Saturday, November 16, 2013

Mathematician's toy or Klein Quartic

Here it is - a mathematician with stuffed toy.

It all started with this animation. Then I learned - it can be made with 24 heptagons. No problem - 24 heptagons were crocheted and then we both started to figure out what would be the best way to put them together. I followed David's suggestion and sewed them together in symmetric way. Then it was put away and when I picked up this fall again I wanted to stress this nice heptagonal tiling, so I put the white lines on the model. Since it is impossible to finish this surface in three dimensions I decided to add handles that would indicate how heptagons will fit together in fourth dimension. David was not happy with the surface because; It is not smooth! - that was topologist speaking. Can you iron it to make it smooth? We do have iron at home and I pressed all heptagons to make them smooth. As soon I finished doing it I regretted what have I done - yes, surface was smooth but it was not holding shape anymore! It looked like wrongly put together afghan.;-( David suggested to stuff it. And here it is - stuffed toy for the mathematician.

What is this surface? Klein quartic is named after Felix Klein who first described it in 1878. It is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed.

Klein's quartic occurs in many branches of mathematics - some say it is one of the most important mathematical structures. Which mathematical branches use Klein quartic? Representation theory, octonion multiplication, homology theory, it was used to prove Fermat Last theorem... Sounds very complex - but you do not need to know all that to enjoy this nice stuffed structure!;-)

I would like one day to see artistic representation of this surface by Halaman Ferguson. His sculpture The Eightfold Way is at the Mathematical Sciences Research Institute in Berkeley, CA. It - is made of marble and was unveiled on November 14th, 1993- 20 years and two days ago!

The acquisition of the sculpture led in due course to the publication of a book of papers (Levy 1999), detailing properties of the quartic and containing the first English translation of Klein's paper.

Thurston explained this sculpture

Halaman Ferguson about creating The Eightfold Way

Mathematician's toy made it to Scientific American blog

Tuesday, November 5, 2013

Robert Lang talks about origami in Cornell

Over the first weekend in November the weather in Ithaca has changed to warnings of winter. Monday, November 4th afternoon is still bright and sunny but with significant chill in the air. Good reason to come to Rockefeller Hall Schwartz Auditorium to listen to Robert Lang who is visiting Ithaca.

He is introduced by Itai Cohen who ends his intro by saying: " And now Robert Lang will show you what can you creatively do with physics degree."

Nobody really knows when origami started - it was (and still is) a popular past time activity for kids - everybody has made a boat, flapping bird or jumping frog at school. Earliest pictures showing folded butterflies are from 1680, pictures in 1734 Japanese manuscript let conclude from the complexity of origami that it was already well developed art. Well known around the world is
Senbazuru origami based on ancient Japanese legend about the thousand folded paper cranes. The most significant in the art of origami happened in 20th century when Akira Yoshizawa developed the language of origami.

The next big step in origami happened when after 80 publications in physics and 50 patents in optoelectronics Robert Lang decided to turn to origami full time. The reason? It was physicists "instinct" that told him - origami must have underlying mathematical properties and it is time to find them. And he did.

There are four rules for creating flat origami:

1. Two colorability - which means if we map necessary creases, we should be able to color all regions in two colors so that no two adjacent regions are in the same color.

2. At each interior vertex Mountains - Valleys = +/- 2.

3. Odd and even numbered angles around the vertex has to be the same amount and add up to 360 degrees.

4. Paper cannot intersect itself.

First 3 rules are easy to create, the fourth one is hard.

Whenever something is hard we want to know how hard. Pleats in origami are not all equal - they can be denoted true/false. That makes digital circuits of the folds, which turns into problem in mathematical logic - "not all equal". It is proved that crease assignment complexity (NAE-3-SAT) in origami is NP-complete.

If the flat origami can be so hard, then what about three-dimensional one? Robert Lang shared with the audience how he develops ideas of making 3-D objects from one sheet of paper - how the amount of paper for flaps is calculated, how the various regions are allocated on the paper, how to get from string figure which represents the object you want to fold to the number of flaps needed. These problems lead to discussions about intelligent design in origami which is connected with nonlinear constrained optimization.

For my delight Robert Lang was mentioning folding of hyperbolic tiling and even 3-D printed hyperbolic paper which he used to fold "hyperbolic crane". Well, that crane came out with two heads...
Once origami even led to "bug wars" - competition among the origami enthusiasts who can fold the most complicated bug. But are there any applications in real world?

Miura-Ori origami folding pattern is used in space exploration. This folding pattern makes tessellation which allows for example to open a map just with one move. Here is how it works.

Origami is used in Solar sail - NASA uses it to observe far away planets. Of course, there is another agency which would love to turn it to the other direction...;-)
Heart implants, origami stents, protein folding, folding membranes... Paul Rothemund used DNA folding to develop cancer drugs which would be folded and unfold only when they have reached a cancer cell they should kill.

Origami is not an invented past time thing - nature knows how to fold - proteins, DNA, earwigs fold their wings.

To learn more and access software for mathematical origami design explore Robert Lang's official website.

Monday, September 16, 2013

DUMBO Arts Festival 2013

The 17th Annual DUMBO Arts Festival will take place from Friday, September 27 to Sunday, September 29, 2013.

One of the city’s largest and most ambitious arts events, the DUMBO Arts Festival will draw an expected 200,000 people to DUMBO to experience extraordinary art by more than 400 artists from around the world.

Set amid the backdrop of the Brooklyn Bridge and the Manhattan skyline in one of the city’s most tech-friendly and culturally rich hubs, performances and exhibits will fill more than 50 stages, while 100 artists’ studios will open their doors as the entire neighborhood becomes an exuberant arts playground.  For the full festival schedule visit
I am very excited to be one of the artists to participate in this festival. I will show not only my works but will also conduct four workshops. Please see schedule of my performances here HyperbolicHyperbolicHyperbolic

Wednesday, May 22, 2013

One of the fastest growing professions

Academia seems like this giant uprooted tree

What is one of the fastest growing professions in US?
Let us see what Google search comes up with.

Here is a list of 50 fastest growing occupations.
I did some counting there. For the first two fastest growing (= finding job) required education level is less than high school. 16 requires high school diploma, 2 some kind of post-secondary and 6 Associate degree ( I do not know what the difference is). 10 on this list asks for Bachelors, 5 for Masters, and 5 for professional or Doctoral degree...

Another list - 30 fastest growing jobs by 2016.
May be because this list was done in Boston, average education requirements are a little bit higher but still 2 of the fastest growing jobs does not require high school diploma, 5 out of 30 does require high school diploma, high school plus 2 more years will be useful for 7 out of 30, this list predicts that bachelors will be needed for 9, masters for 5 and only one job - veterinarian will require post graduate education.

The list of fastest growing jobs by U.S. Bureau of Labor Statistics predicts that the fastest growing need will be
1. Personal care aides2010 employment: 861,000
2020 projection: 1,468,000
Percent growth: 70.5
Median annual wage (2010): $19,640

Education requirement is less than high school diploma.

U.S. News 100 best jobs in 2013 does not mention on the list neither mathematician or college professor. But those two do appear on a list WSJ 2013  best and worst jobs : university professor as #14 and mathematician as #18. On a list of the worst jobs janitor is three steps higher than author (#153 and #156 out of 200).

However in none of these lists I found Adjunct Professor. If you are not familiar with academia closely then you would say - why bother looking through all these lists, it's under the general term - University Professor.

According Concord Monitor adjunct professor is one of the fastest growing and most poorly-paid occupations in America. According to the American Association of University Professors, while a full professor at a public university with a doctorate earns $120,000 per year plus benefits, an adjunct, even one with a doctorate and a full course load, makes $20,000 with no benefits. (Janitor's median salary is slightly over 22K). Few earn what could be considered a living wage, yet adjuncts now teach more than 70 percent of all college courses.

I wanted to write about the life of an adjunct in university for quite some time. It felt to me that people should know how it is. In my talks I have met many people and only those who are adjuncts themselves or have a family member or a friend who is an adjunct, really understand the situation.

Recently a very good article went viral on Internet - Academias indentured servants by Sarah Kendzior where she wrote:
Is academia a cult? That is debatable, but it is certainly a caste system. Outspoken academics  are rare: most tenured faculty have stayed silent about the adjunct crisis. "It is difficult to get a man to understand something when his job depends on not understanding it," wrote Upton Sinclair, the American author famous for his essays on labour exploitation. Somewhere in America, a tenured professor may be teaching his work, as a nearby adjunct holds office hours out of her car.
 What stuck with me most from this article was:
"It is easy to make people work for less than they are worth when they are conditioned to feel worthless."It is easy to make people work for less than they are worth when they are conditioned to feel worthless"
Worthless - it is familiar feeling. You may be surprised hearing it from me. I am not working for less, I am often expected to work for free - because people think that I am well paid university professor.  I am officially retired but I am too young for Social Security. I am lucky - I do have a husband who supports me and hyperbolic crochet. But it does not help in fighting off a feeling of being worthless. My job is unpaid and therefore - worthless, which makes me feel a failure. Writing Crocheting Adventures with the Hyperbolic Planes was possible because of the support of my husband and some of my own money earned when I was lucky to get some calculus courses to teach. ( I am not joking when I say that I have to teach calculus to have money to buy yarn for my crocheted hyperbolic planes.) I was very proud when learned about the Euler Prize for my book. I decided to use an opportunity to speak about adjuncts in my response. There were several people who later came up to me to say "Thank you for talking about adjuncts."  At that point I did not know that later in the same year there will be more talks in media about adjuncts. I missed them because I was in Latvia to take care of my mother for a long period.
Now I looked up some of them.

Last year Sarah Kendzior already published an article The closing of American Academia. At the end of it she wrote:
 I struggle with the limited opportunities in academia for Americans like me, people for whom education was once a path out of poverty, and not a way into it.
My father, the first person in his family to go to college, tries to tell me my degree has value. "Our family came here with nothing," he says of my great-grandparents, who fled Poland a century ago. "Do you know how incredible it is that you did this, how proud they would be?"And my heart broke a little when he said that, because his illusion is so touching - so revealing of the values of his generation, and so alien to the experience of mine.

Here are some more attempts to raise awareness about adjuncts and the situation in higher education:

The PhD now comes with food stamps

A number of PhD on public aide has tripled

Why so many PhD's are on food stamps from NPR

The Crisis in Higher Education

The Caste System in Higher Education

It is late already, enough for tonight. One day I may be brave enough to write my own story how it feels to be adjunct...

update October 1 (well, late update - sorry)
here is  infografik about the adjunct crisis by John Kelle:

<a href=""><img src="" alt="Un-Hired Ed: The Growing Adjunct Crisis" width="500"  border="0" /></a><br />Source: <a href=""></a>

Friday, April 12, 2013

Math Babe on the Weapons of Math Destruction

Feels strange that now I will be writing a blog entry about a blogger. Yesterday Cathy O'Neill or MathBabe gave a talk in Oliver Club at Cornell. In a picture - Tara Holm is introducing Cathy.
When you are not in academics one can give talks with sexy titles and become a public face of mathematics. What Cathy does is best to learn from her own blog Math Babe. What she was talking about yesterday was more like recruiting new math PhD's to her boot camp to learn more about conscious modelling.
Mathematical models are all around us and while mathematicians recognize them, general public does not always see how math is used against them. Starting already in school people are trained that "this is math - you will not understand". This phrase is fashionable to use instead " I don't care to explain it to you even if I understand it." It is so often that a number becomes your characteristic - google search results, your credit score, GPA, SAT score, Predictive models are used based on sometimes questionable data and data interpretations. Those are not oracles but non-mathematicians trust them because - if it math it must be true. Mathematician is always questioning these results. mathematicians are mostly honest people, they easily admit if they are wrong. I proved a theorem! No, you have a flaw here. Oh, really? OK, I should work harder, no hard feelings. This is not routine in other sciences.
Banks are underestimating risks. They are making us to believe that they are following a normal distribution. Investment banker will show you something like this:

In the meantime "banks too big to fail" actually have this:
Quite a different picture! 95th percentile makes everything look good. So what the PhD's on Wall street do? Push the risk in last 5%.
Data are screwed in many places. Let's take teacher evaluation. When Bloomberg wanted to be elected as mayor of NYC, tests were easier and test scores went up. Teachers are often evaluated for things which are out of their control, for example, student attendance. Standard teacher evaluation punishes teachers in tough schools. The value of standardized test scores are so much overvalued because it is a huge business, so those who are in control of preparing tests, push importance of those tests.
Example of data for some teacher evaluation:

Any mathematician will tell you that there is no correlation among these data, but nevertheless administration will pull out "valuable" evaluation data.
So what Math Babe proposed mathematicians should do not to let math be abused and not letting people to be abused by math?
Defend math
Educate ourselves
Sign up to referee public models!
Require transparent evaluation methods
And request to new math PhD's - let's not become economists....

Cathy raised many interesting questions, was open to questions from  the audience, but also was good not really answering some of them. Afterwards though anybody could go and talk to her in more details.
But of course - read her blog to learn more and as mathematicians we should be responsible for educating general public and stop using math as abusive tool in hands of those in power.

Friday, March 22, 2013

Allowing Unexpected: Creativity in Your Classroom

These are some excerpts from my talk today in Western Illinois University annual Teachers conference and also links to some aditional material.
In an ordinary mathematics class, the program is fairly clear cut. We have problems to solve, or a method of calculation to explain, or a theorem to prove. The main work to be done will be in writing, usually on the blackboard. If the problems are solved, the theorems proved, or the calculations completed, then teacher and class know that they have completed daily task. Is this teaching how to think mathematically? Getting to know new mathematical facts and there applications - it is creating new knowledge in students heads - but is it creative thinking?

Everybody at least once has had an experience when you have to talk in a language which is not native for you and it is hard to find correct words to express what are you thinking. Mathematics also has its own language (may be languages?) and we have to listen to another person carefully to understand what he/she wants to say. Are we patient enough with our students? How do we help students to express their ideas in a "foreign" language? We can understand a person talking even with mistakes if we understand the ideas the person is talking about. And no damage is done to these ideas because of some grammar mistakes. Why do we want mathematics to be an exception? Why is formal language so sacred? 

            Here is a quote from George Orwell:

 It is instructive sight to see a waiter going into a hotel dining room. As he passes the door a sudden change comes over him. The set of his shoulders alters; all the dirt and hurry and irritation have dropped off in an instant. He glides over the carpet, with a solemn, priest-like air... he entered the dining room and sailed across it, dish in hand, graceful as a swan.

  What does this quote tell us about the teaching mathematics? What is the purpose of separating front from back, kitchen from dining hall? It is not only to keep customers from interfering with the cooking. It is also to keep them from knowing too much about cooking.
The front and the back of mathematics are not physical locations like dining room and the kitchen. The front is mathematics in its finished form - lectures, textbooks, journals. The back is mathematics among working mathematicians. Which mathematics we are teaching to our students? Of course, the front one. Why? Because we feel safe there. We are teaching facts which are a priori acknowledged and if that happened more than 100 years ago then it is even safer. Looking over mathematics curriculum you are teaching have you pondered about the question which is the newest mathematics we are teaching? Are we telling our students that mathematics is changing all the time even that it is one of the oldest human activities?

Keith Devlin in American Scientist compares learning mathematics to learning how to play piano:
Just as music is created and enjoyed within the mind, so too is mathematics created and carried out (and by many of us enjoyed) in the mind. At its heart, mathematics is a mental activity—a way of thinking—one that over several millennia of human history has proved to be highly beneficial to life and society. In both music and mathematics, the symbols are merely static representations on a flat surface of dynamic mental processes. Just as the trained musician can look at a musical score and hear the music come alive in her or his head, so too the trained mathematician can look at a page of symbolic mathematics and have that mathematics come alive in the mind. So why is it that many people believe mathematics itself is symbolic manipulation? And if the answer is that it results from our classroom experiences, why is mathematics taught that way? I can answer that second question. We teach mathematics symbolically because, for many centuries, symbolic representation has been the most effective way to record mathematics and pass on mathematical knowledge to others.

 A necessary (though certainly not sufficient) condition for significant teaching is the provision of emphases; if everything is important then nothing is important. - Abe Schenitzer

1964 book The Act of Creation ArthurKoestler attempted to develop the general theory of human creativity. His concept of bisociation has been adopted, generalized and formalized by cognitive linguists Gilles Fauconnier and Mark Turner, who developed it into conceptual blending. Koestler defined  bisociation as “the creative leap [or insight], which connects previously unconnected frames of reference and makes us experience reality on several planes at once.” How to realize it? Koestler offered a suggestion in the form of a triptych, which consists of three panels…indicating 3 domains of creativity which shade into each other without sharp boundaries: Humor, Discovery, and Art.

The first is intended to make us laugh, the second make us understand, the third make us marvel Or for short: Ha-ha-ha! – Aha! – Ah!

But there is another word – Oh! – when things go wrong. If math is to be a creative subject then we have to regard it as a subject where it is ok to get things WRONG. If you have never made mistakes, you are never discovering anything new.
As Tomass Edison once said – I made a lot of mistakes. Later I patented most of them.

Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. - William Thurston
If you haven't seen yet then do join almost 10 million viewers of  Sir Ken RobÄ«nson's talk
 Changing Education Paradigms. Another one of his talks is The World We Explore.

Some more interesting talks
Fun to Imagine:

Are Mathematicians Creative?

What Mathematicians Actually Do?

I Want to Be a Mathematician:

Andrey Cherkasov Math Jokes collection

Feynman and Computing:

Mysteries of Mathematical Universe - talk from World Science Festival

Mathmagic with Arthur Benjamin


Friday, March 15, 2013

I found Ueno Masao!

I was looking for something else but stumbled on intriguing picture.
It prompted me to look up who is the artist and I found another picture:
He  called this "Rotating Ellipses" but of course I noticed hyperboloid!
This is how I found Fabuluos works by Ueno Masao like this Spring:

Japanese bamboo artist Ueno Masao (b. 1949) studied architecture in Shibaura Institute of Technology (class of 1972). From 1977 to 1983 studied traditional bamboo art and worked as apprentice with master craftsmen Konma Hazuaki and Kondou Shousaku.

How bamboo is processed for his works

Ueno Masao Swirl

Making Swirl

An Interview with Ueno Masao

Braiding by Ueno Masao

Earlier Works

His works are indeed The Feast for the Eyes:

Tuesday, March 12, 2013

Secrets of Mental Math

 The Kieval Lecture Series (Cornell Mathematics Department) is funded through a bequest of the late Dr. Harry S. Kieval ’36, a longtime professor of mathematics at Humboldt State University in Arcata, California, who died in 1994.
Innaugural speaker in 1998 was John Milnor. The full list of speakers is here.

Today in 2013 Kieval lecture series Arthur Benjamin (Harvey Mudd) was speaking about Secrets of Mental Math.

Arthur Benjamin grew up in Cleveland, Ohio, and earned his B.S. at Carnegie Mellon University in 1983 and his Ph.D. in mathematical sciences at Johns Hopkins University in 1989. Since then he has been a professor of mathematics at Harvey Mudd College, in Claremont, California, where he has served as department chair. He has written three books and is co-editor of Math Horizonsmagazine, published by the Mathematical Association of America (MAA). In 2000, the MAA awarded him the Haimo Prize for Distinguished Teaching.
Arthur Benjamin is also a professional magician, and frequently performs at the Magic Castle in Hollywood. He has demonstrated and explained his calculating talents to audiences all over the world and has appeared on numerous television and radio programs, including The Today Show, CNN, and National Public Radio. He has been featured in Scientific American, Omni, Discover, People, Esquire, The New York Times, Los Angeles Times, and Reader's Digest. In 2005, Reader's Digest called him "America's Best Math Whiz."

While he clearly has an extraordinary gift in the realm of mathematics, Benjamin also considers a lot of his numerical talent (and prodigious talent in general) to be gained through hours of practice. He describes the some practical drills and techniques for left to right calculation. He also explores visual, auditory and mental imagery that can assist in holding and manipulating large numbers in your head. Certain tricks work better for some than others, depending on individual traits and learning styles.
Secrets of Mental Math are revealed to readers, hours of practice now should be done.

 TED talk: Arthur Benjamin does Mathmagic

 TED talk:    Teach Statistics before Calculus

Arthur Benjamin on Colbert Report

How to memorize Large Numbers

Bache Auditorium today was full with excited audience ages 3 to ... well, let us say, very mature (it means including professors-emeritus).
First calculators had to be checked, so four volunteers came forward. You may recognize that one of them is Steven Strogatz.
As expected, these four guys could not use their calculators as fast as Arthur Benjamin used his mental calculation skills.

One of audiences most liked tricks was creating "personalized" magic square - based on your birthdate. Since this was mostly "mathematical" audience one of the questions was - how did you do this? Arthur said that explaining magic tricks takes away that wow! moment but agreed to show how this trick works. Of course, the following question was - how did you come up with the idea of this trick? The answer was - very much like it happens with mathemathematics: you read about something, then start thinking how this can be used in other ways, then some new idea may appear. He said that he was reading some publication on magic tricks, then he decided to try to make these magic squares using person's birthdate and figured out how he could perform it. Important part of magic tricks is the performance. Magicians have to be also actors and psychologists.
Arthur Benjamin also explained how he has memorized digits of pi - this time paper sheet could hold only 60 of them but he can go up to 100.
How could this mental math can be useful? Arthur told us that he got interested in numbers after he learned as a child to multiply three digit numbers by two digit numbers. He figured out that there are many different ways how one can do it and amazingly - result is the same! This fact made him to be excited about mathematics and up to this day he always has the urge to try different ways how to prove or solve something. (Earlier today he gave a talk in Math Department how to prove trigonometric formulas with combinatorial tools. He call it "combinatorial proof" or shortly in Harvey Mudd it is called "cool proof".)
As the "grand finale" he sang us a song dedicated to "pi day" which will be in two days. Melody was borrowed but rhyms were original and cool.
It was fun day with Arthur Benjamin!

Saturday, March 9, 2013

Math in Movies

I tried to take a picture of Toni deRose when he was giving a talk but it did not work. So here is a snapshot of him already afterwards when we all enjoyed some refreshments.On March 6th MoMath organized Math Encounters celebrated two year anniversary and for the very first time were hosted in MoMath own place. It was windy and cold outside, so while waiting for prompt 3:45pm opening of the door for the talk Math in Movies by Toni deRose we were let inside MoMath store. This was the first time I was lucky my visit to NYC coincided with Math Encounters, so I was as promptly there as was suggested by e-mail from the organizers. As it happens when things are set somewhere for the first time - prompt opening time was a little bit late and the talk started late. It had an advantage of catching-up with some people I hadn't seen for some time.
While I was thinking of writing about this event Tim Carmody already did - read it here.

Pixar's policy is to use only in-house stories, they do not accept stories from outside the studio. I found interesting Pixar's 22 rules of story telling. Number 9 on the list - When you’re stuck, make a list of what wouldn’t happen next – is a great one and can apply to writers in all genres.

The road from the story to the film is four year's long and takes about 150 thousand sketches. Technology through the years has improved and actually these days anybody can make an animated movie using free software Blender. As Toni deRose said - we are just waiting when some kid will make a great movie, so we can hire him/her before we are put out of business.

Mostly the math behind the animations is coordinate geometry - positions of key points of movie characters are described by coordinates and then software just translates (addition, subtraction), scales (multiplication), rotates (trigonometry) these coordinates. More interesting math is used in designing the characters - animators use subdivision surfaces, which actually has developed themselves into interesting research field.

More about how Pixar studio is using math:
An Interview with Toni deRose in 2009 and this year in January before his invited address for Joint Math Meetings 2013.

Some of previous Math Encounters can be seen on YouTube:

Keith Devlin on Golden Ratio

Colin Wright on mathematics of juggling

Jeff Weeks on Shape of Space

Eric Demaine on Geometry of Origami

Carlo Sequin on topological sculptures

Some other  Math in Movies:



Math - Bunker Style

Good Will Hunting - math problem

The best list of Math in Movies is by Oliver Knill (Dept.of Mathematics, Harvard)

There was not much time (always not enough time in NYC!) to find some art exhibit connected with math but the one on my list was the exhibit in  Metropolitan Museum Cambodian Rattan: The Sculptures by Sopheap Pich. My favorite was Morning Glory in which can be seen pseudosphere continuing in ruffles. Since it is exhibit, visitors are not allowed to take pictures, so I could not take a picture to show it. More images of Sopheap Pich sculptures are on his webpage.


Wednesday, February 13, 2013

Happy Valentine's Day!


I saw the idea on Cut-the-Knot page  and decided to make my Valentine to all friends of my blog.
Here are the steps:
1. Take two paper strips.
2. Glue ends together with 180 degree twist - yes, make two Moebius bands BUT make sure they have - twisted differently : one is right, the other is left.
3. Now glue these two strips together:
4. Cut the strips in the middle and your mathematical Valentine is done!
As a reading for Valentine's Day something romantic and mathematical at the same I suggest
Paolo Giordano the Solitude of Prime Numbers

During the Nemo storm I finished a piece which will go to the exhibit this year. It has been confirmed that my dream has come true - I will be having joint exhibit in Mattatuck Museum with my friend Gail Rothschild - please visit her webpage! And if today's meeting will have a positive outcome, then there will be another exhibit - in New York...