Möbius Torus from Origami オリガミ製メビウスのトーラス…

8 years ago, I posted an idea about making an origami tube into a torus. Rolling flat origami into a tube

In fact, the folding pattern is too difficult to fold for mass-production and the curvature is very subtle as the pattern is too rigid to curve and creates a huge radius. So this time I made a different model with a simpler pattern.  The section of the tube is designed to be an equilateral triangle. Ignoring the zig-zag pattern of the folds, the torus looks like it is composed of 3 twisted surfaces.  If you trace any of the 3 ridges you can see they disappear before coming back to the starting point, yet the 3 ridges are continuous and part of one envelope. It is like a Möbius strip which has a continuous front and back, but this torus has an additional surface. When you make this tube from paper, the two ends of paper are combined, but the pattern where they meet at the ends is shifted like misaligned buttons holes. That is why the triangle spirals.

8年前に 丸めてチューブにと題してオリガミでトーラスを作るというアイデアを紹介した。実はその時のパタンでは量産するには骨が折れるし、曲がりがゆっくりとした巨大なリングを作らないといけない。しかし別なパタンなら部屋の中に納まるかもしれない。トーラスの断面を工夫すれば捻じれが良く見えるだろう。断面が正三角形になるようにデザインしたのが下の写真。折り目のジグザグを無視すれば、三面でできているように見える。しかし三角の一つの角をトーラスに沿って目で追って行くと一周しても元の点に戻らないで後ろ側へ入って行く。実は一面しかない。すべて一続きになっている。まるでメビウスの帯のみたいだが、帯の裏表の面だけではなくて、もう一つの面を持っている。メビウスの帯を作る時に端をねじってリングを作るのと同じように、ここでも、ボタンのかけ間違いみたいに紙の端部をずらしてジョイントすると捻じれが生じる。

By definition, a symmetrical shape becomes the same shape when it is rotated through certain angles, so a tube can be combined at both ends after spiraling to become a torus. In the case of equilateral triangles it can rotate 120 , 240, 360, 480…. to create a torus. In fact an odd number of corners like a triangle or a pentagon cannot meet at the ends without shifting the “button holes ”. On the contrary, an even number of corners does not have to be shifted to have those ends meet.


Above is an example of a rhomboid section. The ends meet when rotated by 180 degrees. The twist is very slow so that the total length of the paper reaches 8m before it is folded.  A 360 degree rotation will require twice that length of paper. By the way, when tightly folded, both the triangle and the rhombus lose the twist and become a simple column. The spiral happens only when they are expanded (see the photo below).


left; compressed triangular section, right: rhombus section. 左:三角断面、右:菱形断面

In the image below can you see that the triangle spins 360 degrees and comes back to the beginning? The 3 ridges spiral along the envelope of the torus,  which are not the crease ridges of folding. Lets call that ridge the “ modulated ridge“ to distinguish it from the creases of the envelope that are created by folding. Any one of the 3 modulated ridges will come back to the starting point, without crossing each other. This means they are independent and triple helical. A rotation of  120 or 240 degrees will result in the ridges aligning. There is only one modulated ridge, there is no front and back but a continuous surface. ( in the same way the DNA double helix does not cross). In detail, the creases that make up this ridge are also a single continuous surface.


This model is designed to spiral even when tightly folded. This speeds up the twisting along the tube faster than the previous models, and thankfully the torus has a smaller radius. Additionally in the previous models the weight of the paper pulled down the shape creating kinks in the form, the twisting speed is even here so that the influence of gravity is negligible and kinks don’t happen.

このモデルの場合、筒を折りたたんだ時にも その形自身が螺旋を描くように折りパタンを計算した。面白いことに、そのために捻じれの速度が速い。トーラスの径が小さくなるので、大変ありがたい。しかも、重力で垂れ下がる部分が無視できるほど小さく、捻じれの程度がどこでも一様なので折れ曲がりキンクが起こらない。

By the way, suppose these models are 1/100 scale architectural models! The ceiling would be around 15m high. What do they look like inside? Can you imagine? It will be amazing!!





new ORIGAMI hat design at Easter Parade 新オリガミ帽デザインイースターパレードで

The Easter Parade came again this year. We have been creating new hat forms since we joined the parade in 2013. This year we experimented with a series of hats created from a single tube of origami. They are all composed of one sheet of paper (to be precise a tube). Six papers are combined to form a long tube, otherwise one long sheet of paper is cumbersome to handle. Precisely speaking, these final forms are not mathematically “origami”. Making a tube with origami is a precise mathematical process, however when it is bent to become a torus we are using the flexibility of paper. In other words, if the material is very hard it cannot be used to make these hats.


tube before folding. thin folding lines are printed. 折る前の筒、かすかに折れ線が印刷されている

Many people were attracted to these hats and constantly asked for permission to take photos. Some wanted their picture to be taken together with our models. They were amazed with the forms but strangely they didn’t ask any questions. Simply they don’t care how the hats are made? Or they are lost for words as they have not seen this design? Only a friend of ours asked questions to satisfy his curiosity.


I answered  “We don’t use any origami generating program software. We use Computer Aided Drawing software just to draw lines more precisely than hand drawing.”


“We fold for fun and for finding practical applications but also for transcendence through mathematics without equations and formulas.  If any computer program starts generating these forms automatically, I will quit origami.”


Please check different designs posted in the past. 過去のデザインもご覧あれ。





ORIGAMI chair, Science Museum Oklahoma オリガミチェア 科学博物館にて


Do you know about corrugated plastic or coroplast ? It has the same structure as corrugated card board but is made from plastic. It is very inexpensive and used for protection at construction sites, signage and packaging.  We designed a chair using this material a few years ago and since last week it is has been in an exhibition at the Science Museum of Oklahoma. The exhibition looks at many aspects of origami, from the traditional folded animals to modern abstract objects. It is at the Science museum where they have delved into the mathematics of this art.



The mid point of the video above shows the chair as part of their origami exhibition. Please take a look. This “continuum” is made by folding one 1.2m x 3m sheet, quite different from a typical chair which is assembled from legs, a seat, a back… It is very light; you can lift it with one hand but sturdy enough to sit on. This tube-like form is combined at the edges with stitches, generating the “continuum” which has enough strength to accept your weight. You can sit directly on the soft plastic seat but a cushion will make you happier. For more information, please go to a link below.







origami staircase オリガミ階段

We posted some information about a duplex apartment we designed where an opening was mode in the concrete slab and an origami bridge was inserted between the upper and lower floors. I wonder if it was too abstract? In this blog you can watch a video animation explaining the concept. Please take a look. An explanation follows after it.


“This staircase was conceptually inspired by paper folding art principles where a single strip forms the flight of risers. Simple serpentine folding of a metal sheet connects both floors. A thick and hard material like steel requires a totally different technology; cutting with a computer controlled laser cutter and special bending machine. These days,  “3D  printers”  have become a buzz word but a “3D steel printer” has not been invented yet. Instead of cutting out a long zigzag shape from industrial metal sheets and bending the strip 24 times, a more economical, energy saving and much more dimensionally accurate method was developed. First, a large standard sheet was folded just 4 times and after folding sliced “diagonally” into 6 identical modules then finally stacked. This procedure made precisely identical modules which would never have been achieved by bending the strips one by one. This accuracy was essential to accommodate the glass handrail which requires precise alignment.”


Insight into the nature of fabrication and the nature of the material developed and shaped the design. With these principles the design was realized —— reducing the cost, optimizing the use of material and maintaining the conceptual integrity of the design. Throughout this project the idea of “how to make” was a  part of the design,  the design itself  further clarifies how to it was made. Once the concept is created, it is not a matter of “bull dozing through to a solution” instead restrictions on materials and production contributed to inspire the concept … I think this is a good example.


Transform to Duplex メゾネットに変える


We have just finished a project to combine upper and lower floors of adjacent apartments using a void and staircase in an east midtown Manhattan apartment. The biggest challenge was the staircase which connects the two floors. We thought this might be a good application of ORIGAMI. The folds are 3 dimensional and turn back and forth to create the treads. This is quite different from the typical idea of folding to form of a staircase, where the steps and treads are all continuous to create a zigzag shape when viewed from the side. That is structurally rigid but there is no visibility through the staircase when you move up the stairs. Our design of a folded ribbon allows more transparency as you traverse the stair.

ニューヨークのミッドタウン、アパートの11階と12階をつなげてメゾネットにするプロジェクトを最近終えた。中でもチャレンジだったのは床を抜いた後につける階段のデザイン。何とかオリガミのようにしたいと思っていた。良く見かけるのはイナズマ型階段と呼ばれ、 踏板と蹴込板(登るときにつま先が当たりそうになる板)が繋がり、これが上から下まですべての段が繋がっていて、横から見るとイナズマが走っているような恰好をしている。でもこれはもう使い古された形で、面白くない。自分が上っているときは蹴込板に遮られて視界ゼロ、透明感がない。

Our idea was to bend a long strip of steel plate into zigzags that are in the opposite direction from the typical ones. As the result all the verticals are eliminated and it is very transparent. The continuous ribbon of steel is like the fabric streamers that gymnasts use. It is completely cantilevered from the wall so the support is a mystery. How to support the load? It will be up to the readers imagination. It is heavy steel.



Spatial continuity is everywhere, there are no partition walls. Frosted glass divides the bathroom from the bedroom, bringing abundant sunlight from the east windows. The staircase and void provide a continuous connection between the bedroom at the upper floor and the living space downstairs.


It used to be that New York families moved out to the suburbs when they expanded, but these days many families try to keep living in an urban setting. Their lifestyles are changing. This market requires multi-bedroom apartments that are very expensive. Re-selling concerns require architects to combine two adjoining apartments into one large apartment with more bedrooms. As a matter of the fact the owner of this apartment just wants to keep living here not because of accommodating family members, but because of the amazing view, the terrace and the location. This combination is not the adjacent apartment but the one above… very special.


at OrigamiUSA オリガミUSAにて


our table-all from one sheet. 私たちのテーブル、どれも一枚の紙から。

One day out of the blue we got a call inviting us to an origami convention. We have no idea how did they found out we are interested in origami. We thought this would be a good opportunity to discuss our ideas about origami. Instead of just attending the conference we proposed that I could give a lecture and that we also wanted to present our origami works at their exhibition. After sending some images of our work we were quickly approved  and luckily the conference was being held in New York this year. We selected some works to take and prepared images of the pieces that were too big to bring to the conference. We completely packed our friend’s car with our origami and headed to the conference.



Transportation of this chair was the most difficult part. Why didn’t we collapse it? which is the nature of origami!! too late. この椅子を運び込むまでが難事業、折りたためばよかった!

Origami USA is the largest origami organization in USA with many origami lovers. This year approximately 800 people attended the 3.5 day conference.  There are lectures like the one I did and classes on how to make a particular origami object and many gatherings of origami folders. There is no hierarchy between lecturer and  audience; lecturers become the students in other classes and there are many different age groups.



from a sheet of paper!  by Tomoko Fuse R-40cm, 一枚の紙から!布施知子さんの作品 半径40cm

The exhibition highlights many origami works. There are all kinds of origami from simple cranes to unbelievably complicated animals, insects and abstract objects. Some are newly invented by researchers based on mathematics.  Anything seems OK if it is similar to Origami. Even mizuhiki or Japanese ornamental knot works were exhibited, which are beautiful but not folded at all.  (The mathematical link between knots and origami  may be discovered someday.) The organization covers so many concerns but is not split into specialized  sections. The researchers also exhibited some of their insect and bird works.  They simply love origami. Creative origami work attracts people’s attention but they do not stick on the originality. They do not mind if they find that someone has already  created the very same design before them. It is all right as long as you enjoy it. No one has a superior attitude to those who have newly joined and beginners.  Does the organization have this philosophy? or does origami itself possess it?  I was given 45minuutes  for my lecture on Monday the last day. After long  lecture, many people came to talk to me. I hope my lecture gave them some inspiration.

展示会場には、さまざまな作品が五万と展示されていた。ツルのような簡単なものから、信じられないほどの複雑な動物や昆虫の大群、そして抽象的なオブジェ まである。数学に裏打ちされた研究者による新作もある。オリガミならなんでもあり。オリガミとは直接関係のない、ミズヒキの展示や講義さえもある。(将来、組み紐とオリガミの数学的な関連が発見されるかもしれない。)参加者の層や関心ごとが多岐にわたっているのに、それぞれに分化していない。研究者も昆虫や鳥を折っている。彼らは本当にオリガミが好きななんだなと感じ入った。創造性のある作品は、注目されるけれど、みんなオ リジナリティに固執していない。独自作品とよく似ているのが過去に作られていたことが分かっても気にしな い。楽しければいい。だから、先輩面した人が一切いない。これはこの組織の特性なのだろうか?確実に言えそうなのは、これがオリガミの持つ大きな特典だということ だろう。講演にもらったコマは45分。長い講演を終えると、たくさんの人が話掛けてきた。新しいインスピレーションが生まれればとおもう。


from a sheet of paper! designed by Robert Lang. 一枚の紙から!ロバート・ラングの設計

The organization takes care of everything from organizing the space and amenities to registration of attendance at lectures and classes . This huge work is maintained by volunteers. We were supposed to volunteer one hour during the conference. But I forgot about it-it was our first convention and we were unfamiliar with everything. I talked to one of organizers who helped us set up the exhibition.  She said ” no problem! as long as you enjoyed”

組織は会場のセッティングや参加者の登録、手続き、窓口事務、などなど、参加者の便宜を図って支援をしている。これら膨大な仕事はすべてボランティアで維持されていて、講演者や展示者も、会期中に1時間のボランティアをすることになっている。ところが、初めての参加なので何が何だかわからず、アタフタしているうちに忘れていた。面倒を見てくれた担当者(当然、ボランティア)にその旨を伝えたら、曰く、「気にすることはないのよ、楽しかったら、それでいいのよ!」 アメリカだなあと思った。


hundreds of folded units are combined to form a ball. オリガミのユニットをたくさん組み合わせて、球にしてある

Easter Parade 2016 イースターパレード


Here we are on the avenue…for the annual Easter parade. It was wonderful sunny warmish weather and lots of people were out and about. This year we introduced a new line of “circular” hats and remodeled the “Audrey Hepburn” from last year. Always fun to see the reactions and be part of a New York ritual.



The folding lines of the new model are circular so some people may have a problem calling these hats origami. Although you can extend the traditional definition of origami a little – “origami is formed from a flat sheet by only folding without any cutting in principle”. We made joints to make the folding  easier, but  if you have extremely fine fingers, special tools  and the right paper product, you do not need any joints.  The new model  is composed from a cone which is “mathematically” flat or 2 dimensional, or curved but not warped. As a matter of fact the “Audrey Hepburn” is also a cone before folding, like the new model. The two new circular models are slightly different. Can you tell the difference?

今年の新ラインは曲線の折パタンが使われている。毎年折り紙帽子をいろいろ試してきたけれど、ここまで来ると折り紙とは ちょっと言い辛いような気もする。曲面が入っているけれど、それでも一枚の紙を折るだけでできている。伝統的な折り紙の定義をちょっと拡張しさえすればよい。制作上の技術的な理由で、継ぎ目があるけれど、無限に細い指と道具、それに特別製の紙があれば継ぎ目は必要ない。今回の新モデルは円錐面を折曲げて作られている。それは数学でいう二次元的に平坦な面で、ウォープしていない。実は去年のモデルも円錐からできている。今年の曲線モデルの二つの帽子は微妙に形が違うのだけれど、さてどこが違うか?(3つ の去年のモデルも折りパタンがそれぞれ少しだけ違う。)


We were often asked how to make the hats. We explained the procedure, but I guess it does not satisfy their curiosity, but it is so difficult to explain how we came up with this form. First we have a vague idea of what pattern is fold-able mathematically and what the final shape will be. But that does not mean it will be beautiful and interesting.  Imagining the final form we modify within that idea and make several test pieces. We tried not to mimic the traditional hat this time. If you emulate the classical form it easily becomes elegant and understandable. Although we are doing folding why not explore the design mathematically? Why not let the mathematics drive the design and have the hat be a byproduct? It may unexpectedly be very beautiful but we do not want to lose the sense of a hat.. to balance them was difficult.

毎年、みんなからどうやって作るのか質問を受ける。一連の作業のプロセスを説明するのだけれど、たぶん彼らが本当に知りたいのは、どうやってこの形に辿り着くのか?ということかもしれない。でも、それを路上で説明するのかなり難しい。まず、どのようなパタンなら折ることができて、どのような形に折り上がるかという数学的なアイデアが最初にある。しかしそれだけでは必ずしも面白い形にならないので、出来上がりを想像しながら、そのアイデアの中で自由にできる部分を変えてみる。そして、数個の試作品をつくる。今年のラインは、帽子であることは守りつつ、できるだけ、既成の帽子の形にならないようにしている。これまでの帽子を思い出させる ようにすると、比較的簡単に違和感のない優雅な形にできる。けれど、せっかく折り紙でやっているのだから、もしそうでなかったら思いつかなかったような異形ができあがるのじゃないか?しかしそれでも帽子の雰囲気は失なわないようにする、その辺のバランスが難しい。


Thank you models! it was not our intention but they are from Austria, Poland, China, Washington and Canada; so New York…モデルの皆さんありがとう。意図したわけではないのに、彼女たちの出身はオーストリア、ポーランド、中国、ワシントン、カナダからで、いかにもニュ-ヨークらしい。