate category. Since this is a rather typical model, and so is its genre, the genre gets a nice name once again, modular polyhedron (non-cube). Note that it probably makes sense to also name genres such as polyhedron (non-cube) and polyhedron (without any adjective) since polyhedra can be folded from single sheets as well as from multiple. Therefore, a model can belong to multiple named genres, some of them more specific than others. While in other classification systems, modular origami might contain the implicit assumption of subject, here we are quite strict and can tell between designs which are modular (with any subject), modular polyhedra, and polyhedra which are not modular.

What’s coming up

We will encounter some quirks and tough questions that need to be answered when we start looking at the classification close-up. The tough questions pertain both to the system as a whole as well as to the way specific parts of the classification should be organized. The examples above were rather simple and uncontroversial. As we get deeper into the system and look at more interesting cases, the classification it suggests will diverge more and more from the one we use intuitively. Obviously, the system is also not without flaws. In follow-up posts, I will explore some of these issues in more detail.