Colegiul National "Moise Nicoara" Arad

The Applications of the Mobius Strip– A European Comenius Project



                                                            LILIANA NEGRILĂ

                                                                Project coordinator

One of the Romanian team’s task was the application of Möbius strip. The students had to write a short essay in which they had to present the magic of the Möbius strip. In the 21st of January 2012 we met at school and held a small contest. The jury was composed by both students and teachers. While the students had to give 0 points for himself/herself and from 1 to 13 points for the other students (13 points for the best eseay in each vision); the teachers marked each essay from 1 to 14 points (14 points for the best esay in each vision).

Below are the essays presented:

 

  1. Alina Negrila, 11A

Möbius strip is not magical only because it defies the laws of normality, but also because it can be found in everything around us. This strip stands actually at the basis of the whole universe, it can be found in musical works, drawings and paintings, literature, physics, mathematics and in other domains as well. Moreover, we are a living example of this magical strip and the proof stands in our very DNA and biorhythm. We are born, grow up, mature and no matter which paths we follow or how many ups and downs we go through, we grow old and die. Sooner or later everyone gets where he/she has started his/her life.

Hence, many would conclude that it doesn’t matter what we are doing during our lives, but this would be completely erroneous. It counts, it counts a lot how we opt to live our life, irrespective of the fact that it is long or short. I have chosen to enjoy each moment and “twist” of my life, because it is also, just like the strip, unpredictable, magic and unique!

 

2. Andra Jivan, 11C

The magic of Möbius Strip is caused by one simple reason: its existence. Who would have ever imagined that out of a piece of paper you can make so many interesting, yet weirdly looking shapes? By gluing the two ends of a simple strip of paper, after one end was twisted at 180°, the magic reveals itself: one unusual strip with only one side and only one edge, an innovative element in geometry.

The “Crab Canon” by Johann Sebastian Bach is the most marvelous way where this magic manifests itself. When I first listened to this musical piece I didn’t realize how magical it was. After listening to it from beginning to end and backwards,though, it hit me! This song feels infinite, it was almost like I was in some other dimension. As music is one of my greatest passions, I discovered this with an immense sense of satisfaction. The magic is truly out there and it’s waiting to be discovered, you just have to know where to look. If you don’t, it might be out there, looking for you instead.

A couple of weeks ago I hadn’t even heard about this object and its special features: although it strikes you as something quite ordinary, it dazzles you as you discover more and more of its magic. With each experiment I did, my mind got even more sucked into this “game”, as I was trying to find out what the result would be, but the truth is that this strip is unconventional, and my ordinary thinking has got me to loose the game, only to start it all over again. However, the loss always put a smile on my face, because it revealed awesome stuff about this strip.

The Möbius Strip is so magical due to its ability to break patterns, its capacity of making us realize that there are billions of inexplicable phenomena out there. The human body is one of them, as we are the perfect life form in which its properties apply, like DNA. There are many domains in which the Möbius Strip has applicability but I believe that what we feel, what we see happening in front of our eyes as we experiment, is the most charming aspect, because we can touch it even though it seems surreal.

To conclude, I would say one more time that this strip is magical, amazing and I believe its magic should be spread as much as possible, since it’s a new concept to many people, although it was discovered over many years ago. So, Hocus-Pocus, let there be magic a la Möbius!


3. Andreea Sabadus, 11C

“The Möbius strip” is a two-dimensional object in a three-dimensional world. “An anomaly”, a drop of magic amongst so many rigid shapes. We can also say that man is the same; a spiritual being in a material world.

Two out of ordinary “things” amongst so many stereotypes. The distinctive qualities of a “spiritual being” are “wisdom” and “love”. Some kind of argument may be the reaching of an equilibrium of these qualities on the two plates of the “justice scale”. Man can “judge” correctly only when a perfect equilibrium is reached. The harmony of these two characteristics make man a complete being, a complete “spiritual being”, one that is above all the others. A unique person.

“The Möbius Strip”, in my opinion, can be considered the transposition of the “spiritual being” in the material world. “A face” of the strip is “wisdom”, and the other “face” is “love”. These two “faces” combine perfectly to make one.

In conclusion, we can say that these two “things”, “The Möbius Strip” and man, are each unique in their “world”, but joined we discover how alike they can be. The magic of the strip consists of the ideal combination, “homogenization” of two diametrically opposite “things”, forming something unique, superior in this world…


4. Ioana Stan, 11B

My story related to Möbius strip began one sunny autumn. Although it appeared to be one monotonous  day, it was the day I started a new chapter of my life. It all began when my Math teacher told us about a project that would take place over a period of two years. She built the Möbius ‘s strip and slowly managed to captivate our attention with bizarre results from cutting this strip in different ways . I felt engulfed by the magic of Mobius strip as, until then, I had not reflected upon the possibility of the existence of a single-sided tape. Then I said to myself that I really needed to get into this project because it was something different, which was detached from any rules and also a chance to meet new people and make new friends.

Involvement in this project became very important to me. So looking for information on this strip, I discovered that there still were people who had dedicated a good part of their life to discover the magic of the single-sided tape and write whole theories about this. Deepening my study, I found a lot of applications of this formidable strip in all the areas, but reflecting a little more closely, I realized that it had existed in my life all the time, since childhood.

Friendship is a source of infinity and also effervescence like Möbius’s strip. True friendship is based on trust, tolerance and, most importantly, it is characterized by the fact that you feel what the person next to you needs. True friends are always with you and always give you the advice you need.
I believe in everyone’s life there was, is, or there will be a „Mrs. Shears’’ that will try to „cut” our wings and shatter our dreams, but she will not succeed anything else than to make us stronger and more confident.

Also, our life is a Möbius strip as between our past, present and future there is a vital link. We can never escape the past no matter how hard we try, and the future is conditioned by our past actions.
Daily monotony can be considered another practical form of this band.

I think that what best describes Möbius’s strip is adolescence because during this period we feel free, carefree, and assured about tomorrow. When we are young, we are eager for adventure, to try all sorts of new things and even if we get entagled in problems like a clover knot or an eight knot, these things will just help us find our way in life and become immune to everyday evils.

In conclusion, I hope the magic of Möbius strip will touch the souls of all people and they will realize that something that seems so insignificant can also be so complicated and fascinating.

 

5. Georgiana Hategan, 9C

Möbius strip is apparently a normal ribbon, but it has special qualities, for some people even magical. Even if it is easy to make, using a sheet off paper, the result is impressive for those who don’t know the secret of this band. At first I believed it to be a trick too, but after making it I saw that the trick didn’t required much skill.

This band has only one side and edge and is we draw a line with a pen without lifting it, we notice that we reach the starting point in the end. We obtain a continuous line, the end is the beginning of which cannot be seen. It is a circuit with no beginning or end. This left me mesmerized, especially when I thought of it in association with life and time. For example a day starts at 00:00 and it ends before 00:00, another day following. It’s the same with the weeks, months and years. Similar to Möbius strip, life is a circuit that ends where it started:”…for dust thou art, and unto dust shalt thou return.” Unlike the Möbius strip, life and time can’t be manufactured.

At first everything seemed a game to me, but the I understood the complexity of the proprieties of this band and it’s applicability in many areas such as: Phisycs, Maths, Astronomy etc. I’m most interested in its applicability in Architecture, because I want to become an architect and design a building around the idea of the band.                  

 

6. Crina Dascal, 10B

Who would have ever imagined that a simple paper strip, created by a genius mathematician can open so many horizons and minds.

Children and adults, poets and scientists, musicians and engineers, designers and sculptors, fascinated by the magic one surfaced strip, are constantly trying to compete with themselves because of this challenge thrown up by Möbius.

I wonder if he thought that his invention will bring together so many minds I think that both writers and scientists tried their best to become immortal, just like Möbius.

Many people tried, but only a few succeeded!

In a short time, the Möbius strip helped humanity create some useful things like

the conveyor belts, buildings, clothes, jewelery, games and even music compositions.

Many researchers are constantly trying to find new applications to this outstanding strip, every attempt reminding us about the genius who probably was not as appreciated as he should have been during his life.

I also wonder if people who knew him have ever thought that Möbius will become a muse to creators, that he will bring joy to both children and adults.

It is interesting how this strip can delight the soul, the sight and the mind, all in the same time.

It is interesting how this magic makes people competitors and friends, how it makes all borders disappear and how it gets us all together, all talking the same language: ”Möbius”.

 

7. Laura Gligor, 11B

The Möbius strip has been named so in honor of the German-born mathematician, August Ferdinand Möbius, who invented it and studied its properties.

Its unusual attributes make it look more magical than something related to mathematics.

Who can imagine that a piece of paper can only have one face and one edge? It is pretty hard, in a world in which everything has at least two faces.

This strip can be made very simple from a ribbon of paper. The ends of it stick together, but one of them is twisted by 180 °. In this way it will result an apparently normal ribbon, but which, if you want to color „inside” with a color and „outside” with another, you will find that you have no chance. Also, if you draw a line on the middle of the strip you will see that you will reach the place you started, without lifting the pen from the paper, something that, no matter how much we try, we could not do with an ordinary strip.

If you cut the strip along this line you will notice that, instead of two equal length strips, you will get only one of double length than the original. Instead, if you cut at one third of the edge, you will get two intercalated ribbons, one of equal length of original and one of double length.

Due to these properties it goes far beyond the limits of human thought and I believe that its magic consists in our inability to fit it into a particular category, because, although it looks like one of many optical illusions, it is something we can touch.

We could say that the Möbius strip is a practical demonstration of the fact that the universe, although it has limits is infinite.

There is also, in the three-dimensional space, the correspondent of the Möbius strip, the Klein bottle, a single sided object and without edges. It was invented by German-born mathematician Felix Klein.

This bottle can be constructed by bonding the edges of two equal Möbius strips. Theoretically, there would be no problems, but practically, this can be made correctly only in a four or more dimensions space.

Another easily realizable method would be passing an end of a cylinder through its shell and then uniting the two ends. The reason why the three-dimensional representation of the Klein bottle is not very correct would be that what seems to pass through the coating does not actually intersects it.

The Möbius strip has inspired lots of artists and engineers. For example, conveyor belts, in order not to be worn out just on one side, can have the shape of a Möbius strip, doubling so the length of the strips, therefore their lifetime.

There are also multiple applications of it in chemistry, biology, physics or mathematics.

And now that we know what is a Möbius strip, and especially how it looks like, we can associate it to the  recycling symbol, or even to the infinity symbol (∞).

 

8. David Bacos, 9C

The magic of Möbius strip is similar to the magic of life, in my opinion.

Humans are made of matter and soul. After birth starts the long and twisted path of life. During this path we live happy or unhappy events, have good times and bad times and see the world from different angles. These do not represent two completely distinct parts of life, but one, in which the moments of happiness alternate continuously with the moments of sadness. In the end, just like a Möbius band, all humans go back from where they came, that means in the earth, the soul being the only one who remains. So, not the road covered is important, but the feelings and the deeds lived during the lifetime.

Another similarity between life  and Möbius strip is finding true love. That is the time when one’s life is shared with another person and, like the Möbius band cut on the longitudinal axis, there appears one path crossed simultaneously by two soul mates, but whose opinions and impressions do not coincide all the time.

That is why I like to see the magic of Möbius strip as a metaphore of life.

 

9. Bogdan Popa, 9C

The Möbius strip is one of the most interesting things discovered. It can express whatever we want, from a paper strip to a video game or electrical conductors.

The magic consists in the fact that you can take the Möbius strip and have 100 ideas about what you can do with it. For the beginning we can cut the strip in personal and new ways and have a great success or just a ball of paper. No matter what the result is we can say that we helped the scientists to solve the mystery. For example at one of the project meetings I tried to cut a Möbius strip with no logic or planning and of course that the result was a big lump. But after a moment of thinking I thought that if I sprayed my new lump, it would look really nice as a decorative object.

This game with Möbius strip looks like a magic show. Every time you start from a certain point and end up in the same point, like a magician who disappear in a mysterious way and reappear on the stage like nothing happened.

 

10. Oana Catrinescu, 10A

An abnormality of the nature, an incomprehensible object from an alienated world, whereas, for the initiated, just a plain piece of paper. However, what sets it apart from all its lookalikes, what has allowed it to climb the evolution scale is the fact that it only has one surface and one edge.

From the standpoint of a person hearing this statement for the first time the conclusion can only be the fact that this Möbius strip is an undoubted aberration. However, appearances are deceitful and people change their minds over time. No matter how absurd might the aforementioned hypothesis be it can easily be proved right.

Not many are familiar with the multiple “facets” of this miraculous strip, for the simple fact that it does not fit in the school curriculum and it is of general knowledge that youngsters do not find themselves in any supplementary lecture considering the various technological alternatives. Nowadays, the ones who share/show an interest in universal enigmas, in topological paradoxes like the one this Möbius discovered, are in a decreasing number. Time will eventually bring the unknown to the surface for the sake of ordinary people, less and less interested in science and utterly lacking the desire to enlarge their knowledge. Excuses vary. Some say it is the ego that prevents them from being curious, some blame it on commodity, but the thought that there is nothing worth seeking that could change our global perspective is more plausible. It is much more convenient to just glance at some masterpiece, rather than thoroughly analysing every square inch of it.

Due to its short history this amazing strip did not enjoy a proper popularity and it would now seem inconceivable that the world has overlooked it for so long.


11. Mihai Atimut, 10A

The Möbius strip: an invention which travels among us from birth and which is far from being known worldwide. An anomaly of nature, a totally mysterious object or for some people an ordinary strip like the other creations of paper lies in my hands waiting to be further explored.

I am privileged to be part of this project and I pride with this achievement. Now all I have left is to discover its “powers” but I must be prepared with patience and a lot of research. As a wise man once said “Discovery consists of seeing what everybody has seen, and thinking what nobody has thought”, this journey awaits for me and I am very thrilled to explore its hidden abilities.

In course of time everybody will know what is specific and unique about this strip: a single surface and a single edge. At the very beginning I said this was just a pathetic trick or an eye illusion, but this Möbius strip captured my attention immediately, as it surrounds us from the oldest times and we find it in nature: it seems that the DNA is built after the model of the Möbius strip. Every time I access the internet to search and find extra pieces of information about this anomaly, new doors occur waiting to be opened and automatically my brain will be tempted to ask more and more questions. This project also involves communication with other people fascinated about this ribbon. However my main target is related to the development of this unknown object and the fact that all of these years in which I worked with paper really amazes me because being a child, I could say that I was playing and modeling paper continuously, but I had never thought what one could make of it.

The mathematician of Germanic origins, Möbius, discovered this strip and we owe him all the credits, praises and merits because at this time there are several inventions which are based on this principle. To me, only thinking at its simple mechanism, though full of mysteries, makes me feel as if I were an amateur scientist, eager to find more and more curiosities about this strip. Indeed, I am fascinated by this wonder and I predict that in future, leaving time flow, we will be aware of all true mysteries and secrets of this strip!


12. Roxana Unc, 11B

Life is knowledge, curiosity, wisdom, will and trust. Life is a journey that you decide upon according to your own taste, your own powers. Möbius was the man who knew the true meaning of life, discovering its magic which he has given us.

One of his findings was the strip with just one side. Curious, isn’t it? How can a strip have a single side, how can you, by cutting a strip, get one? It seems like magic, but it is not. Möbius strip  is a very practical demonstration of the fact that our universe has limits, but it is infinite at the same time.

Möbius strip is found in everything, even in our lives, in fact we move around a Möbius strip all our life, we walk on  the „two faces”,  our life and the others’, which actually are only one and finally arrive at the same place where we started… Many people say that Mobius strip is something that can not be understood. This is its magic, a thing understood by magicians but misunderstood by the others, a thing that always surprises you, always makes you curious and eager to discover  its  mystery. Trying to discover the secrets of this band is an art, is a detachment from the normal, usual rules, and when you try to undestand, you realize that no matter how hard you try to untie its mysteries you will not be able to end up, because you will always find at least one thing where its principle exists.

This strip opens a new gateway to the beauty of mathematics, it shows us what mathematics actually is but also it always expects us to discover it in all things in our lives, either directly or indirectly, but we are free to choose: to see the magic or let it disappear like a butterfly that comes, shares its beauty with us and then disappears, leaving behind life that soon will be lost.


13. Robert Gal, 9A

 

In my opinion the magic of Möbius strip consists of: friendship, first of all, which might be the most important, the desire to “unravel the mysteries” behind the strip and the desire of discovering practical applications the strip could have. All of them are important and they are the reason this magnificent project started.

The friendship among the four implied countries made the project very successful since the very beginning. The desire of making something together gave us, the children and teachers, the chance to meet “foreign” people across the borders of our country, to socialize and share many opinions, to learn one from another about different ways of thinking in solving problems and so on. Since the “magic” Mobius’ strip hides many possible discoveries, a lot of work has to be done in future and this is possible only by working in teams, by sharing all kinds of opinions, by supporting our mates, etc.

New concepts based on Möbius strip could save some money which is very helpful nowadays. Besides money, energy could be saved too and give the strip an environmental friendly character which leads us to making the strip a new environmental sign such as the recycling one. I personally hope that in the future new, totally different concepts will be found to help other people and to give more need of research. In this way the project could be even prolonged for several years if a major discovery is made which, in my opinion, would be great.

To conclude, Möbius strip implies a lot of work which has to be done constantly during these two years. The magic is surely present when it comes to this strip. Just think how this simple strip brings together many students and teachers from different countries to work together. I consider it very special and I hope it will never end!

 

After the students read their essays, it was the jury time. Crina had to leave earlier from the meeting because of a maths contest and Laura arrived late due to the trafic jam. Hence, they could not give marks to the others priers.

Mrs. Camelia Circa Chirila, a Romanian teacher from the Comenius team, has been positively noticed by the others there. She explained the students the reason for which they had received certain marks and she gave them some advice regarding the next linquistic competition.

The results of the contest are represented in the following chart:

 

Name

David

Robi

Mihai

Oana

Alina

Andreea

Andra

Bogdan

Georgi

Roxi

Ioana

Raluca

Pusa

Cristina

Cami

Points

Alina

6

6

11

11

0

11

11

13

13

7

12

10

13

5

13

142

Andra

4

12

10

13

10

12

0

8

12

8

10

8

8

9

8

132

Andreea

5

11

12

12

5

0

9

9

7

11

11

3

9

12

10

126

Ioana

9

7

5

6

13

6

13

6

2

12

0

12

11

10

12

124

Georgi

12

10

8

7

11

9

12

10

0

3

3

9

4

6

14

118

Crina

1

1

13

8

3

5

8

11

6

9

7

5

14

14

11

116

Laura

7

5

4

9

8

7

5

2

11

10

8

6

10

4

3

99

David

0

13

7

3

9

4

6

5

8

13

9

7

7

1

6

98

Bogdan

10

3

3

4

12

2

10

0

9

4

4

13

6

7

9

96

Oana

8

8

9

0

6

13

3

7

1

6

6

1

5

13

4

90

Mihai

3

2

0

10

7

10

7

4

5

5

2

11

3

8

7

84

Roxi

2

9

6

5

2

8

2

3

3

0

13

2

12

11

5

83

Raluca

11

4

1

2

4

1

1

12

10

1

1

0

2

2

2

54

Robi

13

0

2

1

1

3

4

1

4

2

5

4

1

3

1

45

 

 

 

 

 

 

 

 

The winner was Alina Negrila. She won a chocolate and she shared it with her colleagues.

The atmosphere was relaxed and the good mood was permanently present. The students and teachers from the Comenius team were really joyful to work together. They have always felt that this project is like “a different school” while working arduously, with a lot of enthusiasm.

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