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。

Origami Hepburn hat (continued) ヘップバーンのオリガミ帽子(続き)

At the Easter Parade some people questioned ” Where did the hole in the center of the heat come from? You cut it off after finishing folding right? “……a very good question. As we have been saying the hats are made of one sheet of folded paper, …. yes this is a cone, like the left one in the picture below. It becomes totally flat but the edges are continuous. Unlike a sphere, doughnut or saddle this surface does not have a bulges or any torsion. It is curved but flat and the layers of concentric circles that fold back and forth are similar in form.

前回の記事で、イースターパレードをしているときに質問があった。「帽子の中央の穴はどこから来るの?おりあげた後で切り抜くんでしょう?」… なかなか鋭い。オリガミでできた帽子は一枚の紙から折られてできているのだけれど、実は下の写真左側のような円錐状の紙からできている。押しつぶせば平らになるのだけれど、端が繋がっている。この面は球やトーナッツ、鞍点等と違って、膨らみや、捩れがない。曲がっているけれど、正真正銘の平面、平坦な面と言うことができる。


Our Easter hats are all folded this way as a cone and the pinnacle was pushed up and down as shown in the model above. We designed the hats to generate a spiral with the identical folding patterns at each level. The direction of the sweep is all in one direction, but the folding pattern of each layer can be switched to the mirror image in the middle of the process (the rotation direction can be flipped from the right hand to left hand and vise versa). The rule that I adapted of having 6 lines converges at a vertex gives total freedom for folding which makes the folding condition very flexible.  In the case of 4 lines converging it is not like this, the folding  condition between layers is not independent.

By the way, the central axis of the cone does not have to be aligned. Every layer can be pushed off center or  pulled up with a different axis like the model below.

イースターパレードに使った帽子は上の写真のように、このコーンの細っている部分を中に折り込んだり、また出したりすることでできあがるバージョンだった。上のモデルは最も簡単なコーンとその折り返しを繰り返したもの。各層がお互いに相似形になっている。オリガミ帽子になったときにスパイラルを描くように、回転方向をすべて同じにしたけれど、反対方向に回転してもあって も折が成立できるように見える(換言すれば鏡映パタンが含まれていても)。6つの線が交点に集まる線の数が6であること、4つの場合と異なり、折れ方の条 件を決定的に自由にする。4つの線の場合は、各層の間で折れ方が独立に決められない。



The lines on the paper are from recycled paper, not folding lines. 表面の線はリサイクルした紙にたまたまあったもので、折れ線にあらず。

This looks difficult to fold. what about the next one below?  Looks totally impossible. Is it really so? every flipped surface is a flat curve, which means you do not need to stretch the paper to fold it. So I think we can call this ORIGAMI too.




The Audrey Hepburn(s) オードリ・ヘップバーン達


Mesdemoiselles!! may we take your picture?  マドマゼル!写真撮って良い?


This years Easter parade on 5th Avenue was a beautiful spring day. We did not have so much time to prepare this year so we decided to reuse one of the existing design  from  last year’s parade and make 6 variations of  it. They are all similar but slightly different…so we  call them the “6 sisters”. Our friends and their friends helped us by being models. Some people may say the 6 sisters look like the petals of a rose. It is very appropriate for this parade which celebrates the start of spring.


The pattern before folding is exactly the same but by revising the order of the Valleys and the Ridges you can generate many variations,   V-R-V-V-R-R or V-R-V-R-V-R and so on. We picked the 6 most interesting forms among 6 x 6 combinations to create the hats. To be considered ORIGAMI they have to be made from a single sheet of paper, then the form becomes highly symmetrical like a flower. Actually it may be the way other around—-flowers are smiler to this folded hat… I mean the symmetry, periodicity and self-similarity you can recognize in this hat reflects the mathematical rules that may have something in common with the development of blossoms from a rule imbedded within the embryo….

折る前の紙にひかれた線のパタンは6つとも、すべて同じで、山谷を入れ替えることでバージョンができている。中心から、谷-山-谷-谷-山-山とか谷-山-谷-山-谷-山…などなど。6の二乗ほどの組み合わせの中から、6人の姉妹たちに合わせて面白そうな形になるものを選んだ。折り紙と呼べるには、一枚の紙から切り張り無しでできていないといけない。一点に集まる線の数は6本に制限して、対称性の高い形になっている。花の形にするためにこのオリガミを考え出したのではなくて、花を思わせるような形になるものを選んだに過ぎない。むしろ反対で、花がこの折り紙に似ているのではないか?ここにあるような規則性や対称性、自己相似性はシンプルな規則からできたもので、同じように胚から花弁に至る発達には、シンプルな法則が関与していているはず。それはこの折紙の折の規則に似ているかもしれない… もしかして。


Every year we are asked lots of questions but this year many people asked if we sell the hats or the hat  design, and they recommended that we do so, even if we do not sell them now. We did not have this question until this year. If someone were to help us launch the business, we might be able to start folding for a larger audience. But for now we fold them as art pieces. By the way did Audrey Hepburn wear a hat similar to this in a film? or do we just think she would look fabulous in this hat…..


Easter Bonnets 2014 イースターの帽子


This is the 2nd Easter Parade that we have brought our hats to. Easter this year was quite warm and sunny. Maybe that is why we thought there were more people sauntering down Fifth Avenue than last year.


The scheme for our bonnets this year used 6 lines converging at a vertex point. We have previously been developing 4 lines converging to a vertex (means 4 planes) and wanted to develop more complex patterns. Actually any even number of lines that converge to a point can be fold-able theoretically , but more than “8 lines converging” is technically very difficult to fold by hand. A ” 6 line” scheme is maximum for us to try folding and turned out quite different from the “4 line” patterns, there is more freedom. To understand the essential difference in these configurations is the current concern for me now. Please take a look at them. Anyway every hat was folded from one sheet of paper (or from 2 sheets to make a closed envelope ).
僕等の今年のテーマは各頂点に6本の線が集まる(6面が集まると言っても同じこと)折れ線パターンで帽子を折ることだった。これまでは4本が集まるパター ンばかりを追っていた。実際には偶数ならば理論的には折ることが可能なのだけれど、8本以上になると角度が小さすぎて、手で折るのは酷く骨がおれる。僕ら には6本スキームは実際上の折れる限界。でも4本スキームと違ってかなり自由度が上がるように思える。このパレードには3つの帽子が「6本スキーム」でできているのだけれど、目を凝らさないと判らない。本質的に違うようにみえる。何がそうなのかを理解したい。ともあれご覧あれ。どの帽子も切ったり張ったりしないで一枚の平たい紙から折り出されたもの。(閉じた形の場合は封筒のように周囲が繋がっている。これを 一枚と数えるか、それとも二枚と数えるかはお好みで。)


Not an exaggeration or self  admiring,  the hats were very popular, every few steps people asked us if they could take pictures of us, “Of course !”


IMG_5682s IMG_5696s

But we hardly moved ahead. We realized we took more than 2 hours to go a few blocks. After walking to Rockefeller center from St. Thomas church we celebrated Easter the Austrian way with hard boiled eggs, ham, and horse radish on bread.

でも、おかげでなかなか前に進めない。気がついたら、聖トマス教会からロックフェラーセンターまで、たったの数百メーターを行くのに二時間あまりかかっていた。パレードは切り上げて ランチにしよう。オーストリア式に、固ゆでの卵とハムをパンの上にのせてイースターを祝う。薬味のホースラディッシュはお好みで。



FOLDING PAVILION – big origami 大オリガミパビリオン


A half year ago we made this video of an interview from an exhibition at Fordham University in NY.  To respond to the  suggestions that ” the chat with the designers is not enough, we want to see  more of the  real work” we  made a different version focusing on just the big Origami (approximately 9ft high by 14ft wide). We are developing this as an architectural enclosure. We have been working with various materials and ways of joining, like the one at the beginning of the video. To watch the interview about how this was developed, go to the link at the end:

フォーダム大学でのオリガミの展覧会の中で行われたインタビューを半年前にビデオにしました。作者のおしゃべりはどうでもいいから、作品をキチンと見たいと いうご意見に答え、今回、その中の大きな作品、折紙パビリオンだけに焦点を絞ったビデオを作りました。音楽はインタビュー版と同じ、南アフリカのヴォラ ン。ご高覧ください。 いろんな機会を捉えて、この作品をビデオの巻頭にあるような建築的なスケールで展開したいと考えています。作品の説明は、半年前のインタビュー版を、ページ最後のリンクからごらんください。

Also please see  an extract below from the exhibition, 以下に、展覧会ポスターからの抜粋。

“ What comes to mind when you think of Origami? Probably insects or birds made of folded paper. Given a piece of paper, you can fold it into small triangles and figures come to life. For many of us there is an “aha” moment as we watch a two-dimensional sheet become a three dimensional object. What will happen if this piece of paper is combined with another to make a larger sheet? This new sheet cannot be folded into the insect or bird any more. You cannot even make two birds attached. The original paper is uniquely appropriate in size to create those objects.  ( You can, however, double the size of the first piece of paper while retaining the proportions, but after all it will be exactly the same bird, only larger. ) What are the criteria that allow you to combine an infinite number of sheets of paper and still fold them? The patterns must have mathematical laws or rules behind them that define the physical manipulation of the material. These rules, of course, will be abstract, but they will become visible in the folded pieces. Symmetry, one of the abstract concept/elements of the high mathematics underlying the folding art, becomes visible through the folding of the paper. In other words, the abstract becomes physical.”

折り紙というと、鳥や昆虫の形を思い出します。正方形などの紙を細かく折ってゆくわけです。仮にこの正方形にもう一つ正方形を継ぎ足してみたらどうなるでしょうか?そうすると もう元の動物は折れなくなります。元の正方形がその動物を折り出すのに過不足なく適当な大きさだったわけです。(もちろん二倍の大きさの紙で始められるけれど、結局全く同じ形の鳥になるだけ…。)では、2枚、10枚…100枚…と無限に繋いでいっても、破綻なく折り畳める性質を保つにはどうすればいいか?そのためには元の紙の山と谷の折れ線のパタンにその可能性が内在していないといけないわけです。そこに何か美しい法則が現れてこないだろうか?そしてその数学的な法則は目に見える美しさになって現れてくるのではないか?紙がペッタリ平らに折り畳めるということは当たり前のようだけど、これは私たちの住んでいるこの三次元空間の性質を如実に反映してるからに他なりません。紙を折る事ができるという事実は(大げさだけど、)この宇宙の構造を直接に反映しているといってもいいかもしれません。展覧会より

Link to the interview version :  インタビュー版へのリンク :

Design Bureau デザインビューロー誌


Design Bureau, a publisher based in Chicago featured an interview with us in their July issue. We discussed origami, its 3 dimensional quality and how we incorporate that into our architecture… It is always interesting to try to talk about your work, what inspires and what is important and in the end the process is still a bit of a mystery.
The magazine features all kinds of design; fashion, furniture, etc, but this issue focuses a lot on architecture. We are not sure how they found our firm. That is also a mystery. Please take a look at a major book store ( Barnes & Nobles has it ) if you have chance.

シカゴの雑誌社、デザインビューローの7月号にぼくらのインタビュー記事が載った。折り紙とその3次元的な性質についてと、それを建築デザインにどう生かしそうとしているか話をした( と言っても、諸々の理由から、なかなか折紙と言えるとことまでは行かないげど… )。自分自身の作品について話すのはいつも当惑する。アイデアがどうやって浮かんだのか?何が大事だと考えていたのか?やっているときはあまり考えてないから。デザインプロセスはいつも少々ミステリー。


from Manhattan Residence


Movie making is a little bit similar to designing architecture. The budgets are enormous and someone who pays for the production can’t see the final product except with sketches and limited information. The production process is also similar; a huge number of  professionals have to work together. The final product is the  result of an intensely working team with a common goal.  For a movie there are script writers, costume designers, composers, musicians, camera men, art director, actors… and the director. For architecture, facility engineers, structural engineers, lighting designers, acoustic designers, theater consultants, landscape architects, contractors … and the architect. For both groups the final product must be public work (even a project for an individual house, as it will contribute to the urban space). A big difference is that architecture does not have such fancy ceremonies like the Oscars or Cannes! Actually small works like those below exist which do not have anything to do with the things I mentioned above.  Please check this video.

映画制作は建築デザインに似ている。予算が莫大で、金を出す人は最初はスケッチや限られた資料を見ることができるだけで、成果品を見ることができない。製作過程も似ている。専門家の共同作業が必要で、成果品はチームの産物。映画であれば、スクリプトライター、衣装、美術、俳優、振り付け、…作曲家、音楽家そして監督。建築なら、設備設計、構造家、照明、音響、劇場コンサル、工事施工者そして建築家。できた作品が多くの場合、公共の財産となるのも似ている ( たとえ個人の施主のための建物であっても、それは都市空間を構成する。 ) ただし、建築ではオスカーやカンヌのような鳴り物入りの授賞式がない。まったく全然違う!もっとも、以下のような小さくて、上に挙げたようなこととは関係の無い作品もある。

As you know there is a category in the Oscars for editing. When I made the video of Space Tessellation I realized that editing a movie is an independent art form. The piece is very tiny, but making it good required huge work. You first decide the sequence, arrange the frames, time intervals/duration, figure out suitable music, cut and paste both the music and the images and make the whole thing smooth. This process repeats until the final product becomes impressive enough to transmit what you want to say. Please also visit the link below for the original version.

オ スカーで編集の賞があるのは、いかにもだと思う。前のヴァージョンのビデオを製作した時、そう思った。映画の編集は一つの独立したアート の形式 だと感じた。たった二三分しかないのに、面白く見せるのはとても大変。画像のシークエンスを決め、各フレームの動き方や時間を按配し、イメージに合 う音楽を選 び、画像と音楽を切り張り、見せたいものがキチンと見えるよう、それがスムーズに見えるようになるまで すったもんだを繰り返す。ここでついでに前のヴァージョンも 。


I said that only the rhombic dodecahedron and truncated octahedron exist which fill space. I don’t know if it has been proven or not but it does not seem that someone will say someday ” I found another one!!”. Until now when looking for another type of space tessellation people assumed that the object is composed of flat surfaces. But why can’t the surfaces be warped?  then it’s a totally different story. An infinite number of space filling objects may exist.


Our Easter Bonnet Video イースターの帽子 ( ビデオ )


As we mentioned one post before the last one, we made a video of the Easter Parade with our origami hats. It  took so long to finish because we were searching for just the right music for it. Hope you like this! The music is ” re-composed ”  by German composer Max Richter from Vivaldi’s Spring in The Four Seasons. Each bonnet was folded from a flat sheet of paper. The hexagonal hats and cylindrical one can be compressed and flattened, so no need for bulky hat boxes to store them.

前々回に載せたイースターのパレードの様子をビデオにしました!良い音楽が見つからなくて遅くなりました。この曲はマックス・リヒターによって ” 再作曲 ”  されたビバルディーの四季から春。どの帽子も平たい紙から折られています。6角形のと、筒の帽子はさらに押すとペシャンコになるので帽子箱が要らないのです。ご高覧あれ。

We have launched a Facebook page for our firm. Please check this too!!
link to our facebook

Easter Bonnets イースターのオリガミ帽子


photo by Patric M Bernet.  Click for bigger image

photo by Patric M Bernet. Click for bigger image

Last Sunday was the Easter Parade. We did not bring our hat from last year instead we  designed several origami versions this time. Fifth avenue was packed with people and after the service at St. Thomas, once we donned the hats we were continuously asked to pose for  photos, our change to be fashion models!  We could not move away from the church.  We were enjoying  our 15 minutes of fame. We had the chance to talk with many people out enjoying the spring weather and viewing the amazing creations that New Yorkers come up with. Some of these images here were taken by one of them. We finally wandered up Fifth Avenue an hour later.




Now here is a quiz; the 4 images below show the hats when they are opened and  flattened. Which one corresponds to which of the 5 hats?
We are going to post a video taken from the parade next week.



SPACE TILING(3d) 立体空間タイリング

It is called “tiling” to fill a 2 dimensional plane with densely spaced identical objects.  Can we do a similar thing in 3 dimension space?  Stacking cubes or rectangular solids is the easiest solution. Adding height to a flat tiling pattern is the simplest way, yet this is essentially still 2 dimensional. So far only the truncated octahedral and  rhombic dodecahedron have been identified as pure 3 dimensional objects that fill solid space. Really no more?

これまでに掲載したオリガミの例は平らな平面を同一の図形で埋め尽くすものでした。タイルで壁を隙間無く張るのに因んでこうゆうことをタイリングと呼びま す。同じことが立体の空間でもできるでしょうか?3次元空間同じ形の物体で埋め尽くすことができるか?6角柱のように、2次元の平面の上に乗っている図形に高さを加えて立体にするのは一つの方法ですが、本質 的に2次元です。長方体もできますが、当たり前ですね。これまで3次元空間を単一の形で充填できる立体は切頂八面体菱形十二面体の2つしか見つかっていません。本当に他にはもうない?

The object above  is the product of our investigation of this question. Believe it or not this cone can be stacked one on top of the other in a densely packed 3D space in every direction without any gaps in-between. By allowing each flat face of an object to be twisted, the form achieves solid tiling. Folding  approximates the envelope of the object and it is made by combining 4 sheets. It is very close to the mathematical form, but the more folding, the more accurately it will approximate the real form. You can not believe  it?…. see the  movie