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!!





Women’s March 女性の行進

I have attended the women’s march for the last 3 years and I am always recharged and encouraged by the energy and hope— we all seem to think we can make things better.

There was a lot of whining this year about how there were fewer people and there was not a consolidated idea behind marching- but I really wonder if there needs to be. It was great to see a truly diverse group of women out there, and some fantastic signs, raunchy, funny, serious and so disrespectful sometimes.


今年は人が減ったことや、行進の背後にある統合された考えがなかったことについて多くの意見が寄せられたが、その必要が本当にあるのでだろうか? 多様な女性のグループが出ているのを見るのは本当に素晴らしかった。

I joined up with the women designers, engineers, architects, and landscape architects- any women in a design profession (or not) was welcome in the group. There have been several newspaper articles lately about why aren’t there more women architects? Lots of reasons and women architects are still often not paid as much as their male colleagues, but there were over 73 women willing to come out and march together under the banner of #women build.

私は女性のデザイナー、エンジニア、建築家、造園家と一緒にいた。デザインの仕事をしている女性は誰でも大歓迎。 最近、なぜ女性建築家が増えないのかについてのいくつかの記事新聞に載った。 そこには多くの理由がある。女性建築家が彼らの男性の同僚ほどまだ給料が支払われていない。一緒に出て行進したいと思う女性たち#women buildのプラカードの下に73人以上いた

One of my favorite groups that I shared several blocks of the march with seemed to be friends from high school. They were a completely racially mixed group of women, black, yellow, white and every shade in-between; they were singing and dancing and eager to change the world.

数街区を共に歩いたいくつかのグループの中で、私が気に入ったのは、完全な人種の混ざり合ったグループで高校時代の友達らしい、黒人、黄色人、白人、そしてそれらの混ざり合ったさまざまな色のグループだ。 歌と踊り―――そして世界を変える情熱。

There were, of course politically focused signs and many volunteers to sign up anyone who was not registered to vote.

Someone stopped me and asked “can’t you build us a better White House?”  I said “we sure are trying….”


見ていた人が尋ねてきた。「あなた方は私たちのために良いホワイトハウスを作れるわよね?」…… 答えて曰く、「頑張ります。」



Bowne House ボウンハウス

For this years “Open House New York “volunteer assignment I was sent to Queens and on my way home I discovered this wooden house, the oldest house in Queens, preserved for its significance in the history of New York.  John Bowne was the author of the declaration that was the first step in confirming religious freedom for Americans and his family lived in the house for 9 generations.

毎年オープンハウスニューヨークのボランティアに参加しているが、今年はクイーンズにある大きな庭園とその中にある古い家 ジョン・ボウン ハウスを選んだ。クイーンズ地区に現存する最も古い家屋で、ニューヨークの重要な歴史の証人として保存されている。米国で最初に宗教の自由の宣言書を書いたジョン・ボウンがここに住み、その後9世代が住んでいた。

A member of the family was the founder of one of the first nurseries in NYC who collected trees and plants from around the world. He is the one responsible for the wonderful garden and the collection of specimen trees.  Many of their descendants have been spread across the country from this garden in Queens.  The garden is huge compared with the typical suburban houses and the street is lined with large oak trees planted by the family. It was originally farmland but now it is surrounded by masonry low-rise apartments, so out of place in this dense environment.


In this garden I met two interesting women, one had been a professional gardener and the other was an architect who is also the caretaker of the house. The caretaker lives in the house and lovingly cares for the garden. The house is part of the New York City Parks Department system,  but like most of the parks in the city the parks department cannot maintain it by themselves and they rely on volunteers. She was looking for more help to prepare for the winter


I brought another volunteer with me today. We are here to prune and trim trees. As some of the plants are rare and have historical value as part of the original planting, you have to be careful how to cut. This additional helper amputated even the largest branches. The architect looked a bit worried. Did he meet her expectation?


There is no photo showing it but there was a huge pile of trimmed branches. You will see the results of our work next spring. We were invited to come again.




AllKB on the waterfront ウォーターフロントのAllKB 



Every fall the Alliance for Kip’s Bay has been planting and cleaning up Glick Park, located just south of  the United Nations. But this year it is closed for renovation, so this fall for the “It’s My Park Day”  we traveled outside the park. “It’s My Park Day” is a twice a year event sponsored by the NY Partnership for Parks, who supply tools, bulbs and this year bright pink spotted garbage bags. Volunteers gather throughout the city to work on their local parks.



Our neighborhood is changing- we have a new park, new buildings and and a ferry service connecting 35th street to the Bronx, Brooklyn and Astoria, all taking advantage of our location on the waterfront. Our clean up was focused on the area  by the ferry terminal, part of the East River Greenway where we worked on the tree beds south of Glick Park. Our efforts focused on making this piece of the Greenway, a greener place to relax and enjoy the waterfront. Our work also contributed to a special project;  as a memorial to the victims of 9/11 this year 500,000 daffodil bulbs are being planted throughout the city.



We organized volunteers  from NYCares (an volunteer organizer), a local school and the local men’s shelter.  We dug up the rock-hard ground, planted 1,000 bulbs out of the 500,000, and discovered snoozing worms.





The ferry route was expanded this year. It has been welcomed by the community, bringing more pedestrians and other New Yorkers into the neighborhood. This transit node has been demonstrating the potential that a continuous  bike/pedestrian Greenway would create along the East River. The Greenway is a potential loop around the entire island, although it has a gap. Currently the Greenway north of the ferry terminal  disappears at the UN which you can see in the background of the photo. North of the UN, the section from 60th down to 53rd  is ready to be closed, but still the UN will remain one of the biggest barriers to continuous waterfront access. We keep advocating for a continuous route but it seems a distant goal. I finished our work hoping this will help to make this stretch of the waterfront an welcoming accessible neighborhood amenity.



We are all looking forward to the spring to see the blooms on our waterfront and then get back to Glick Park.


Bike Tourism, Shikoku Henro 四国自転車ツーリズム 遍路道


The river ahead and the city road beyond. この先は川があり、その先は市道

Today we are bicycling further inland following the pilgrimage route of the Shikoku 88 Temples. Most of the original ancient routes are developed with wide modern paved roads except for the very tiny paths in the mountains. Sidewalks can be used as bike lanes as no one is walking these day, but actually the sidewalks are not continuous so we had to cross the traffic constantly switching from one side to the other. Some parts of the old roads remain and the main traffic doesn’t use them anymore, but they are very short, interrupted by the new roads and you never have a really relaxed ride.



A rural road in between the mountain and the river. 山と川に挟まれた田んぼ道

If you ride a parallel route, for instance on the opposite shore of a river which is on the main route, you can have fairly nice travel. These photos are a good explanation…. We found that this side of the river is much quieter and has much less traffic. These narrow streets are almost like dedicated bike lanes as only local people use these meandering roads which discourage fast driving and actually are not needed by the local people. Very few bridges join this area to the other side. Why don’t we publish a map that carefully connects these local routes and call it a continuous “ bike route”? There must be a lot of similar locations like this.


This cul-de-sac road ends at the mountain. 反対側、山があって行き止まり




We are now in a village. Unfortunately this potential bike route hits a heavily trafficked car route. There is a temple in this village “Byodo-ji” the 22nd of the 88 temples. Traffic passes just in front of it. It is very quiet for someone who came from a city but you cannot be totally relaxed. A parking lot is provided by the temple.

村の中に入ってきた。「自転車散策道」は残念ながら 村の中に入いる地点で車の行き交う道路と合流する。この村には四国巡礼88寺の一つがあって、門前は車がよく通る。都会から来れば十分閑散としているが、それでも車への注意は欠かせない。お遍路に駐車場さえ用意されている。


Byodo-ji main hall. This unsophisticated garden is not like in Kyoto but is very charming. 22番札所 平等寺本堂 この洗練されていない庭がいかにも四国らしくて良い。

To follow our original path back would not be boring but it would be daunting, it would be a long distance and there is a tough hill in between. If we can take a train from this village it would be great for our tired legs but our bikes do not fold and we have no covers which are both required by the railway company, even in an empty car.  It does not make sense, no one would be annoyed, and it helps maintain the passenger deficit by excluding cyclists. This rule is a inheritance of the golden age when trains were always packed. The weather is getting dark. Let’s take a quick visit to the main building and have a panoramic glance at the village, and trace our way back.




orange: this trip, yellow: last trip オレンジ:今回の旅、黄:前回の旅

Bike Tourism in Shikoku 四国自転車ツーリズム

between fishing villages 漁村から漁村へ

Every country is having difficulty to maintain their train system outside of the big cities. Cars came to the world just over 100 years ago, and quickly became something we cannot imagine being without. Google map ignores railway lines. In North America cars dictate the materialistic world and also the spiritual world—-world view, someone criticized it as Ford-ism. Train systems in the countryside are becoming extinct.


In such a global situation, a new idea for train systems is emerging, updating the old idea of trains being just transportation from one point to another, it is now entertainment with train rides introducing beautiful views and interesting towns. This reminds me of the cross continental trains or ocean cruises. Those tickets sell  out quickly. If this is effective even in the country side, local lines may be reactivated. Nowadays the local routes are losing passengers and therefore service and as a consequence they lose more passengers…


Today, we are 20km from Tokushima  and about 100km from Muroto. There was a cyclist on the bank of the beach. He was wearing racing gear on a super light bicycle equipped with GPS. He takes a bicycle tour every weekend and typically rides about 200km. He says some bike routes are well separated from car traffic like the one we are on but mainly he shares the road with cars. We are amazed with his passion as  touring with vehicle traffic cannot be pleasant. If he can skip heavy traffic in and around the city by bringing his bike on the train, his tour would be fabulous. He also could have a much longer bike ride.


What if we can launch cycling tourism? Many more people would visit the country side by easily carrying  their bikes onto the train.  You can get off at a station and get back on again from a different station, even a different line. This would be a long day but a real trip. Actually we do this with the New York city subway. The direction of travel is always opposite to the rush hour commuters, particularly on the weekends when the subway is not crowded.


The other thing we can do is to organize routes which are safe and have fewer interactions with car traffic. A dedicated bike path is ideal. The route can zig-zag, it is not the shortest ride, but we can enjoy the beautiful scenery and simply ride; visiting towns and villages. The shoulder of the road can be widened and defined as a bikeway. Sidewalks may be shared with bikes if pedestrian use is them very rarely.


Because it is not easy to get to the local towns it means that an undeveloped simple environment is still there. Once it is developed for tourism assuming car access it will bring more traffic and end up with wider roads and bring more traffic.  This bad circulation will threaten the quiet environment which was the charm of that location. On the contrary, train tourism preserves the environment but requires a sustainable number of passengers. Some weekend-cyclists carry their bikes in their car to reach areas outside the city, but they have to come back to the car. It is not a bike tour. They have to go through heavy traffic again to come back on Sunday evening. If bicycle and train travel are combined, more people can easily access many places keeping the environment and culture there and along the way. In order to do this, trains and platforms need to be easily accessed. Not only professional cyclists but recreational cyclists like us can easily travel very far, even on the way home from work. The current health boom may spur bike tourism. What about touring to the 88 Temples of Shikoku by bicycle…..


ALLKBA at Glick Park グリック公園

In our neighborhood there are public parks and streets which need flowers and trees, but the ground is so dry and hard – it has been left forgotten for so many years. KBNA is a not for profit organization that takes care of the flowers and plants at Glick Park and also the tree pits near the park that are part of the bicycle lane system.


One of the activities our group supported was posted in this blog, we proposed a service roadway to become a real public amenity, street cafes , seating a kids play area and an event space where local musicians can mingle with residents.  The streets should become more lively and active, unfortunately our proposal was  denied by people who live in the adjacent property. It should have benefited them but they were not convinced.






Our targets are any kind of park/garden and streets which will contribute to making urban living better and enhance community life. 私たちの活動の目的は、どんな公園でも構わず、地域コミュニティーを活性化して、より良い生活の助けになるようにすること。




One of them is this ….. park, with it’s fantastic view of the East River, a full panorama although the park is not well recognized . We want to make this park more beautiful and popular. Today we planted flowers and shrubs  and our next task is to keep them watered.


The neighbors who believe this park will make the community better got together to take care of the garden. They came from many different backgrounds. An Estonian who  recently arrived noted that the medical system is terrible here, a Chinese family and so on. What made them to come to join this activity? Look at their kids full of curiosity and having fun. Their contribution is so great. Why not more kids? Why not more adults?


They came from many different backgrounds—-old migrants and new immigrants. An Estonian who recently arrived is working hard complaining that the medical system is terrible here. What made them to come to join this activity? Look at their kids full of curiosity and having fun. Their contribution is so great. Why not more kids? Why not more adults?.