Celebrating the Conjugate Twins

Today is the birthday of Richard Schoen born 1950 in Celina, Ohio. Schoen is a notable mathematician that has made contributions in the area global differential geometry. In a collaborative effort with Shing-Tung Yao, he proved the fundamental positive theorem in general relativity.

Today's quote is from the cat in The Cat in the Hat by Dr. Seuss.

"... 'I will show you

another game that I know!'

and then he ran out.

and, then, as fast as a fox,

the Cat in the Hat

came back in with a box.

A big red wood box.

It was shut with a hook.

'Now look at this trick,'

said the cat.

'Take a look!'

Then he got up on top

With a tip of his hat.

'I call this game FUN-IN-A-BOX,'

said the cat.

'In this box are two things

I will show to you now.

You will like these two things,'

Said the cat with a bow.

'I will pick up the hook.

You will see something new.

Two things. and I call them

Thing One and Thing Two.

These Things will not bite you.

They want to have fun.'

Then, out of the box

came Thing Two and Thing One!' ..."

I teach the idea of conjugates quite often in my classes. Conjugates are two arithmetic expressions that involve two values. As an example, let's suppose I use the two values, 9 and 5. Conjugates using these two values would be: 9 + 5 and 9 - 5. Conjugates are simply two expressions that are formed by adding and subtracting the values in the same order. Conjugates can be generalized to a + b and a - b whose product becomes a^2 - b^2 which math geeks refer to as the difference between squares. A few examples of conjugates that are utilized in mathematics are: confidence intervals (statistics), complex solutions (analytical algebra), and rational expressions containing square roots (limits/calculus).

Today has always been a special day for me. As a child, this day signified the beginning of the holiday season. I marked these dates on my calendar: October 23, October 31, November 22-28, December 25, and January 1. I am sure that four of the five dates listed are obvious to the reader and many may presume that with a mathematics background that I am championing today's date as Mole Day, but the reality is today is my brothers' birthday. I have brothers that are identical twins.

In fact, my brothers are mirror twins. Bruce is right-handed and Bill is left-handed. When I study their facial features, Bruce resembles my mother's side of the family and Bill, of course, my father's. On their birthday, my mother would make a white angel food cake for one and a chocolate devils cake for the other. I was in culinary heaven, two slices in one day! The holiday season had begun!

I have always felt their relationship is one of the strongest I have every encountered. My sons have often mentioned that when Bruce and Bill are together, the sum is much greater than the parts. In my classes, I use stories of my family to form connections to mathematics. I refer to Bruce and Bill as the conjugate twins.

In my blog, The Fabulous Five, I noted that the interactions we have with our siblings shape our personalities. My grandmother would describe to me the befuddlement that overcame me as a toddler when my mother came home with twin babies. I don't really know how much I understood of the complexity that just entered my life as a 18-month-old child, but I would agree their births changed my life significantly from that moment. I was transformed in a short period of time from the baby of the family to something different, not necessarily the middle child but not exactly the oldest either. Like Thing One and Thing Two, two creatures had entered my domain that would bring a new thrilling energy that was a mixture of frustration, fear, and sheer enjoyment.

My brothers became my playmates and my rivals. In the next 16 years, I learned to be competitive, cooperative, assertive, compromising, creative, imaginative, empathetic, and manipulative. We were explorers, warriors, athletes, and builders. We crawled through culverts, shot imaginary intruders with sticks and broken baseball bats, and played endless games of baseball and football. Our teams were easily decided--the twins versus me. We adjusted rules so that two-on-one games were possible through a series of arguments and heated discussions. Any disputes evaporated in our nightly dreams and we began each day anticipating a renewed adventure.

Today, I will celebrate the birth of the Conjugate Twins, Bruce and Bill. I will raise a toast in their honor, give them a call, and enter the holiday season thankful for the impact they have had in my life.

Happy Birthday Sis!

Today is the birthday of Boris Bukreev born 1859 in Lgov, Kursk gubernia, Russia. Bukreev worked in the areas of algebra, mathematical analysis, calculus of variations, differentiable geometry, and complex variable functionality.

Today's quote is from Persi Diaconis relating his transition from a magician to a mathematician. He recalls, "I thought I could do anything...So I bought William Feller's Introduction To Probability Theory and Its Applications and I thought I would just read this book. And I couldn't read it. I didn't know calculus, or at least not enough."

Persi Diaconis

I first discovered Persi Diaconis when I was assigned to give a presentation on a current mathematician in a History of Mathematics course I was taking. I was immediately caught up in his life's journey. He was born January 31, 1945 in New York, New York. His parents were musicians and as a young child he studied violin at Juilliard School in New York. He was interested in mathematics but also highly motivated in the area magic. He used mathematics in many of his magic tricks. He was on track to graduate high school at the age of 15 but dropped out a year earlier to pursue a career as a magician. He eventually left his career in magic, went back to school, and received a doctorate in statistics from Harvard University. Diaconis mathematical pursuits are widely divergent. He has written books and papers ranging from group representations in probability and statistics to Markov chains. In 1992, with Dave Bayer, Diaconis proved that the maximum number of shuffles need to riffle shuffle a deck of cards is seven. By comparison, the overhand method of shuffling would take 2500 shuffles to randomize a deck of cards.

Today is my sister's birthday. Marilyn is my only sister and although she looks younger than I, she is seven years my senior. In fact, for much of my preadolescent years, she raised me. I believe her structured guidance was due to my mother's needed attention to my younger brothers. I was a challenging child. I had three noticeable characteristics. I was defiant. I was stubborn. I was argumentative. These notable traits did not go unnoticed by my teenaged sister and many times we butted heads.

I have written previously about the sibling bond and my sister did instruct and direct me to paths that I still travel on to this day. She taught me how to twist to Chubby Checker and she spent an entire afternoon with me detailing the intricacies of riffle shuffling. Most importantly, the summer before my ninth grade year, she pulled me aside, and we had a long talk about what to expect in high school. I have forgotten many of the details of that conversation but one bit of advise has impacted me to this day. She suggested that I get involved in school. She felt that although academics were important, being involved in school activities outside of the classroom would energize me, create an enjoyable experience, and build healthy relationships. I think that her suggestion planted the seed that later grew into my desire to be a teacher. As I reflect on our conversation, it was a touching moment between a twenty-something woman and a teenage boy. I think she was aware that I was a socially awkward introvert that needed a gentle push or more appropriately, to be placed in her hands and riffle shuffled, not seven times, just once.

We receive many gifts as we journey through life. Some gifts come wrapped in bows and boxes on holidays. Some gifts pass us as we wander unaware in this passage of time until we pause and gather our senses. Today, I celebrate one of my gifts.

Mona smiles at a wrinkle in time

Today is the birthday of Jurgen Moser born 1928 in Konigsberg, Germany. Moser was a mathematician proficient in techniques applied to Hamiltonian dynamical systems and generated the "Moser Twist Stability Theorem".

The quote for the day is from Madeleine L'Engel in A Wrinkle in Time. "I don't understand it any more than you do, but one thing I've learned is that you don't have to understand things for them to be."

I recently traveled to France, spent some time in Paris, and had the opportunity to journey to the southern regions of the country. This blog will be a summation of my observations through the tinted lenses of a mathematics cheerleader.

One of the stops touring Paris was the marker, Point Zero. Point Zero is the spot in which all distances from Paris are measured. The distance from Minneapolis, Minnesota to Paris, France is 4203 miles.

Point Zero

The path of a flight from Minneapolis to Paris is an arc that is a portion of a great circle. This arc is called a geodesic.

A great circle is the concept of a line in spherical geometry. Spherical geometry has applications in navigation and astronomy. This type of geometry is termed Non-Euclidean geometry which considers the Euclidean axiom of parallel lines.

Another perspective that I have of point zero is the concept of the origin. The term origin can conjure many meanings. On a Cartesian coordinate system the origin is where the horizontal line (x-axis) and the vertical line (y-axis) meet. This point is designated by the ordered pair (0, 0). Similarly, the Earth has a horizontal line (equator) and vertical line (prime meridian) which intersects in the Gulf of Guinea in the Atlantic Ocean. This point is designated as 0° latitude and 0° longitude.

                           Prime Meridian                Equator

The Mathematical Origin

I also visited the Louvre. The Louvre is massive and impressive but I found a treasure chest of mathematical and scientific wonderments in rooms that displayed items owned by King Louis XIV.

Compasses and Calipers      Directional Compasses

Miniature Sundials              Globes and Telescopes

Microscopes                               Scales

Sextants                         I Have No Clue :-)

The painting, Mona Lisa, created by Leonardo da Vinci hangs in the Louvre and is one of most popular sites. Not only was Da Vinci an artist but he also was a scientist and mathematician. In mathematics, a controversial numeric value is the "Golden Ratio". The Golden Ratio is computed from the following proportion: (a + b)/a = a/b which simplifies to the equation a^2 - ab + b^2 = 0. The solution for this quadratic equation is (1 + sqr(5))/2 or 1.61803398875 (this decimal is an approximation since the golden ratio is an irrational number). This ratio is historically called the ratio of beauty and most famous works of art and architecture supposedly have that particular ratio embedded within them. A "Golden Rectangle" has its ratio of longest side to shortest side as an approximation to that ratio. A "Golden Spiral" is a graph of a logarithmic equation that has the Golden Ratio as its growth factor.

The "Mona Lisa" at the Louvre

The legs of the blue triangle originate in the bottom corners of the painting and bisect the width of the top of the painting. The Golden Rectangle is placed on the left side of the painting with its width progressing across the top of her head. The end of that segment lies on the right side of the triangle. The Golden Spiral frames her face.

When I was in sixth grade, I bought the paperback version of A Wrinkle In Time by Madeleine L'Engel. In fact, I still have it. A quote from the book has stayed with me for a long time. A character in the book, Charles Wallace, explains a tesseract in the following manner, "Well, the fifth dimension's a tesseract. You add that to the other four dimensions and you can travel through space without having to go the long way around. In other words, to put it into Euclid, old-fashioned plane geometry, a straight line is not the shortest distance between two points."

Recently, I watched the movie, Interstellar. In this movie, an astronaut explains the concept of a wormhole. In the explanation, two points are drawn on a piece of paper and a straight line connecting them. The paper is bent so that the two points coincide and a pencil is punched through both holes. That punch represents a wormhole, a portal, which connects one dimension with another.

Towards the end of my stay in France, I had the opportunity to visit an old friend from high school. Jean was an exchange student from France my senior year. We became good friends and previously had seen each other about 20 years ago. There was no rekindling of our friendship, we are still great friends. Our conversations were like we had corresponded consistently for 39 years but in actuality, we hadn't. I have had a few friends in which time has placed a wormhole, in which the tapestry of our lives was folded and 1976 and 2015 became one. The shortest distance between two points is not a straight line.

France: Day 17- Cedric Villani

Today is the birthday of Alexis Bouvard born 1767 in Contamines, Haute-Savoie, France. Bouvard is famous for the mathematics he used to discover Neptune.

The quote of the day is by Cedric Villani. In his book, Birth of a Theorem, A Mathematical Adventure, he writes, "Far from moving swiftyly between two points, in a straight line, the mathematician moves forward haltingly, along a long and windy road. He meets with obstacles, suffers setbacks, sometimes he loses his way. As we all do from time to time."

Cedric Villani

Cedric Villani was born 1973 in Brive-la-Gaillarde, France. Villani is currently working on partial differential equations and mathematical physics. He was awarded the Fields Medal in 2010 for his work on Landau damping and the Boltzmann equation.

In the April 14, 2015, edition of The New Yorker, an article entitled The Lady Gaga of French Mathematicians Comes Stateside by Thomas Lin, Lin quoted Villani, "We (mathematics) are the most hidden of all fields. We are the one that typically interact the least with the outer world. We are also the field which is the most emblematic of revulsion in school."

Geoffroy Clavel wrote in the May 7, 2014 edition of the Huffington Post, an articled entitled Cedric Villani, 'The Lady Gaga of Mathematics' Wants To Bring The Joy Of His Discipline To Everyone. In the article, Clavel describes Villani as the current ambassador of mathematics. Villani not only loves mathematics, "he also wants to convince the wider public that this dry subject can be fascinating - - as long as you know how to talk about it."

I have another item to add to my bucket list, meeting Cedric Villani, perhaps on my next trip to France. Passez une bonne journee mes amis, jusqu'a la prochaine fois.

France: Day 9 - Chatelet

Today is the birthday of Blaise Pascal born 1623 in Cleremont-Ferrand, France. At the age of 19, Pascal invented the first mechanical calculator that was sold commercially. In physics, he proved that air had weight and that vacuums are possible in nature. In mathematics, he developed an early form of integral calculus and cofounded with Fermat, probability theory. Eventually, Pascal lived as an informal hermit producing two works of religous philosophy: Provential Letters and Thoughts.

Today's quote is written by Voltaire in a correspondance to King Frederick II of Prussia. He wrote that Chatelet "was a great man whose only fault was being a woman."

Emilie du Chatele

Emilie du Chatele was born December 17, 1706 in Paris, France. Chatele was a noblewoman who made contributions in the areas of philosophy, natural science, and mathematics. 

In her writings, she challenged John Locke's philosophy. She was adamant that knowledge could only be verified through experience.

In natural science, she specifically focused on fire. She predicted that there was a special light that emanated heat on objects. This special light is now known as infrared radiation.

In the mathematics, she is most well known for translating Isaac Newton's Principia Mathematica. This translation led French scientists to discard Cartesian physics and adopt Newtonian which was highly controversial at the time. She also corrected Newton. Newton had shown that energy of a moving object was proportional to the mass times the velocity of the object. Chatele demonstrated that energy was proportional to the mass times the square of the velocity of the object. Chatele's work on the relationship between energy and velocity inspired Albert Einstein to formulate the equation, E = mc^2.

Emilie Chatele was born during the Age of Enlightenment. Her father was a courtier for King Louis XIV and her parents used this advantage to educate her in languages, mathematics, and the sciences. Her mother strongly encouraged her to question any stated fact. This encouragement helped foster an independent thinking daughter. Her independence was present in her intellectual works as well as her personal life. She had an intellectual and intimate relationships with Voltaire and the philosophers, Maupertuis, and La Mettrie. She was independent and original woman. She was her own person.

France: Day 8 - Fermat

Today is the birthday of Frieda Nugel born 1884 in Cottbus, Brandenburg, Germany. Nugel was one of the first women to receive a doctorate in mathematics in Germany.

Today's quote is by Pierre de Fermat. He said "I am more exempt and more distant than any man in the world."

Pierre de Fermat

Pierre de Fermat was born August 17, 1601 in Tarn-et-Garnone, France. Fermat, a lawyer, was influential in the early developments of calculus and made significant contributions in the areas of analytic geometry, probability, and optics.

Thirty years after his death, a handed written note by Fermat was found in the margin of a book entitled "Arithmetica" written by Diophantus.  Fermat wrote, "It is impossible for any number which is a power greater than the second to be written as a sum of two like powers.  I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." His quote means that there are no three positive integers, x, y, and z, where x^n + y^n = z^n and n is an integer greater than 2. This "marvelous demonstration" the Fermat refers to, was a mystery for 358 years, spawning a branch of mathematics called algebraic number theory and the modularity theorem.

Two examples I can give are: 3^1 + 4^1 = 7^1 and 3^2 + 4^2 = 5^2 and there are infinite number of examples for powers of 1 and 2 but none greater than 2. In 1994, Andrew Wiles successfully proved this mysterious conjecture that had baffled mathematicians for almost four centuries.

Fermat had made claims of original proofs on many of his theorems but only a few of those proofs are in existance. Many mathematicians doubt that those proofs existed due to the difficult nature of his theorems and the limitations of the mathematics at his time.

What type of man was Fermat? His contributions to mathematics are immense. He and Descartes are considered the emminent mathematicians of their lifetimes. Why did he choose to write that note in the margin of his father's book? Did he feel a need to overexagerrate his abilities? Was he concerned about his legacy and wanted a theorem that would live for centuries? Was he a prankster and some how knew that his brief commentary would drive future mathematicians to the edges of their own sanities? For myself, Fermat is the true mystery.

France: Day 2 - Germain

Today is the birthday of Paul Guldin born 1577 in St. Gall, Switzerland. Guiding made contributions in the areas of volumes and the center of gravity.

Today's quote is by Carl Friedrich Gauss. In a letter to Sophie Germain, Gauss wrote, "The enchanting charms of this sublime science reveal to only those who have the courage to go deeply into it. But when a woman, who because of her sex and our prejudices encounter infinitely more obstacles than a man in familiarizing herself with complicated problems, succeeds nonetheless in surmounting these obstacles and penetrating the obscure parts of them, without doubt she has the noblest courage, quite extraordinary talents and superior genius."

Sophie Germain

Sophie Germain was born April 1, 1776 in Paris, France. When she was 13, the Bastille fell and as a result she was required to stay inside. During this isolation, she taught herself Greek, Latin, and mathematics. Her parents disapproved of her passion for mathematics which at this time was considered an inappropriate field of study for women. Her parents attempted to restrict her studies by eliminating the fire in her room and by removing her clothes. However, Sophie's parents relented when they found her asleep with a frozen ink horn in hand and a slate of equations on her desk. 

Sophie was not allowed to attend to attend an university but was able to obtain lecture notes. She started send comments on the lecture notes under the pseudonym Monsieur Antoine-August Le Blanco. Using this pseudonym, she established a relationship with Carl Gaus and Joseph-Louis Lagrange, two prominent mathematicians. 

Germain made contributions in the areas of elasticity theory, differential geometry, and number theory. A Germain prime number and her additional work on Fermat's Last Theorem enhanced the exploration of the subject for hundreds of years.

I became aware of Sophie Germain through the movie, Proof. I wonder if the gender bias that has been prevalent in the STEM occupations has improved. Our mathematics department consists of 9 teachers, 5 of which are female. I wonder about nonteaching occupations and the pay inequity that exists. I know of at least 10 female, former calculus students that are deeply involved in STEM occupations. I wonder how they are coping?