This is one of my earliest original designs, yet even to this day, it remains one of the works I am most satisfied with—primarily because of the successful shaping of the head. It is also likely the design I have folded the most times. When I created it, I did not reference any specific species of bat; instead, I relied entirely on my own impression of bats. As a result, while the anatomical structure may not be entirely accurate, the model nevertheless captures the striking and charismatic essence of a bat.

At the time, my understanding of systematic origami design theory was still limited, so this model was created entirely using classical methods. In particular, I applied grafting and splitting techniques to modify the traditional bird base in order to establish the fundamental structure: grafting extends the wingspan and provides a larger surface area for the wings, while splitting was used to create the legs and the tail from one flap of the bird base. The remaining details (especially the head) were gradually refined through the process of trial and error.

這篇文章 Impressionist Bat, 2013 最早出現於 Tsai Origami。

The inspiration for this particular origami model came in a rather amusing manner. As many of us are familiar with the clichéd scenario where a specialist is humorously asked to perform stereotypical tasks related to their field by someone less knowledgeable, my experience echoed this theme. It all began when a layperson, upon learning of my origami expertise, inquired whether I could make a traditional pinwheel.

To be perfectly candid, I felt a twinge of offense at the question, as a pinwheel doesn't involve origami in the true sense. It doesn't require folding; instead, it involves cutting and bending the paper, which is fundamentally different from the art of origami. However, after I explained this distinction, a creative spark ignited within me. I wondered, why not challenge myself to design a pinwheel using only folding techniques, entirely devoid of cutting or the use of glue?

Fortuitously, I had already formulated the foundational concepts of my edge-river method (ERM), which adeptly addresses the challenges posed by such non-uniaxial designs. The crease pattern and the ERM map for this particular model are illustrated in the following figure.

In this map, the four lakes serve as the surfaces for the pinwheel. The gaps between each pair of lakes organically give rise to the river structures and the peninsulas, resulting in a spacious central area on the sheet that can be utilized to craft the spinning axle. Additionally, the entire layout is strategically tilted to facilitate the creation of locking components at the four corners, while strategically ensuring a straightforward and manageable initial folding sequence.

A commonly asked question regarding the initial folding sequence revolves around the significance of the 21/41 reference point. Let me clarify this intriguing detail. The core of the pinwheel structure necessitates an octagon with dimensions specified by the blue lines in the figure above. As previously mentioned, this octagon must be tilted to accommodate the locking flaps. While it's possible to adjust the tilt in a way that it occupies a larger area within the square (at the cost of shorter locking flaps), doing so would require locating all eight edges of the octagon, which would be rather cumbersome.

Instead, I opt to tilt the octagon in such a manner that its eight edges align precisely with the angle bisectors of the red triangle. By doing this, in conjunction with the tilted blintz fold, I only need to locate two reference points. It just so happens that the red triangle that fulfills these criteria corresponds to the 20,21,29 Pythagorean triangle, hence the magical ratio of 21/41.

However, one question lingers: why did I opt for this specific octagon rather than a regular one? The answer lies in the delicate balance of the spinning axle's width. My experimentation revealed that subdividing the center region into a 16\times16 grid made it too thick, while an 8\times8 grid wasn't narrow enough. As a compromise, I settled on a 12\times12 grid. Because of this choice, I decided that I should choose an octagon of integral proportion, so that the precreasing can be done simply by using the edges as references.

The entire folding process typically takes around an hour and a half, making it a somewhat tight timeframe to cover everything within the allocated two-hour session. Fortunately, we managed to achieve just that. I'm genuinely delighted to witness how this design of mine, which has endured over the years, has now found appreciation and enjoyment among others.

這篇文章 Pinwheel without cuts, 2016 最早出現於 Tsai Origami。

Reason for recommendation:
Best overall single-authored instructional origami book I have seen, judging by the quality of designs, folding sequences, diagramming, etc.

No, I'm not saying that I've seen a lot of instructional books (definitely not as much as those book collectors at least), but it's hard to imagine an even better book than this one.

Instructional books are meant to let people enjoy practicing origami, and there are several factors that determine how enjoyable they are.

The first thing is the quality of the designs themselves, and that also breaks down to several factors. Are the models aesthetically pleasing? They surely are. Is the idea behind those designs novel and inspiring? Definitely yes, most notably as Kato made famous his "tilted grid system" through various applications in this book, which has since inspired many new designs by other origami artists. Do the models provide enough challenges for enthusiastic folders? Certainly, many models in this book are among the highest complexity of all sequentializable designs.

The second thing is the quality of the folding sequences, and this is perhaps the most impressive part of this book. Many designs in this book are typically considered not sequentializable, but Kato managed to sequentialize them nevertheless, and his folding sequences are considerably fair to follow. I would say he has pushed the boundary of sequentializability.

Finally, we have the quality of diagrams and other ingredients in the book. Kato's diagrams are textbook examples of what good diagrams are about: clean lines, just the right amount of offsetting of edges, good use of shading, and amazingly well-drawn 3D steps. Especially after I've done my first set of diagrams earlier this year I am astonished by the time and effort he must have put into this book. He also incorporated many of the best elements to make this book excellent: recommended sheet sizes, the relative sizes of finished models, beautiful photos of finished models, and the silhouette indices near the corners. Kato's book is no doubt a role model for all origami authors.

Order Now!

這篇文章 Origami Nature Study by Shuki Kato 最早出現於 Tsai Origami。

Recently I finally completed my first computer-drawn diagram featuring one of my most queried models, Seigaiha without cuts. It was quite an interesting task and I really learned a lot from it. In this article, I would like to share some of my thoughts on drawing diagrams.

The diagram is for sell on my Ko-fi shop, but you can preview the results here.

Inkscape

I drew this diagram using Inkscape, which, despite it being free and open-source software, is less chosen for professional diagramming as much as I know. Perhaps that is due to the fact that many designers have found it difficult to use for various reasons. In contrast, commercial vector drawing software such as Adobe Illustrator or Affinity Designer are more popular choices. But Adobe apps are well-known for their relatively higher prices, while Affinity Designer, though much cheaper and are promoted by a group of origami designers, simply doesn't feel intuitive enough to me (it definitely could be just me, since I didn't spend much time digging into it either). I was hoping that a completely free solution could exist, but when it comes to free and fully-featured vector drawing apps, Inkscape is pretty much the only one on the market, and previously I also would agree that it just doesn't cut it.

However, I would say that things are about to change. Recently the development of Inkscape has become really active, and many improvements have been done in the upcoming version 1.2 (still in beta as of this writing), including performance (Inkscape used to be REALLY slow; not anymore!), multiple page support (and can be saved as multi-page PDF directly), and various bug fixes. Although the beta version still contains bugs (which I am constantly reporting myself) and it still lacks a few features that I think will make it even more suitable for origami diagramming, I could pretty much testify that version 1.2 beta is good enough for creating professional diagrams.

You can download Inkscape 1.2 beta here. It works for all desktop platforms from Windows, Mac OS to Linux.

Basic configuration

In order to make Inkscape suitable for drawing origami diagrams, the most important setting is the rotation snapping as shown in the following picture. By setting the value to 22.5 degrees, the lines we draw will automatically snap to an angle that is a multiple of 22.5 degrees when we hold Ctrl key. You could also change it to 30 or 15 degrees when your model is 30-degree based.

However, Inkscape unfortunately doesn't provide 11.25 degrees as one of the options. I personally found the easiest way to create a line in 11.25 degrees is by first drawing a vertical line, rotating it by exactly 11.25 degrees (using the rotation transform in the Transform dialog), and then resize it to the length I need (by holding Ctrl as I resize it). For this method to work, it is also very important to keep the "Scale stroke width" option unchecked, so that resizing a line will not affect its width.

You would also need to set the bounding box to geometric mode for this to work. Otherwise, the direction of the line will actually change slightly as one resize it.

The perfect mountain-fold line pattern

It is conventional to draw the mountain-fold line in a chain pattern (dash-dot-dash, or sometimes dash-dot-dot-dash), as shown in the next picture. The first one is the line pattern one will find in most of today's literature, where the "dot" is really a tiny square of the same size as the width of the "dash" part. To me, that's simply not quite satisfying. I want the dot to be a circle instead, and it should be slightly larger than the width of the dashes, as shown in the second line pattern.

The good news is that Inkscape does provide a function called "Pattern along path" that enables us to create such a line pattern. To use it, we need to first create a "sample pattern" as shown in the next picture. Notice that it is a single path object (not a group of objects!) consisting of multiple subpaths (you could use path-union to create such an object), and it must be a filled path without stroke width.

Once you create the sample, copy it into the clipboard. Next, draw a line as usual, and then go to Path → Path Effects to open the Path Effects dialog, click the plus-sign button to add an effect, and select "Pattern along path".

Now click on "Link to path in clipboard" (which is the upper-right button in the next picture) and select "Repeated, stretched" in "Pattern copies". This will create exactly the type of line we want (make sure that the line itself is filled with black, and has no stroke color). It is also helpful to check the "Hide width knot" option so that only the end-point knots are shown when we edit the path. Of course, there's no need to repeat this process every time we want to draw a mountain line; once we created one, just duplicate it as we need.

One bonus upside about drawing patterned lines this way is that the pattern is "stretched" automatically, which means that no matter how long the line is, it will always contain exactly an integral number of repetitions of the sample pattern (as shown in the next picture). In our particular case, that means the line will always end in the "dash" part and never in the "dot" part. I've found this property to be extremely satisfying, as I would often need to adjust the "pattern offset" of a line by using the native stroke pattern mechanism.

Drawing arrows and symbols

The template file also contains the symbols I used in my diagram, and you can access them from the Symbols dialog.

One important note: as of this writing, due to a bug in Inkscape (which I have reported here), if you copy and paste a line that has a marker, it will create a duplicate entry in the marker list, which is pretty annoying. In order to avoid that, the workaround is to use "duplicate" (hotkey Ctrl + D) and then move it to the new location instead of copy and paste.

Creases stop before reaching the edges

It is conventional that creases, usually drawn in thin lines, stop before they reach edges (folded or raw). I think the way most people do that is by manually shrinking each line slightly; I think so because I often observe that they missed adjusting a few lines here and there. Doing this type of manual adjustment has at least three downsides:

1. It is quite inefficient, obviously.
2. It is hard to make the gap width consistent.
3. By doing so, the crease lines no longer intersect the edges. Those intersections could otherwise be quite useful for snapping.

A much better way to do this is by using clip paths to make the crease lines invisible near the edges. It only takes a few operations to handle all creases in an edge polygon at once, with consistent gap width, and the intersections are all kept for further use. In Inkscape, this can be done by the following steps:

1. Select an edge polygon and duplicate it (default hotkey Ctrl + D).
2. Use the inset command (default hotkey Ctrl + )) to shrink it. I usually perform it twice for most edge polygons, and once for smaller polygons.
3. Select one of the crease lines and the shrunk polygon (making sure that the polygon is above the line), and perform Object → Clip → Set. (This command has no default hotkey. I set personnaly it to Shift + C.) Doing so will create a clipped group.
4. Select the rest of the crease lines and move them into the clipped group in the Objects and Layers dialog.

Alternatively:

3. Select all crease lines in the polygon and group them (default hotkey Ctrl + G).
4. Select the group and the shrunk polygon, and set the clip the same way. This creates a new group around the said group. You can then ungroup the said group to simplify the hierarchy (however, I've found that keeping the extra grouping is helpful for future editing as well).

Make use of clone objects

It is commonly the case that later on we find some mistakes that took place in a much earlier step, and the error was copied and pasted throughout all the following steps. Clone objects could make our lives easier. It keeps the clone identical to the original object (can be a group) even if we modify the original object later on, so if we find any error, we only need to make the correction once. Clone objects can be created by the hotkey Alt + D.

Layer management

I use at least three layers in my document, which I call base, action, and reference. The base layer is where most of the drawings (paper layers, edges, existing creases, and text) take place. The action layer includes things that represent the operations in the current step (mountain/valley folds, arrows, and symbols). I put these into a separate layer so that I can easily skip those objects when I copy and paste the previous view to the next (all I have to do is temporarily lock the action layer, and drag-select everything in the previous step). Finally, the reference layer is where I put photos of my test piece for tracing complicated 3D steps. I will set the photos to about 30% opacity, lock the layer and trace it. After I'm done, I can just make the entire layer invisible and continue.

Styling

I use A4 size pages for this particular diagram, which is probably not ideal for physical publications, but it certainly works for digital ones.

I use three different line width:

• 0.1mm for existing creases.
• 0.2mm for edges.
• 0.3mm for things in the action layer.

This is roughly based on the suggestions given by Nicolas Terry, and having them in a progression makes it easy to change the storke width by just clicking the plus/minus buttons in the Fill and Stroke dialog.

I use 3mm Arial Font for normal text, filled with black and without stroke. Then bold font for the step index.

The color side of the paper is set to 30% gray, and the shades are black with 30% opacity.

這篇文章 Diagramming with Inkscape 最早出現於 Tsai Origami。

Some of you might have noticed that I recently wrote a series of articles about my old works dated years ago. I did that in preparation for an origami exhibition that I'm participating in for the first time, which takes place in Miaobei Art Center from March 2nd to April 17th. The exhibition is orchestrated by the Taiwan Origami Association and exhibits hundreds of models by more than 20 artists around Taiwan.

The plan for this exhibition was announced by chairman Chang of TOA at the end of last year when the art center contacted him. With 2 months available for preparation, one would expect me to fold more models for the exhibition. The problem was, my first daughter was also born around that time, so I was simply too busy to fold more. Plus I am super stubborn that I insist on displaying only my original designs, but then another problem is I didn't keep all my folds throughout the years. Besides my most recent folds, the best I can do is to select some of my older models that I still keep in good shape to enter the exhibition. In the end, I selected a total of 8 designs, just enough for one display platform. Not much, but good enough for me.

When it comes to origami, I'm arguably more interested in the theory of designing than in the final execution, and I think it would be a great idea to present the thoughts behind each of these designs, so I came up with the idea of showing the corresponding ERM maps next to models, together with QR codes linking to articles about them on this website. I choose ERM maps over the traditional mountain/valley CPs, because I think those are better for conveying the structure of the models, especially for non-uniaxial models.

On March 1st, I went to the art center setting up my own exhibits as well as helping other members of the association get everything ready. I must say, getting an exhibition ready is a lot more work than one would expect, but the members are all awesome people and working with them was quite an enjoyable experience. I also learned a lot by chatting with them about folding tips and thoughts on holding an exhibition. By the way, it was also my birthday actually, and there's no better way to celebrate it than this.

If you have seen the exhibition by TOA last year, we have more exhibits this time which I'm sure you won't want to miss. Be sure to visit Miaoli county and go check it out!

這篇文章 First experience of origami exhibition 最早出現於 Tsai Origami。

In goldfish terminology, "fancy" means double-tailed. They were first documented in the Ming Dynasty (C.E. 1368-1644) of China. From an evolution perspective, such mutation is detrimental as it makes them incapable of swimming faster and thus less competitive for natural selection. Goldfish breeders, however, do love such appearance, and various breeds has been kept ever since due to articifial selection. This design of mine was roughly based on a certain breed like the one show in the next picture.

This model was my first fish design and an example of applying my early knowledge of ERM to the 22.5° system. I still kept my old note back then, which contains my hand-drawn abstraction and the ERM map for this design, as shown below. One may compare the map with the actual map shown later and see how much they resemble each other. (In the sketch I only put one anal fin; I later realize that fancy goldfish actually have two anal fins as well, so I fixed it in the final version.)

One of the major difficulties with fish designs is that the dorsal fin tends to consume a large area due to the additional rivers in its front and rear side. This typically will cause the lakes corresponding to the body to separate far apart at least on one side, leaving huge space there that is difficult to justify. However, the doubled tail of this subject cleverly justifies the space on the rear, and it also leads to a symmetric layout that is easier to fold.

這篇文章 Fancy Goldfish (Carassius auratus), 2016 最早出現於 Tsai Origami。

This design wasn't exactly based on particular species of Cockatoo, but it more or less resembles the sulphur-crested cockatoo, yellow-crested cockatoo, or the Major Mitchell's cockatoo (without the color changing effect). To be honest, I didn't thoroughly study cockatoo species and anatomy before I work on this design, so I made several mistakes on my first version. It was Robert J. Lang who pointed out those mistakes and drove me into fixing them as much as I can.

There are still some anatomical imperfections about this final version, but from a design perspective, it's not a bad design at all. It has a fully closed 3D body, a 3D head, and a very efficient ERM-inspired layout. The idea of allocating the crests as middle flaps may sound crazy, but it turned out that doing so nicely balanced the outflow and also results in a very interesting folding sequence.

However, as mentioned, there's still room for improvement. Arguably the biggest issue is that I didn't make the color change at the crests and the layer distribution made it look like the crests are coming out from the back of the head (whereas they really all come from the front). Another structural issue is that I used a bird base to create the claws since I initially believe that cockatoos, like most birds, have a 3-to-1 claw formation, but only later did I learn that they actually have a 2-to-2 formation, and it's much harder to shape a bird base nicely into that formation. I would certainly do these differently should I revise this design.

這篇文章 Cockatoo (family Cacatuidae), 2017 最早出現於 Tsai Origami。

I had mentioned this work in my previous article Senbazuru from a single uncut square, and in this article, I would talk more about the background of this design.

In the history of origami, the earliest known publication dedicated to recreational paper folding (as opposed to ceremonial folding) is Hiden Senbazuru Orikata (meaning "the secret of folding a thousand cranes") by Gido Ichien (pen named Rokouan) in 1797. The book contains 49 designs in the category of what is now known as Renzuru (connected crane), each folded from a square or rectangular sheet of paper with cuts. The book shows only the CP and the folded result of each design, with no instructions. In many of its designs, the cranes (or technically orizuru) are connected only by a tiny piece of paper and need to be folded very carefully to not tear the connection. A full list of its designs can be found on the website orizuru49.com.

One of the designs from this book is titled "Seigaiha" (meaning "the wave of the blue sea"), which is a name borrowed from one of the traditional Japanese decoration patterns. The folded result is basically 9 orizuru in a 3 × 3 formation, connected only at the tips of the heads, tails, and wings.

This, of course, is done by cutting slits on the paper (refer to the link above for instructions). Even with all the modern design techniques for super-complex models, it is still pretty much infeasible to fold to exactly this result from a single uncut square, as those places where the orizurus connect will be ridiculously thick in practice.

However, if we compromise by accepting a "base plane" to exist underneath these orizurus, the same formation is then completely feasible from a single uncut square.

In the ERM map shown above, we can see that the concept of this model is really simple. The lakes become the base plane, connected by rectangular rivers, and the circles forming the orizuru are placed in the spaces in between.

Despite its simplicity in concept, its folding is perhaps even more tricky than the original Seigaiha. In my note, I mentioned something like "96 closed sinks and 144 spread squashes", and that's only a tiny part of the process. It took me four days to fold this model, and after that, so far I've never folded it a second time.

Update

I later fully diagrammed the folding sequence of this model, and as a by-product, I did fold this model again. I highly doubt that I will ever do it once more, but who knows?

The full instruction of this model is now for sell on my online store:

Buy the full instruction diagram of Seigaiha without cuts

這篇文章 Seigaiha without cuts, 2015 最早出現於 Tsai Origami。

A general rule of thumb for origami design is that long, thin appendages should be allocated on the corners or the edges of the sheet for better efficiency. For spider subjects, most designers would choose to allocate all legs on the edges in an octagon formation, since using the corners would lead to an asymmetric layout.

Such a formation, however, tends to leave a relatively large empty space (comparing to the leg length) in the center. For spiders with larger bodies, such as tarantulas, we can still fully utilize the empty space for the body, but for slim spiders such as orb weavers, the space in the center is usually too large to be fully utilized.

To solve this problem, I looked into books on spider species, trying to find one with more features on the body so that I can fully justify the space in the center. And this particular spider soon caught my attention. It has a total of six spikes of its abdomen, a perfect choice for space utilization. The name "Micrathena" means "micro Athena", and "sagittata" means "arrow shaped".

In this design, I deliberately allocated the tips of each leg slightly off the edges of the sheet, making them strong enough to hold the body without installing steel wires. I also faithfully reflect the actual leg length differences in the design.

In retrospect, this design can still be improved quite a lot. The method I used to form the abdomen is overkill, and the last pair of spikes should be made longer (in some individuals, those spikes are really long). I also didn't make the color changing on the abdomen. I'm sure I would revise this design someday.

這篇文章 Micrathena sagittata, 2017 最早出現於 Tsai Origami。

This model was in some sense a tribute to John Montroll, who is the first person in origami history to accomplish the same subject from a single uncut square, with the correct number of major appendages, which was published in Animal Origami for the Enthusiast (1985). I try to use the smallest possible grid size to achieve all major features comparable to Montroll's design while making it 3D and optimally efficient. I managed to do all that in a grid size of just 25 (with some half-unit structures). I wouldn't say it's not possible to use an even smaller grid, but very likely the result would be off in proportion.

Anthropods are especially suitable subjects for uniaxial box pleating, hence there's no need to bring ERM into this design. Still, I present the map in the figure above with lakes to show where the body is allocated. One thing about uniaxial box pleating is that in many cases there would be no land at all.

The crease pattern of this design uses a variation of GOPS near the head, which was originally discovered by test folding. The studying of that particular pattern later leads to the discovery of the general transformation rules of GOPS.

這篇文章 American Lobster (Homarus americanus), 2016 最早出現於 Tsai Origami。