This final article on the Polygonal Letterfolds of Shuzo Fujimoto presents the Pentagonal Wrap that started my quest. Fujimoto was inspired by this design (by an author unknown to him, whom I propose was Kazuo Haga) to create square, hexagonal, heptagonal and octagonal letterfolds, which he called “wraps.” These four original designs by Fujimoto were published recently in The Fold, and links to these earlier articles appear at the end of this essay. Annotated diagrams for the Pentagonal Wrap with instructions to fold Fujimoto’s version and notes on Haga’s version can be downloaded from the link in the caption above. Complete instructions for Haga’s version can be found here, on pages 62-71.
I first discovered the “anonymous” Pentagonal Letterfold in a YouTube video, and it became one of my favorite origami models. In June, I decided to make this elegant design more widely known by publishing it in The Fold — if I could discover the creator and get permission.
My first step was to post a photograph and rough diagrams I’d made to the Spot the Creator Facebook group.1 Paula Versnick replied that she and Michel Grand had done some research and discovered that the design was published by Shuzo Fujimoto in “Twist Origami 2” (1983),2 a book of hand-drawn diagrams. But Fujimoto states “the creator of the regular pentagon is unknown.”3
Paula also mentioned that Vicente Palacios had made drawings of the letterfold in 1981, attributing the design to Kunihiko Kasahara. Michel told me (in an email dated July 15, 2024) that the Palacios diagrams are unpublished.
My next step was to ask Mr. Kasahara if the design was his. I had corresponded with him before with the help of Nobuko Okabe, who agreed to translate and send off my current query. Kasahara replied that he was unfamiliar with the model (and also Fujimoto’s “Twist Origami 2”).4
Michel and I continued to correspond by email. Michel found evidence that Kazuo Haga may have originated the design. He discovered several references to a pentagonal letterfold in articles by Haga: a mathematical article from 19855 and an ORU article from 19966 on the mathematics of folding a regular pentagon from a square (this included diagrams for two pentagonal letterfolds). But Fujimoto’s book was published in 1983 — before either of these sources — so I still had no evidence that Haga’s design preceded Fujimoto’s diagrams.
I then asked Nobuko to draft a letter to Haga asking about the origin of his design. The email bounced. Next, I asked on the Origami List if anyone had contact information for Haga. I was sad (and embarrassed) to be informed by Miranda van de Beek (in an email dated July 23, 2024) that Haga had died in December 2023. Miranda had found an article in Wikipedia Japan indicating this.7
Finally, I combined information from two books by Haga to discover the origin of the Pentagonal Wrap.
Michel found diagrams for Haga’s Pentagonal Letterfold in Haga’s “Have Fun With Geometric Shapes Through Origami: Fold Paper To See the Math!” (2014). Here is Haga’s introduction to his Pentagonal Letterfold from that book:
My First Origami Project
I was already in my forties when I plunged into the world of origami and tried to find the mathematical phenomena it illustrated. I have completed only a handful of works — a dozen or so. If I list my origami works chronologically, this can be said to be Number 1.
At that time, I was using a microscope to observe egg slices of a strange small insect, Thysanoptera [thrips]. The work made my shoulders stiff. I noticed the square note paper on the table, so I tore off a sheet and folded it randomly. Instantly, I felt that origami was very healing.
One day, I began folding a sheet of paper as usual and unintentionally produced a simple pentagon. I continued to fold paper, and after some time, I came up with the letterfold illustrated in Figure 1 [the Pentagonal Letterfold]. You can write any text you want on the inside, and after you finish folding the paper, give it to someone.
I wrote down the steps of how to make the letterfold and sent these instructions to Mr. Kunihiko Kasahara. I also sent him an example of the finished work. Once, I had stood in a bookstore reading Mr. Kasahara’s origami book but felt the lack of physical examples to help me learn.8
So now I will introduce this project! Let’s make the Pentagonal Letterfold!9
This tells us that Haga designed the Pentagonal Letterfold when he was in his forties, or between 1975 and 1985, so not necessarily before Fujimoto published his 1983 drawings.
A few days after reading the passage above, I found an English translation of the Introduction to Haga’s “Origamics.” (Published in three volumes, 1996-2005). Here again, Haga tells the story of working on thrips and using origami as a diversion. He gives the year as 1978, so he definitely designed the Pentagonal Letterfold before Fujimoto’s drew his diagrams:
My first mathematical findings on origami were done in 1978. At that time I was a biologist majoring in arthropodan morphology, and observing tiny insect embryos under a microscope, however, there was no relation between origami mathematics and insect egg study. As the microscopic study needed much time with mental fatigue and eye-strain, I often had a recess and folded a piece of paper torn off from a small notebook for refreshment.10
According to the Wikipedia Japan entry on Haga, he discovered a new genus of thrips during his research, which he named Pentagonothrips, certainly after his first origami discovery.11
Fujimoto’s diagrams differ from Haga’s in some minor points, but the basic structure is the same. The diagrams I’ve created present Fujimoto’s version and conclude with an illustrated explanation of how the two versions differ. Since I have been unable to find Haga’s heirs to get permission, I am not publishing his diagrams in full. I do not know how or when Fujimoto encountered Haga’s design, but I believe he did, somehow.
I hope you will enjoy folding both versions of this model as much as I do. As Haga exhorts us: “Let’s make the Pentagonal Letterfold!”
Endnotes
1. Spot the Creator (private YouTube group). Thread begun June 2, 2024. https://www.facebook.com/share/p/1DuUBaJ11c/. [back]
2. Shuzo Fujimoto Twist Origami 2 (Okazaki City: Second edition. Privately reprinted by Satoko Saito in 2007 from the 1983 first edition), 46-47. [back]
3. Fujimoto, Twist Origami 2, 47. Quotation translated by Nobuko Okabe. [back]
4. Email from Kunihiko Kasahara translated by Nobuko Okabe, dated July 16, 2024. Surprisingly, Haga says in his book Have Fun With Geometric Shapes Through Origami: Fold Paper To See the Math! that he sent a copy of his pentagonal letterfold to Kasahara. This must have been in 1978, but perhaps it never arrived. [back]
5. Kazuo Haga, “Making a regular pentagon from a single square with Your Bare Hands,” in Mathematics Seminar (Nippon Hyoronsha: May 1985), 82-89. [back]
6. Kazuo Haga “Single-Piece Folding Into Geometric Shapes,” ORU Magazine 13 (June 1996): 58-61. [back]
7. “Kazuo Haga,” Wikipedia Japan, last modified December 2023, https://ja.wikipedia.org/wiki/%E8%8A%B3%E8%B3%80%E5%92%8C%E5%A4%AB. [back]
8. As mentioned above, Mr. Kasahara does not recall seeing the Pentagonal Wrap. [back]
9. Kazuo Haga, おりがみで楽しむ幾何図形 紙を折れば数学が見える! (Have Fun With Geometric Shapes Through Origami: Fold Paper To See the Math!). (Shinso: December 2014) ISBN-13: 978-4797349573. [back]
The English translation in this article is based on the Chinese edition, 摺紙玩數學 - 日本摺紙大師的幾何學教育 (Origami Playing With Mathematics - Geometry Education from a Japanese Origami Master (Taipei: Future View Technology: 2014): 62-69. ISBN 978-986-92837-0-0). It is available online here: https://archive.org/details/origami-mathematical-geometry-2016/page/62/mode/2up. Excerpt translated by Valerie Samson and Tao Peng.
10. Kazuo Haga, ORIGAMICS - Mathematical Explorations Through Paper Folding, English edition (World Scientific Publishing Co., September 12, 2008), xi-xii. ISBN: 978-981-283-490-4. [back]
11. “Kazuo Haga,” Wikipedia Japan. https://ja.wikipedia.org/wiki/%E8%8A%B3%E8%B3%80%E5%92%8C%E5%A4%AB. [back]
Bibliography
Fujimoto, Shuzo. Twist Origami 2. Okazaki City: Second edition, privately reprinted by Satoko Saito in 2007 from the 1983 first edition. The book is available online as a free download here: https://archive.org/details/twist-origami-2.
Haga, Kazuo. おりがみで楽しむ幾何図形 紙を折れば数学が見える! (Have Fun With Geometric Shapes Through Origami: Fold Paper To See the Math!). Shinso: December 2014. ISBN-13: 978-4797349573.
Haga, Kazuo. Origamics (English edition). Singapore: World Scientific Publishing: 2008. ISBN:978-981-283-490-4.
Haga, Kazuo. 摺紙玩數學 - 日本摺紙大師的幾何學教育 (Origami Play Math: Geometry Education From a Japanese Origami Master). (Chinese translation of Have Fun With Geometric Shapes Through Origami: Fold Paper To See the Math!) (Taipei: Future View Technology: 2014) ISBN 978-986-92837-0-0. The Chinese edition is available online as a free download here: https://archive.org/details/origami-mathematical-geometry-2016/mode/2up.
Haga, Kazuo. “Single-Piece Folding Into Geometric Shapes.” ORU Magazine 13 (June 1996): 58-61.
Kosmulsky, Michał. “Shuzo Fujimoto — First Origami Engineer?” The Fold 72 (September-October 2022): /thefold/article/shuzo-fujimoto-first-origami-engineer.
“Kazuo Haga,” Wikipedia Japan article. https://ja.wikipedia.org/wiki/%E8%8A%B3%E8%B3%80%E5%92%8C%E5%A4%AB.
“How to Fold an Origami Letter.” You Tube video, https://www.youtube.com/watch?v=yEMhkzKnnKk.
“Spot the Creator,” private YouTube group. https://www.facebook.com/groups/spotthecreator.
Acknowledgements
I am grateful to Hitoshi Fujimoto and Shumi Okada, Shuzo Fujimoto’s heirs, who have made their father’s work openly available and have specifically allowed me to recreate the Polygonal Wraps. Paula Versnick read my request in Spot the Creator and, along with Michel Grand, led me to the pages in “Twist Origami II” that contain all the Polygonal Wraps. Michel tracked down many of the puzzle pieces I needed to discover Haga’s creation of the Pentagonal Wrap. Nobuko Okabe was a tireless translator of Japanese emails and other writing. Valerie Samson, Dawn Erickson and Tao Peng worked on Chinese translation. Miranda van de Beek found the article indicating that Kazuo Haga died in December 2023. Ian McRobbie and Judith Peterson reviewed my diagrams and Laura Rozenberg and Val Landwehr reviewed the text. Finally, I must thank Michał Kosmulski for bringing so much of Fujimoto’s work before the public. Please see his article on Fujimoto in The Fold.
Other Articles in This Series
Diagrams: Square Wrap
Diagrams: Hexagonal Wrap
Diagrams: Heptagonal Wrap
Diagrams: Octagonal Wrap