If one takes the narrowest possible dictionary definition of the term, a “crease pattern,” or CP, is nothing more than a set of lines that is a representation of some subset of folds in an origami shape, real or imagined. But it can be, is, more; the deeper question is, what does a crease pattern represent to some person? And that answer varies with the person.
At one extreme, and at the most personal level, to an origami designer, it is a form of external memory: a way of recording one’s ideas and the relationships among those ideas in a level of detail too great to carry around all at once in one’s mind. In this form, it carries a meaning that is expressed to no one else in precisely the same way: it is a reflection of the unique pattern of synaptic connections and chemical potentials within the mind of its creator.
But a CP can certainly have meaning beyond its creator. To a fairly small community of folders, it is a guide for how to fold an object; a CP does not include all of the creases in the folded artwork, so it is something more than a hint, but something less than a full plan, and not all that many people can make use of it in that way. To those who can, though, the crease pattern is the trail of clues to a finished work. With suitable interpolation, it can be followed to create a representation of the creator’s original concept.
Then, moving outward to a broader community, which includes both folders and nonfolders, it can be something else, a “proof certificate”: an indication that a folded object really is what it claims to be. Even though the observer can’t make the connection between specific lines in the pattern and folds in a figure, he or she knows that such a connection exists and that any given line, if investigated deeply enough, would correspond to some feature in another representation of the subject of the fold. The proof certification can be appreciated even if one does not see the folded object or even if the object does not yet exist: there is the knowledge that it is connected to something immanent. Along the same lines, a CP serves as yet another imprint of an object or concept, just as the folded form is, itself, an imprint of an object or concept.
And then, to the largest community, the lay observers, a CP is a beautiful pattern whose subpatterns and symmetries evoke memories and associations, an evocation that is made richer by the knowledge that the lines are not random, but in fact follow some inner order and logic. The observer may not know the rules of the inner order, but at some level, he or she can perceive that an order is present, and that knowledge can enhance the overall visual experience.
And it almost goes without saying that these roles for a CP are by no means mutually exclusive; any given person might well experience the same CP on multiple levels. I’m probably the only person who experiences my own CPs in that first sense, but I also experience it in all of the other ways as well to varying degrees — and, I imagine, those who’ve seen my CPs in exhibitions probably experience them in ways I haven’t anticipated here. And, I expect, other artists would describe their own relationship to their and others’ CPs in different ways as well.
CPs as folding guides have a long history in the world of origami, but once their texture and patterns reached a certain level of richness and structure, they began to stand alone, not simply as internal tools of the creative folder, but as artworks worthy of independent contemplation, with diminishing, tenuous, or even nonexistent connections to their associated folded form.
In 1999, I was asked to create several origami-based artworks for the downtown Santa Monica redevelopment project spearheaded by Brailsford Design. This included a set of four bronze sculptures (which you can see here) and four crease patterns to be carved into the pavement of the four intersections where each of the bronze sculptures were installed.
The crease patterns used were the patterns for the origami figures whose bronze likenesses were to be found immediately adjacent on the sidewalk. There was a multiple-step connection between them: pattern in concrete—paper crease pattern—paper folded form—bronze sculpture, but in the installation, the two middle steps were not present, allowing visitors to make their own connection—if indeed, they made any at all.
These CPs, like many published CPs, showed only a subset of the crease lines in the folded form to which they corresponded. A CP showing every fold line in the model would be a dense, unmanageable clutter, and so when creating a CP for display, one selects a subset: the most significant lines, or a set selected for their aesthetic value.
A few years after this project, I assembled an exhibition of origami for the Lindsay Wildlife Museum in Walnut Creek, California. In addition to the folded paper artworks on display, I created four large wall-mounted CPs.
Again, the lines chosen in these works were a subset of the creases in the finished fold, but here I chose to convey more information than I did in the Santa Monica patterns, by showing mountain and valley folds in two contrasting colors—a practice I continued in my book, Origami Design Secrets (2003, A K Peters, Ltd).
I also began coloring the backgrounds of CPs. For those designs based on circle packings, inclusion of the circles and/or rivers in the background creates an echo of the mental processes that led to the original design. These curly shapes do not correspond to folds in the base; they are, instead, a reflection of a mathematical concept associated with the folded form. Toshiyuki Meguro, who independently invented circle packing in Japan, and I each began showing circles and rivers in our CPs in the early 1990s. The figure below, which appeared in an article I published in the magazine ISIS-Symmetry in 1994, is a typical example.
Circle/river coloring adds aesthetic level to a CP that goes beyond the fold lines and it can be combined with the use of background colors to lend richness and texture to crease patterns. I combined the use of color in the crease patterns with the depiction of circles and rivers in CPs published in Origami Design Secrets (2003) and on my website in 2004.
When I was invited in 2008 to contribute artwork to the Museum of Modern Art’s “Design and the Elastic Mind” exhibition, I selected several crease patterns, decorated in a range of styles, for the show.
This concept of encoding information into a CP graphically has long been attractive to me. Aesthetically, we take information out of the full crease pattern when we select subsets of crease lines for presentation, but at the same time, we can inject different levels of information and meaning by creative use of color and pattern. When I wrote TreeMaker 5 from 2004–2006, it presented an opportunity to explore different ways of encoding information into CPs via color and pattern.
(Side note: TreeMaker is a program for origami design that I first released in 1993. Every few years, I updated it by incorporating new functionality, more or less as I discovered the corresponding theory. By version 4, released at the end of 1997, it had become a practical design tool. I began working on version 5 in 2004, with release 2 years later.)
My LWM and ODS CPs used color to represent mountain/valley/unfolded status of crease lines, but of course, that distinction has long been made by line pattern: dashed for valleys, dot-dot-dash for mountains (with variations over the years, as you can see here). In TreeMaker, I began using color as a proxy for different information, namely, structural information. In the theory of uniaxial bases, creases fall into categories characterized by their position within the base, and I assigned different colors to the different categories of crease.
In fact, TreeMaker moved the origami pattern beyond a representation of mere folds, because the underlying data model contained many linear features, not all of which were crease lines in the origami figure. Between any two points in the pattern there was a line, called a path, and each path was associated with a condition, called a path constraint. Even before we construct folds, we must look at paths in the origami design. Paths have a minimum length; those that are too short are invalid; those that are as long or longer than the minimum are valid; and those that are precisely the minimum length are active. In TreeMaker, I decided to match colors to these meanings, making invalid paths red (for warning), valid paths yellow (for caution), and active paths green (because these could proceed to full-fledged creases). A pattern with just the paths colored is shown in Figure 8.
This path-colored pattern, too, is a representation of an origami design that looks superficially like a crease pattern, but in this case, only the green lines become actual creases in the finished result.
With those colors set, I looked elsewhere in the spectrum for colors for the other types of lines that might appear in the pattern. In the theory of uniaxial bases, crease lines fall into several categories that correspond to different structure roles: axial, gusset, ridge, and hinge. I associated colors with these categories, setting axial creases to black, ridge creases to red and hinge creases to blue.
I called this color scheme structural coloring. Structural coloring is complementary to the traditional mountain/valley coloring of CPs. The latter decorates fold lines according to their fold angle; structural coloring decorates fold lines according to their position and function.
I use color, rather than line pattern, as the information vehicle for these lines because of its much greater range of expression. In principle, the expressible palette of line pattern is large—one can define arbitrarily complex line patterns—but in practice, the palette is fairly small, due to the limits of human visual perception. Traditional origami diagrams use dashed lines for valleys and dot-dot-dash lines for mountains, which is all well and good when there are only a few such “action lines” in a diagram, but for large-scale CPs, two different line patterns become well-nigh indistinguishable, as the example below shows. So, in Origami Design Secrets, I tried out color (black and brown), rather than pattern, to distinguish mountain and valley lines.
In retrospect, the contrast between black and brown was not all that much better (it looked a lot better on the computer screen than on the printed page), and for the second edition of ODS, I went to yet another scheme, using both color and line pattern together, with better results for mountain/valley distinguishability.
In structural coloring for uniaxial bases, at least 4 distinct types of lines are required: axial, gusset, ridge, and hinge (5, if one includes pseudohinges), and in TreeMaker 5, I was representing still more linear features as lines: paths (valid and invalid) along with the tree from which the design sprang. Only color offered a sufficiently wide palette to represent all such features in a readily distinguishable way—at least, distinguishable to most of humanity. (To the 5% or so with some form of color vision deficiency, I offer my apologies!)
Prior to TreeMaker, I had focused my coloring activities on the lines of a CP, but graphically, TreeMaker constructed and tracked three levels of object: vertices, creases, and facets, all of which were linked via incidence and adjacency relationships. Like creases, the facets (the regions between creases) could have information associated with them, and as with creases, I could use color to represent this information.
A simple attribute associated with each facet in a CP is its color orientation: is it color-side-up or color-side-down in the folded form? (Assuming the paper is white on one side and colored on the other.) When I met Toshiyuki Meguro in 1992, he pointed out the “cool fact” that every origami CP could be two-colored and, when so colored, all of the facets of the same color would be oriented the same way in the folded form with respect to color direction. (A bit of reflection reveals why this must be so; whenever you cross a folded crease in the CP, you flip the orientation of the adjacent facet in the folded form.) And so, a logical coloring for the facets themselves would be this two-coloring. I built this capability into the 2006 release of TreeMaker 5; an example is shown below.
If we envision the folded form of a base as lying in the xy plane, then the surface normals of all facets point in the z direction: either +z or –z, depending on whether the colored side of the facet faces up or down. Thus, the two-coloring of the crease pattern specifies the z-orientation of each facet.
The visual representation of facet orientation is a purely aesthetic thing; it provides no significant assistance to one’s attempts to fold a base. There is no value, beyond the aesthetic, of appealing to perception with color based on z-orientation. But it is significant: colored or not, the underlying z-orientation of facets turns out to be crucial to determining the mountain-valley assignment of the CP.
Mountain-valley assignment of TreeMaker crease patterns (and of uniaxial bases, or even general bases) has long been a tough problem. In 2004, I was invited by Professor Erik Demaine at MIT to be Artist in Residence at MIT for 2 weeks, during which time we spent many hours batting around the problem of crease assignment for uniaxial bases. That trip was the first of what became an every-two-year visit. During the two years after that first visit, I focused on writing the newest upgrade to TreeMaker and used it to test out several hypothesized algorithms, and by the time of the next visit, had cracked the nut: we had an algorithm that appeared to work for all possible valid circle packings (and we’d identified an obscure additional necessary condition for its construction and validity). Crucially, this algorithm relied upon the z-orientation of the facets. It also relied on yet another bucket of information that could be attached to the facets and that could also be expressed as a linear pattern: the facet ordering graph.
A facet ordering graph is an expression of relationships between any two facets of a crease pattern. If one facet could be identified as being “on top” of another in the folded form, this relationship could be described graphically by drawing an arrow from the one on top to the one below (or vice-versa). We found that the layers in a uniaxial base could always be oriented in such a way that the facet ordering graph contained no cycles, that is, there could never be three facets A, B, and C such that A→B→C→A. Such a graph is said to be sortable, and it is possible to express the information contained in such a graph in a condensed form, called a reduced ordering graph (ROG). In TreeMaker 5, I implemented an algorithm for constructing the reduced ordering graph, and, again, provided a visual representation within the program.
From the ROG and the facet z-orientation, it is possible to construct the complete, full, mountain-valley assignment for the TreeMaker crease pattern. For depicting MV-assignments, I adopted a hybrid representation that maximized visual contrast between mountains and valleys, using all three of color, saturation, and line pattern to distinguish the two types of lines (or actually, three types, because I showed unfolded creases as well). Mountains are dark, colorless (i.e., black), and solid; valleys are lighter, saturated-color (magenta), and dashed. (Unfolded creases are thin and light gray, so that visually, they “fade out.”)
Now, the algorithm for constructing the ROG is not something a pencil-and-paper designer might readily carry out; it involves the construction of computational geometric objects called “spanning trees” and the mathematical cutting and gluing together of various graph-theoretic trees, bits of the ROG, and pieces of the crease pattern. But for purposes of origami design, it isn’t actually necessary to carry out a full MV assignment to the very last crease. If we can merely get close to a valid assignment, then we can start to collapse the base and will see during the collapse what the proper crease directions must be (and can choose crease directions where multiple options are available to us). Since most of my designing was, and continues to be, of the pencil-and-paper variety, rather than TreeMaker-based, I developed yet another labeling scheme, which I called the generic form assignment. In this form, all axial, gusset, and pseudohinge creases are assigned as mountain folds; all ridge creases are valleys; and all hinges are unfolded. This assignment is almost never flat-foldable, but it is always close to a flat-foldable assignment, and in practice, it is close enough that one can use it as a guide to the collapse.
Is two-coloring the only way to assign color to facets based on information content in the pattern? Certainly not. In fact, one can assign color to facets based on primarily aesthetic criteria: see, for example, Sipho Mabona’s colored crease patterns here. But I continued to see artistic potential in the use of color as a proxy for some sort of information that is buried within the design, and in 2008, I began to explore this idea further.
Two-coloring exploited the z-orientation of facets in the folded form, but there is other information associated with the folded form facets—and, for that matter, with the folded-form creases. During the middle of the first decade of the new millennium, I worked on marrying the decades-old design style of box-pleating with concepts from tree theory and uniaxial base design into a theory I call “polygon packing.” Two flavors of polygon packing are uniaxial box pleating —the first offspring of this marriage—and uniaxial hex pleating, more about which in a minute.
In uniaxial box pleating, the familiar axial, ridge, and hinge creases from tree theory are present, but they are joined by a family of axis-parallel, or “axial+N” creases. (These are described in detail in Origami Design Secrets, 2nd Edition). Axial+N creases begin with axial+1, followed by axial+2, and so forth, for steadily increasing values of N. The integer N describes the elevation of the crease, which is its perpendicular distance from the axis of the base; they are integers because the elevation is quantized. The axial+N creases form a potentially infinite family, but in practice, typically only the first few levels, sometimes just one, are needed.
Axial+N creases are lines of constant elevation on the crease pattern, like contour lines on a topographical map, and in fact, the correspondence is quite apt; in polygon packing design, one constructs axial creases by first constructing contour lines, and only late in the process do we identify some contours as creases and others as unfolded lines (which can be indicated as creases, or just simply erased). The addition of axial+N lines to the pattern creates a new opportunity for coloring, for now, each level of the axial+N family can be assigned its own unique color, as shown in the example below (taken from ODS2e).
Elevation provides a rationale for coloring lines, but we can also find a rationale for coloring facets. Each facet has its own z-orientation, which relates to its position in the xy plane in the folded form, but a facet also contains information about its position within that plane: its position in two dimensions, and its angular orientation relative to the axis. It was this last quantity that I began to explore.
In uniaxial box-pleated designs, the positions of vertices and creases are quantized, lying on a grid. In the same way, the orientations of creases and facets in the folded form are quantized. We can carry this information from the folded form back into the crease pattern. While lines have any of 4 different orientations: up/down, right/left, and the two diagonal directions, facets have only two orientations: up/down, and right/left.
In a design I did for the OrigamiUSA 2008 design challenge, “Irish Elk,” I tried out a facet coloring method based on orientation, leaving the up/down facets white, but tinting the right/left facets, as shown here.
This pattern incorporates several color algorithms: the circles from its circle/river packing are shown, and I have combined the generic form crease assignment with structural coloring of the lines. The structural coloring brings out an elegant feature of these patterns: although the ridge creases are constructed independently for each polygon in the pattern, the endpoints of adjacent ridge creases line up so that they form closed loops. Within each loop, all facets have the same orientation, so that even though the xy-orientation colors of the facets are assigned individually, groups of similarly-colored facets form contiguous polygons that are entirely outlined by ridge creases. These polygons give the overall crease pattern an appearance reminiscent of the Rorschach ink blot test patterns. The patterns of colors in such decorated crease patterns do not particularly contribute to the understanding of how to collapse such a pattern into the base (in fact, they might even impede it); instead, like the Rorschach tests, they serve to tickle the perception of the viewer, calling to mind associations that stem from both our stored visual experiences and the mind’s inherent response to pattern and line.
In 2009, I learned of two other peoples’ explorations of colored crease patterns: Sipho Mabona’s beautiful renderings of his own work, and plagiarist Sarah Morris’s uncredited “appropriations” of others’ work. Sipho’s artistry triggered further explorations on my part of the relation between facet color and information inherent in the relationship between crease pattern and folded form. That was also right around the time I began exploring more deeply a new variant of polygon packing, based on hexagonal symmetry, that I dubbed “hex pleating” (this also described in ODS2e).
Hex pleating, like box pleating, is based on an underlying grid, but this grid is composed of equilateral triangles. That gives the facets of hex pleating three distinct orientations in the folded form relative to the axis: 0°/180° (i.e., up/down), 60°/–120°, and 120°/-60°. Color-mapping the facets to these directions requires three distinct colors, and there is a natural fit to the three primary colors of human perception: red, green, and blue.
But we can do more that just color-map the orientation; we can also map in the z-orientation by adding the dimension of light/dark. Each facet gets assigned a base color, depending upon its orientation, which is then selectively darkened based on whether the facet is color-up or color-down in the folded form. My first attempt using this decoration scheme is shown below.
In this pattern, I also kept the structural coloring of the creases and used the full MV assignment of the lines; then, for good measure, I threw in the underlying grid. The result is a pattern that contains many levels of information.
Paradoxically, the more information we put into a decorated crease pattern, the less useful it becomes as a folding guide, as visually, the various levels compete with one another in perception. It is now quite difficult to perceive the original tree-based structure: hinge polygons, which correspond to flaps in the base, are still present, but can only be discerned with close scrutiny. Fold lines, which are the actual lines on which action takes place, are far overshadowed by the larger regions of facet color. Instead, we see the larger regions of common facet orientation as distinct entities, which is somewhat ironic, in that these regions have no particular significance in the collapse to the folded form. Visually, however, they trigger associations, and so, expand the appeal and meaning of the pattern to all of the communities of perception.
The mapping from orientation to the hue dimension of color is particularly appropriate, I think, because orientation and hue have the same topological structure: orientation varies continuously from 0° to 360° and connects to itself, and in color theory, hue can be mapped onto a circle with the same topology. Thus, it is possible to map any facet orientation uniquely to a hue in a way that preserves both closeness and continuity. So, instead of mapping both 0° and 180° facets to the same color, as in the two examples above, we could map them to two distinct colors, and similarly for other orientations.
In a uniaxial base, any given flap can be oriented in two diametrically opposite directions. If the axis runs up/down, then a flap can point up or it can point down. For a given base, we can choose the orientation of every flap relative to the axis (a good practical choice is one in which the layers stack evenly). I call such an arrangement a fully oriented base. TreeMaker assumed full orientation whenever it computed a base, using an algorithm whereby one figuratively picked up the base by a single flap and let all the other flaps dangle loosely to determine the pointing directions. However one chooses the orientation of the flaps, once a base is oriented, every facet in the base has a unique orientation between 0° and 360°; this orientation can be used to choose a hue.
By this notion, a box-pleated base has, not two, but four distinct orientations that its facets can take on, and we can assign facets one of four colors, depending on the specific orientation of each facet. An example of a scarab beetle with this four-color facet assignment is shown here. Again, I have combined the hue–orientation linkage with z-orientation, structural colors, MV-assignment, and grid.
In mathematics and physics, the concept of “position plus direction” is described by the mathematical object called a vector, and so I call this form of facet coloring vector coloring. Every facet can be described by a vector in 3-space that includes its x, y, and z orientation relative to its position in the crease pattern. That 3-vector can then be mapped to a discrete position in color space, which is used to decorate the pattern.
We may also entirely decouple the vector-coloring mechanism from polygon packing, and even from uniaxial bases. For any flat-foldable origami figure—or, for that matter, even a fully 3D figure—we can define some sort of reference frame that is the generalization of the axis and reference plane of a uniaxial base. Every facet in the folded form can be defined by 6 degrees of rigid-body motion: three of translation, three of orientation. One can then project that 6-vector down into the 3D space of color perception, and use the result to color the underlying crease pattern. The choice of projection function is up to the artist; the resulting pattern, then, will be a mixture of algorithmic and aesthetic choice.
But what good is such a pattern, anyhow? Does the projection and color-mapping really convey information useful to the folder? Or is it, in the wonderful neologism of Edward Tufte, just so much “chartjunk”? In fact, it is neither, in my view. As I noted in the introduction to this article, to most of the world, a crease pattern is not necessarily either the personal record of mental process that it is to the creating artist, nor a guide to the collapse of a base (though for a certain minority, it is definitely that). To most of the world, decorated crease patterns are a glimpse into something else: the intersection of origami, art, math, perception, and their own experiences. And toward that end, the various decoration schemes, whether algorithmic, aesthetic, or a mix of the two, establish those connections, uniquely for each viewer.