Systematizing Origami

Abstract

I analyze some aspects of the systematization of origami presented in (Chorna, 2012) and propose some improvements, as well as additional organizational rules and structures. I contend that Chorna’s genres of origami expression should be reclassified according to more natural and mutually independent attributes of origami; I suggest seven of such attributes. Complementing the physical separation of origami designs into modular components, I argue that separating a design into conceptual parts also adds insight to its classification. I also propose using the interlinking mechanism between modules as a classification criterion. Finally, I propose the analysis and indexing of origami designs by decomposing them into their physical and conceptual components.

1. Introduction

A few days ago, O. S. Chorna from the Kyiv National University of Culture and Art notified the Origami Mailing List (O-List) about her Ph.D. thesis titled “Origami as a modern art phenomenon: systematization attempt“. I briefly introduce her article by quoting (Chorna, 2012):

The purpose of this study was to make in order the conceptual and categorical apparatus of the origami art, to analyze its type and style  from the aspect of the art history, to mark out the significant criteria of their possible classifications (by shaping, functionality, content, artistic expressiveness, etc.) and to systematize the origami as an art according to the detected criteria…

… the following rules of division (*) were used:
(i) the division must be done by a single criterion;
(ii) the division must be proportionate and comprehensive;
(iii) the components of the division are required to be alternative and mutually exclusive;
(iv) the distribution must be uninterrupted and one-level…

… The proposed below reduced systematization has a two-dimensional structure and has a function of a scientific forecasting of origami art development. This is the main useful function of this system.

I often think abstractly and cannot accept a concept unless it has a certain “natural internal order”. I’m still trying to figure out what is “natural” for me, but it seems that the criteria after (*) have described concretely some of my (unconscious?) requirements.

The result of  her classification and categorization are the following tables (Chorna, 2012):

Table 1: Expression genres of origami

Table 1: Expression genres of origami

Table 2: Origami classification

Table 2: Origami classification

Regarding terminology, the word “genre” has been used differently in both tables. To avoid ambiguity, I call Table 1’s genre as “expression genre” and Table 2’s genre as “subject genre”.

While I don’t really have the background to comment on the categorizations based on “content and function” or “historical features”, I felt slightly uncomfortable at the division of origami into the expression genres of Conventional, Plastic OrigamiTessellations, Pleating, Tachi’s OrigamiCorrugationsCurvilinearCrumpling, Modular and Synthesized Origami. On closer inspection, my discomfort stemmed from a perceived violation of “rule of divison” (i) and (iii). Hence, I propose an alternate systematization, and I show where Chorna’s system fits in it. In Chorna’s footsteps, I try to give examples for every category I come up with, but the examples have an abstract/geometric/algorithmic slant as such origami forms the scope of much of my limited experience in origami. For brevity, I introduce Figures only when I reference lesser-known works.

This article is an informal preliminary analysis of origami classification, which will hopefully stimulate some academic discussion on this topic. I enjoy analyzing these things because I do have a real-life tendency to organize things abstractly 😉

2. “Continuous” attributes of origami

In the spirit of “dividing via a single criterion” (i), I identify a few attributes ascribed to origami which, *in a rough sense*, can be regarded as varying over a one-dimensional continuum:

  1. Coarseness/fineness of creases in the crease pattern
  2. Coarseness/fineness of regions (faces – planar or curved – of the crease pattern)
  3. Degree of balance between expressiveness of creases vs. regions
  4. Degree of presentation of the medium as a surface or structure
  5. Design “direction”: was it top-down, or bottom-up?
  6. Relative level of emphasis between the medium and represented image (compare with the concept of image-creating art in (Choyrna, 2012))
  7. Degree of “rigidity” in the form: is it more rigid or more organic?

This list is not exhaustive, but it’s all I can think of for now. In an abstract sense, I am plotting origami designs along seven axes; each axis characterizes a continuum of variation of a corresponding attribute of origami. Each origami design can be thought to occupy a certain point in that “7D space” (subject to some uncertainty due to subjective opinion), and unoccupied points represent areas where origami art is yet to develop. Such points should be marked with a “?” in the spirit of “scientific forecasting of origami art development” (Chorna, 2012).

This “attribute space” system is not completely incompatible with Chorna’s genres of origami; each genre has some characteristic tendencies in some of these attributes, which is why the genres were defined in the first place. Hence, each genre occupies a portion of the attribute space, and if the genres are “mutually exclusive” (iii) and “comprehensive”  then the portions corresponding to the genres should be pretty much a partition of the attribute space. Proportionality (ii) requires the portions to be “of comparable size”.

The attribute space should have a lot more “?” blanks than Chorna’s charts (Section 1); while that seems good for “scientific forecasting”, it’s harder to design origami with the end-product attributes in mind than using specific and familiar genres. We could fill in the “?” blanks by identifying which genre has those attributes (i.e. which genre portion portion contains the 7D coordinates of the point) and design from that genre, but that would only fill in the “?” blanks in Tables 1 and 2 from Chorna’s classification system.

2.1. Tendencies of each origami genre

“Scoring” some origami genres on the attributes can illuminate some of the considerations behind each attribute, as well as the tendencies of each genre.

Chorna’s categories of Conventional, Plastic and Crumpled are roughly in increasing order of fineness of creases and regions (1, 2). I would argue that the forms of origami polyhedra are expressed mainly through regions, while origami tessellations and representational origami have a balance between creases and regions (3).

As Chorna points out, origami is unique from sculpture because its volumetric (3D) form is produced by manipulating a 2D surface. This aspect of origami also allows forms to be expressed in terms of surfaces and structures simultaneously, which may also meld into and interact with each other. This unique and important “duality” is applied later in Section 4. Tachi’s Origami uses the Origamizer program by Tomohiro Tachi to generate crease patterns that fold to mimic user-specified surfaces (4). The design process is top-down (5).

Figure 1: A 3D teapot can be folded (right) from the crease pattern (middle) constructedfrom a teapot represented by a triangle mesh (left)

Figure 1: A 3D teapot can be folded (right) from the crease pattern (middle) constructed
from a teapot represented by a triangle mesh (left)

Representationalist origami, by definition, has a heavy emphasis on the represented image, while abstract origami such as Paul Jackson’s work focus on the paper medium (6). Most of Conventional Origami and much of geometric origami use rigid forms and folds, while Curvilinear Origami and “paper sculpting” techniques such as wet-folding are more organic (7). “Paper sculpting” includes the “shaping” stage of base-oriented origami design, and much of the much-celebrated “lifelike” origami of the late Akira Yoshizawa and Eric JoiselGiang Dinh’s work, despite not being strictly representational, lies more on the organic side of this scale.

3. Physical and conceptual components

I feel that Chorna’s classification of multimodular origami is quite satisfactory, but I would like to extend her definitions slightly to include conceptual components of origami forms, rather than considering only physical components. This reduces Origami Tessellations from a category of origami in itself to simply (c0nceptually) homomodular origami. Consider the following examples of conceptual modularity:

  • Homomodular surface: Origami Tessellations
  • Heteromodular surface: origami fractals
  • Homomodular structure: Origami Corrugations
  • Heteromodular structure: grafting (Lang, 2011, pp.129)

Grafting deserves special attention, for it encompasses a humongous range of origami design, with modules of all kinds (body parts, textures like scales) combined in various ways such as those listed in (Lang, 2011, pp.129). In particular, Joel Cooper’s “tessellation-masks” may be conceptually separated into a Portrait and an Origami Tessellation; this instance is reflected in Chorna’s categorization. Extrusion origami explores the algorithmic extrusion of forms (often polyhedral) from the inner space of paper so that the surrounding paper lies flat. That surrounding paper is then treated as a new sheet to extrude more forms. This composes the (conceptual) modules together with a single sheet of paper. As the modules are frequently drawn from some fixed family of geometric solids (e.g. cubes (Ovadya, 2010) or right frusta (Cheng et al., 2012)), this form of origami occupies a middle ground between (conceptually) homomodular and heteromodular structures. To illustrate this possibly unfamiliar concept, Figure 2 demonstrates how extrusion origami can combine a pyramid module and cuboid module to form a conceptually heteromodular structure from one sheet of paper (Cheng et al., 2012).

Figure 2(a): The crease pattern of the first module has been drawn.

Figure 2(a): The crease pattern of the first module has been drawn.

Figure 2(b): The first module  - a pyramid  -has been extruded by folding the crease pattern.

Figure 2(b): The first module – a pyramid -has been extruded by folding the crease pattern.

Figure 2(c): the crease pattern of the second module has been drawn.

Figure 2(c): The crease pattern of the second module has been drawn.

Figure 2(d): The second module - a cuboid - has been folded, completing the conceptually heteromodular structure.

Figure 2(d): The second module – a cuboid – has been folded, completing the conceptually heteromodular structure.

However, beware of a gray area: when does one decide if a piece is a whole or a sum of conceptual modules? Can’t we break a representational origami piece into its functional parts, like the body parts of an origami animal, and call it conceptually heteromodular? I’m not sure how to resolve this (in fact, I’m tempted to consider the molecules used in circle packing, base design (Lang, 2011, pp. 345), Tachi’s Origami, or other algorithmic crease pattern designers as conceptual components!), but as a rough rule of thumb I guess we could say that parts of a design are distinct components if they are somewhat “independent” of each other, whatever that means.

Chorna draws a sharp distinction between monomdular origami and multimodular origami, restricting his studies of expression genres to monomodular origami, and treating multimodular origami as something like a “different species”. However, I feel that it would be more natural to consider mono- and multimodular examples of design within each expression genre, so modularity has some measure of independence from the expression genres. In fact, modularity could be a candidate for an eighth attribute!

3.1. Links between modules

One aspect of modular origami which I feel that Chorna should not have omitted is how the modules link together. The flap-into-pocket (e.g. Thomas Hull’s PHiZZ Unit) is the most common linking mechanism, which prevents separation by “welding” pieces together. In contrast, the modules of polypolyhedra are prevented from separating through weaving, or if you will, by using “the laws of Knot Theory“. Hence, I propose dividing the links between modules to be either intrusive (e.g. flap-into-pocket) or nonintrusive (e.g. Jean Pederson’s weaving of paper strips). Figure 3 compares two physically homomodular dodecahedra whose modules and linking mechanisms are very different, despite their outwardly similar appearances.

Figure 3: (Left) A homomodular dodecahedron from flap-into-pocket assembly of 30 PHiZZ Units. (Right) A homomodular dodecahedron from weaving 6 rings of paper.

Figure 3: (Left) A homomodular dodecahedron from flap-into-pocket assembly of 30 PHiZZ Units. (Right) A homomodular dodecahedron from weaving 6 rings of paper.

However, all those are the links between physically separated modules. What about conceptual modules which inhabit the same sheet of paper? This is of more interest to me because one has to study the behavior of the crease patterns of the modules as they approach the borders between adjacent modules – extra creases may have to be made at the border to adapt the “interface” to become “compatible” with the modules on both sides. This is how Tomohiro Tachi links his tucking molecules together. More often, though, the modules are deliberately designed such that their creases are compatible with those of their neighbors. This is almost always the case for the “unit cells” in Origami Tessellations, and in fact for most conceptual homomodular origami. However, this incompatibility is still an in issue in conceptual heteromodular origami; for example, it is not trivial to extrude multiple forms from the same sheet of paper simultaneously, i.e. when the forms cannot be sorted into a clear sequence.

4. Analysis via decomposition

The predominant form of an object depends on the scale at which it is observed. Due to the aforementioned potential of origami to evoke both surfaces and structure in a very natural manner, surfaces may consist of substructures (e.g. Kawasaki Rose tessellations and other Origami Corrugations) and structures may consist of sub-surfaces (e.g. Satoshi Kamiya’s Ryujin, a scaly texture folded into the shape of a dragon). Similarly, structures can consist of sub-structures (e.g. Jeannine Mosely’s Business Card Menger Sponge). A more general class of examples are bases consisting of mini-bases “glued together”, such as the family of blintzed bases. To pick an example, the Frog Base has four Bird Bases as sub-structures. I don’t have an example for surfaces that consist of sub-surfaces; perhaps you could give me an example, or perhaps it should be marked with a “?”.

This decomposition by changing the scale you view the origami design at can apply to some of the “continuous attributes”  from Section 2, such as top-down designs consisting of individually bottom-up design elements, a case in point being algorithmic crease pattern design using molecules; the overall design is “on demand” so is top-down, but each molecule is a “base” and is bottom-up. Organic forms may also be decomposed into rigid components, for instance looking past the organic faces of Joel Cooper’s masks to focus on the individual rigid facets of the tessellations.

Sometimes it can be hard, or even meaningless, to “score” an origami design on the continuous attributes; this can occur when different parts of the design exhibit varying levels of the same attribute. Again, Joel Cooper’s masks mix organic and rigid forms. Decomposing designs into their physical or conceptual modules can isolate parts of the design that are relatively “pure” in a certain level of an attribute (Figure 4 shows a plan of the decomposition). This leads to a more nuanced and appropriate understanding of the qualities of a design rather than trying to find a meaningless average “score” (I did say the “continuum” is a very *rough* analogy).

4.1. A technical analogy

An origami design may be regarded as a tree, or a node (or point or vertex) branches off and links to other nodes (called its “children”), which further branch out “downstream” to more nodes. The root, or the node where all of the branching off starts from, represents the entire origami design. Each branching corresponds to a decomposition, physically or conceptually, into modules. Each module is an origami model in itself, represented by each child, which is itself a root of a smaller tree. Thus we may recursively decompose origami designs into their components, sub-components, etc. Each node is associated with seven values corresponding to the seven “continuous attributes” from Section 2; there is a *rough* parallel to seven functions from the nodes of the tree to real numbers.

The collection of children of a node may also be given additional structure to represent the way the modules link to each other. Flap-to-pocket linking could be represented by a graph where lines join pairs of children whose corresponding modules are linked. More complex “woven” configurations of modules can be represented using a link.

Be reminded that going “downstream” on a tree, that is, following the direction of branching out, decomposes the origami model into smaller and smaller components and zooms into a smaller and smaller scale. The decomposition should prove particularly effective on Synthesized Origami.

Figure 4 shows an example of such a “decomposition tree” constructed from a Joel Cooper mask.

Figure 4: The decomposition tree for a Joel Cooper mask.

Figure 4: The decomposition tree for a Joel Cooper mask.

The mask as a whole is decomposed into two sub-surfaces (heteromodules), the face and the beard. As a tessellation, the beard can be further decomposed into multiple copies of the sub-units (homomodules). This decomposition model can apply to most generic Joel Cooper masks, so in some (rough) sense the masks are “isomorphic”. Each node of the tree is associated with various values of the attributes from Section 2, which are written next to them in ascending order (from (1) to (7)). Figure 5 shows a decomposition tree for Five Intersecting Tetrahedron.

Five Intersecting Tetrahedra is decomposed into the individual tetrahedra which were woven together. Each tetrahedron is further decomposed into the “strut” modules, each of which actually provides the shape of a strut using their surface. This decomposition model can be applied to most polypolyhedra, even exotic ones.

Figure 5: Decomposition tree for Five Intersecting Tetrahedra

Figure 5: Decomposition tree for Five Intersecting Tetrahedra

It seems that this idea of a decomposition tree can be applied to any art form, but if it’s really that useful, the concept should already exist 😉 It shouldn’t be too hard to write this “tree and functions” analogy out “rigorously”, but what we have now is probably useful enough. It would be a fun exercise, though.

5. Conclusion

I have proposed seven attributes of origami that seem to vary across a continuum, so as to replace Chorna’s enumeration of expression genres with a more natural system that organizes origami designs by their “position” as described by their 7 coordinates along the 7 attribute axes. Expression genres then correspond to portions of the 7D space. Chorna’s enumeration was unsatisfactory because his expression genres varied over too many attributes, thus violating the “rules of division” (i) dividing via a single criterion and (iii) mutually exclusive genres.

I complemented the considerations of physical modularity with conceptual modularity; this extension is useful as it adds on to the existing modularity classification harmoniously, and conceptual modularity has examples as important as grafting. I also explored the linking mechanisms between modules and suggested the intrusive/nonintrusive distinction. Finally, I proposed the decomposition of origami designs into their modules and submodules, representing the result as a tree to aid in analyzing and indexing origami designs.

As for future work – well, ideas for further discussion – are the expression genres listed by Chorna proportionate and comprehensive? My sense is that trying various strange combinations of 7D coordinates may identify some portions of space not covered by any expression genre. What other attributes can we also think of? What is the scope of the “?” blanks? Does the concept of decomposition tree already exist? One important factor in origami expression is the paper itself – texture, color (in particular, the important family of duo-color origami), shaping techniques like wet-folding etc. Both Chorna and I have omitted this from our investigations.

A final disclaimer – this was a rather hurried exposition; I didn’t really take the time to verify the comprehensiveness of my ideas or the level of independence between the attributes. I hope the readers can correct me on my mistakes and raise anything that I may have missed. It would be great if I could be alerted to some examples to my concepts that aren’t abstract/geometric/algorithmic origami design, so that the argument is more convincing.

(Wow, this post got pretty long, I really thank those who made it through!)

Bibliography

(Chorna, 2012) Chorna, O. S. Origami as a modern art phenomenon: systematization attempt. Ph. D. dissertation, Kyiv National University of Culture and Art (2012).

(Lang, 2011) Lang, R. J. Origami design secrets: mathematical methods for an ancient art. 2nd ed. Alamo: CRC Press (2011).

(Cheng et al. 2012) Cheng, H. Y. & Cheong, K. H. Designing crease patterns for polyhedra by composing right frustaComputer-Aided Design 44, 331-342 (2012).

(Ovadya, 2010) Ovadya, A. Origami transformers: folding orthogonal structures from universal hinge patterns. M. Eng. dissertation, Massachusetts Institute of Technology (2009).

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