by Ushio Ikegami
Edited by Tom Hull
The crease pattern of a tree curve.
We are publishing the article as a PDF. Below you will find an introduction by the editor and a foreword by the author’s friend.
Download “Fractal Origami for Beginners.”

From the Editor

“Fractal Origami for Beginners” by Ushio Ikegami is unusual but important in mathematical, geometric origami design. Speaking personally, whenever I see an origami artist refer to a model of theirs as a “fractal” I have to smirk inside. In math the word fractal is used to describe geometric objects that have unusual properties with their length, area, or volume. For example, a “fractal curve” would be a curve in the 2D plane that has infinite length but surrounds or contains a finite area. A circle does not do that — it has finite perimeter and surrounds a finite area. Thus, fractal curves have to be very strange. They often look bumpy and jagged. In contrast, there are many geometric origami models that are called fractal even though they do not have this infinite perimeter property. Rather, these so-called origami fractals possess a type of repeated symmetry (often called self-similarity) that is also present in many fractals. But simply having this type of repeated symmetry does not make something a fractal.

On the other hand, Ushio Ikegami’s origami fractals truly are fractals. His first success at making a real origami fractal was his Fractal Pyramid, which was used for the logo of the 4OSME conference in 2006, but which was so complicated to actually fold that diagrams were not published until 2021, here, in The Fold. In fact, these diagrams are more of a method for attempting this model. Why? because the infinite perimeter versus finite area aspect of fractal curves (which, incidentally, provides another solution, the Tree Curve in the article, to the famous Napkin Folding Problem means that achieving more and more detail in an origami fractal design requires altering the crease pattern quite a bit at each iteration, something drastically different than what one sees in more simple self-similar origami designs.

In other words, a non-traditional approach to origami is required when trying to fold origami fractals. Therefore, it makes sense that when trying to communicate his ideas, Ushio decided to present them in a non-traditional way. Enjoy!
— Thomas Hull


This paper, though I would rather call it just a note, is a discussion about the most esoteric and least-studied origami research field: recursive origami of multi-directional growth. Although Ushio, the author, refers to it as fractal origami or fractal folding, the actual theme in these strange pages is how to construct a multi-directional recursive folding system. However closely related these two are, they are not the same thing. Despite his lack of awareness of academic preciseness, I am truly willing to acknowledge that in every aspect, this paper is a remarkable milestone in this field and will live a long life as the one and only available reference for all the people interested in this topic both casually and seriously. It has everything that one needs to know: theory, approach and basic exercises leading the reader to the top-notch projects that no one, even the author himself, I believe, has ever dealt with.

Reading through his writing, you will soon notice that his sense of date is oddly wrong. The publication of the diagrams of Fractal Pyramid in The Fold was not “in May” of this year. It was in May of 2021. This is because I had received this manuscript from him a couple of months after the publication of the diagrams and was waiting his reply to my editing request through 2021 and 2022. (I still am.) To answer questions from curious readers and to assuage my grief, I shall recount what happened during our exchange over the manuscript.

Although the significance of Ushio’s paper was clear enough, it was difficult to publish it even on a web page. His writing was quite informal and tended to get metaphysical in places, unlike a technical paper. Furthermore, Figure 21 showed nothing more than a sketchy drawing; it should have provided readers with a reasonable amount of information to convince them to follow the fundamental change in the design structure. I asked Ushio to revise the writing entirely with details for Figure 21. He returned, to my surprise, an email next morning, simply saying “I would prefer not to.” I sat awhile in perfect silence, rallying my stunned faculties, after which I promptly replied asking the same thing in a confused manner. “Do you not see the reason for yourself?” he indifferently replied. It seemed to me that while I had been asking him, he carefully revolved every statement that I made; fully comprehended the meaning; could not gainsay the irresistible conclusion; but, at the same time, some paramount consideration prevailed with him to reply as he did. Utterly perplexed, I asked once again what he meant by that. He did not answer and has not contacted me since then.

During the tough time of the pandemic, this matter almost faded into oblivion. It was not until recently that I found a copy of the manuscript I had printed out in 2021. In a brief tranquil moment staring in the tired sheets of paper in my hands, I told myself that for the sake of the future folding culture, I must not let them remain undiscovered, and I consulted with our mutual friend, Dr. Thomas C. Hull, for a possible means of publication. Under Dr. Hull’s careful examination, the manuscript is being released intact in The Fold, as it was when I first received it. (Except for the last passage running some 80 words, which was completely off topic — we had to delete it.)

What the author says in this paper is astonishingly simple and viable. Hasty folders would get ready in 40 seconds to jump into its exercises or their own creations. Some might even see things people would not believe (like fires off the shoulder of Orion or C-beams glittering in the dark near the Tannhäuser Gate). However, regardless of how empowering it is, anything you do in this field requires an immense amount of dedication. Going through the first batch of problems from Exercise 1.1 to Project 1.3 could easily encompass an entire four years of university study. By the time you get to Project 4.3, you will have spent at least a solid one score years out of the three and a half we all are supposed to have received at birth. Even strong fourscore years shall not make any difference. It is a grave life decision. Of course, there is always a chance of an ingenuous novice nonchalantly sweeping them all, like Jack Smurch, the greatest man in the world, did in the history of aviation on July 17, 1937 — but it also cost his life despite the immediate hospitalization right after his perfect landing on Roosevelt Field. Though someone gallant might still give it a go, what is expected most likely is decades of awful wandering and a sheer dead end, where the author may be lying dead (stone-still, wax-white, paper-thin). It is not worth risking your life like that, otherwise guaranteed happy and well. To tell you the truth, my conscience is still in doubt whether this paper should be published. What is the point of disclosing a horrifying black hole to the world? In a sense, it would be a fair statement that I am only presenting what I witnessed through the 20 years of our friendship: the author’s will to pursue what he longed for.

It was in the lounge of the mathematics department at Princeton University, filled with the warm sunlight of spring or early summer, that I met Ushio for the first time. He was with Chris K. Palmer, who was visiting a friend of his, Dr. John Horton Conway. There were a couple of colleagues at the table. I had known Ushio was staying in the U.S. for a while through the courtesy of OrigamiUSA and Makoto Yamaguchi. I took advantage of that unexpected opportunity by casually joining them, and we have been friends ever since. Ushio did not talk much then but I recall his showing a biaxial flasher and a four-piece modular icosahedron. His choice of study during that rather short stay abroad looked reasonable, as opposed to making origami fractals a prime focus, ruining any chance of the world seeing any other new designs from him. On the contrary, after going back to Japan, he started working exclusively on fractal subjects. I used to warn him not to do so for fear of narrowing his potential creativity and harming his future career in origami by going too far into a minor field. Rumor says that Mr. Yamaguchi warned him of the same thing. Although fortunately Ushio was able to keep achieving new results in the fractal field, the versatility that I saw in the U.S. did not really flourish back in Japan. I must confess to having developed unaffected scorn towards his short-sighted stubbornness during that period. However, if personality is an unbroken series of successful gestures, then there was something gorgeous about him. It had nothing to do with that flabby impressionability, which is dignified under the name of the “creative temperament” — it was an extraordinary gift for hope, a romantic readiness such as I have never found in any other person and which it is not likely I shall ever find again.

After all, no matter what field or profession you are in, or are going to get in, you will be asked the same question over and over: “Are you going to stay in this field, or not?” Either choice is right, as long as you are fine with it. Ushio said yes in his youth; and said no at the midpoint of his life. That is all fine. He will find a next life sometime soon. I do not mean the circle of transmigration here — it is the same with the rest of us struggling going through everyday life. I assume this paper proposes not only challenges in origami but also an insight into the question forever facing mankind: Are you ready to stay or not? The question is spoken, heedless of our readiness. Time goes by faster and faster leaving behind us mortals doomed to die, like a wind impossible to chase after.

John Ray, Jr., Ph.D.
December 5, 2023

P.S., If you ever see Ushio someday somewhere, maybe on social media, since the net is vast and limitless, please tell him I still have his jar of mayonnaise.

Copy editing by Jane Rosemarin.