O Tannebaum

Today is the birthday of mathematician, Mary Cartwright, born 1900 in Aynho, Northamptonshire, England. Cartwright was the first woman mathematician to be elected to the British Royal Society.

Today's quote is from The Littlest Christmas Tree: A Tale of Growing and Becoming by Janie Jasin. "Thank you, Dear Creator, for Life. Thank you for Dreams. Thank you for Ideas and Thoughts and Feelings. Most of all, thank you for choosing me to grow - just for today - and to know the wonder of Your World and its many Possibilities."

A tradition that has been established in our family is the assembly and decorating of our artificial Christmas tree on the day after Thanksgiving. Early in our marriage when my wife and I were hanging the lights on the tree, I noticed I struggled with the project. My placement of the lights as we circumnavigated the string about the tree just wasn't quite right. I had too many lights in the back of the tree or the plugin wasn't situated in accordance to our electrical outlet. As my children grew older, I happily abdicated my role of string assistant to more capable and eager hands.

This year my eldest son, Jacob, asserted his birthright, erected the tree, and hung the lights. I assisted. I had been on administrative leave for about 25 years and I thought now is the time for redemption. I thought, "I can do this!" I also searched for another moment to atone for those instances that I was less than the parent I should have been. Jake is my practice child. What I mean is that I practiced being a parent on him.  There are several classes of students that graduated from Lake City Lincoln High School that I was allowed to practice on prior to my first teaching job. I did not have that convenience with Jake. I made mistakes. Mistakes that I still regret to this day. Our relationship has become more than father and son. We have become friends. He has become my confidant. I value his opinion. I look forward to our time together. I look forward to moments of redemption.

He had progressed to the finality of the tree construction when I began this annual event. Quickly a philosophical discussion emerged between my wife and son as to where to start the string of lights. 

"At the bottom," my wife insisted, "closer to the power source. The plug should be on the bottom right."

"At the top," countered my son, "next to the star. It can guide me to that source."

 My wife relented. Sometimes perseverance can appear as stubbornness.  Jacob and I weaved the lights counterclockwise about the tree. This weave appeared as a dance - a dance of a couple in which one is trying to lead and the other, not confidant in the placement the of step or the rhythm, is trying to follow. After a few rotations, a pirouette, and a few string entanglements, Jacob stopped me. "Dad, you're hanging them too low and you need to wind them around the tree so the strings rise and fall. They should make a wave", he instructed. I grew silent. "Dad, are you ok? Are you upset?", he inquired.

"No, I'm ok", I responded. He paused and placed the bubbler in the center of the tree. He regrouped, "Are you thinking about math?"

I smiled slyly. He sighed.

When we were done, he began hanging his ornaments. The ornaments that he was carefully placing on the tree marked his life. My wife and I choose ornaments for each member of the family that signify an event that we view as significant for that year. These ornaments have accumulated for the past 29 years.They can represent moments as exhilarating as birth and marriage or as humbling as rowing a boat across a lake after the motor stopped running.  Each member has their spot on the tree where their bauble is placed. These assortment of symbols, delicately removed from their containers and carefully hung above and below the periodic function of string of lights that winds itself, top to bottom or bottom to top, (depending on your view) around the tree.

His phrase "rise and fall of the strings" resonated in my mind. I have never thought about the mathematics in a Christmas tree. Years ago, I taught seventh graders. During this time of year, I taught prime factorization I would croak the tune "O Tannebaum" and insert the words "O Factor tree, O Factor tree, How beautiful are your factors!". The tree is obviously conical in shape and and the Greeks had described parabola, circle, ellipse, and hyperbola as planes intersecting a cone, but those words, "rise and fall" kept haunting me.

I searched for mathematics in the seemingly infinite Google search engine and found Treegonometry, a series of mathematical formulas that would compute the "perfect" amount of trimmings for ideal Christmas tree. These formulas were derived by students at the University of Sheffield in the United Kingdom. The formulas are as follows:

1) Number of ornaments = Square Root of (37) ÷ 20 x (height of tree in cm)

2) Length of tinsel (cm) = 13 x pi ÷ 8 x (height of tree in cm)

3) Length of lights (cm) = pi x (height of tree in cm)

4) Height of star/angel (cm) = (height of tree in cm) ÷ 10

A six foot tree would 37 ornaments, 919 cm of tinsel, 565 cm of lights, and a star that is 18 cm tall.

I realized in my blog, The Circle of Geometry, I had described the helix as my best symbol of life and there was helix of lights that wound about the tree but this helix, as life, is more complex.

Above, I have provided a top view of the lights and a three dimensional view of the lights traversing the tree. The top view is a spiral. The second view is a conical compression spring often referred to as a tapered spring. I have seen these springs in faucets, holding in the gaskets. The advantage of these types of springs, is the stability they provide when placed on structures such as battery contacts or push buttons. Does the spring start at the bottom or the top?

In my son's description of "rise and fall" and "make a wave", he was expressing the sinusoidal curve which can be shown by stretching a spring.

Unlike the helix, I described in a previous blog, this tapered spring is a better symbol of life, with one addition. I would include "the rise and fall" of the spring as it rotates about its axis. The "ups and downs" that we encounter each day add to the overall wisdom we gain as each year passes.

The tapering of the spring represents the acceleration of time. When my children were born, my wife and I were excited about the brief celebrations of solid food, walking, no diapers, and eventually, no daycare. The time before each of those events were anticipated and calculated.

As a child, the days before the holiday break and ultimately, Christmas, could be excruciating long. I marked the days on my calendar and again, those days were anticipated and calculated.

My first year of teaching was, in my mind, the longest of my career. I made the mistake of marking each day of school and highlighted my last day of the year. Calculated? Yes. Anticipated? Very much so.

Now, I don't measure the day. I do not look forward to the last day of school. I don't calculate and I avoid anticipation.

My parents are in the twilight of their lives. They have often remarked about how time has quickly passed. As the tapered spring reaches the apex, time is rapidly revolving.

I believe the wavy tapered spring is life. As each year turns, and each experience raises the rotation, the circumference of the circle becomes smaller. Time shortens. Opportunities to mend torn moments pass by. Children become adults. The time spent with them becomes rarer and more treasured.

The Dark Side of Math: A PHS Gopher Tale

Today is the birthday of Lene Hau born 1959 in Vejle, Denmark. Hau is famous for her experiments to slow down light. Her team was able initially slow light to 17 meters/second and, in 2001, were able to stop light to one-thousandth of a second.

The first quote of the day is from Steven Wright. He quipped, "It doesn't matter what the temperature of the room is, it's always room temperature."

The second quote is from Brian Holland, Lamont Dozier, and Eddie Holland.

"It's like a heat wave

Burning my heart (it's like a heat wave)

I can't keep from crying (it's like a heat wave)

It's tearing me apart"

When I was in high school, I was a self absorbed student that waffled between nonconformity and the compliance of peer expectation. I liked mathematics and was a motivated student in that subject, but I was rather fickle in other subject areas--my motivation was dependent on the connection I formed with my teachers.  I made strong connections with my Language Arts teachers but not so much with my science teachers.

I also had a fear that I was underprepared for the rigors of college and that fear drove me to take an array of academic classes. One of those classes, German, I studied my junior year. I was really not aware of what I was getting my self into and very quickly, I was in a conundrum. Learning a foreign language was, well, foreign to me. I was already struggling with the verbal portion of the English language. Why did I think that speaking German would be any different? In addition, I had to memorize. I don't learn by rote. I develop connections that lead from one point to another. Some people believe these connections are linear, but I view them as a web. When I encounter a new idea it vibrates a single string and the entire web pulsates. That pulsation allows me to find the connections to other concepts. This process wasn't working for me in German. I struggled with Der, Die, and Das. I was briefly gratified to find out that these articles are assigned to the gender or neutrality of the subject but was as quickly mortified to realize there are a seemingly endless assortment of exceptions to the rule.

In addition, I was required to join German Club.  German Club was more than a social club for the students. The underlying purpose of the club was to market world language and a student exchange program, AFS. I am not typically a joiner and I was a reluctant participant. This reluctance was transformed to paralyzing dread when I found out I was a founding member of the Schottische Dancers. Our first public performance was surprisingly witnessed by a large audience that included my younger brothers and girlfriend.

This combination of attitude, self prophesying  experiences, and personal characteristics that included cantankerousness and defiance contributed to a student that simply did the minimum requirements. I was complacent but had an underlying frustration that was waiting for its moment to surface. That moment came when the class discussion moved to the measure of temperature.

 Temperature was not new to me. In my mathematics classes, students were already taught how to convert Celsius to Fahrenheit and vice versa. My mathematical education also include that each resulting formula was an inverse of each other, a reflection over the line y = x, and an intersection at -40°. I ask you to examine the table of temperatures.

My German teacher was demonstrating how to convert Celsius to Fahrenheit using the formula, F = 9/5C + 32. She was selecting nice, positive values that were divisible by 5, 15°C = 59°F, 10°C = 50°F, and 35°C = 98°F (body temperature). She was on a roll but then a mistake. "As you notice students", the youthful teacher quipped in a confident, almost "cocky" air, "the Celsius temperature is always a lower number than Fahrenheit." My head perked up. I raised my hand. She smiled in the achievement of drawing me in, yet another mistake. "Could you convert -40°C for me", I inquired. "Negative numbers?", she murmured, somewhat shaken. "I can do this", she responded energetically. "Uh, the answer is ... -40°F that's not possible, I must of made mistake. Give me another one." Eagerly, I made my play in the game. "Oh, how about -50°?" I inquired, again. "-50°C is ... oh, my, a -58°F, how can that be?" she responded with sweat on her brow and a quivering lip. The blinds rattled  ominously above the closed windows. I leaned back, closed my eyes, and let the piranha feast.

Our German teacher wasn't in class until the following Monday. I don't know if her absence was the result of the failed lesson plan on temperature, the loss of credibility with our class, or a preplanned absence. On Monday, she retaught the lesson and explained how negative numbers had impacted her calculations. She did admit she needed a brief lesson from a resident math teacher. I am sure that with each preceding year, she used this experience as a building block to advance her teaching skill.

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?

France: Day 1 - Descartes

Today is the birthday of Charles Reyneau born 16546 in Brissac, Maine-et-Loire, France. Reyneau published one of the first influential textbooks on the newly invented calculus.

Today's quote is by Rene Descartes entitled, Cogito ergo sum. "... Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry, I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts (presentations) which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all objects (presentations) that had ever entered into my mind when awake, had in them no more truth than illusions of my dreams. But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be somewhat; and as I observed that this truth; I think, therefore I am, was so certain and of such evidence that no ground of doubt, however extravagant, could be alleged by the skeptics capable of shaking it, I counted that I might, without scruple, accept it as the first principle of the philosophy of which I was in search."

As witnessed when I watch a recent news segment, I am on "assignment" which means I am on vacation, and will be writing about my five favorite French mathematicians; Rene Descartes, Sophie Germain,  Pierre de Fermat, Emilie du Chatelet, and Cedric Villani.

Rene Descartes

Descartes was born on March 31, 1596 in La Haye, France. As a student, he studied rhetoric, logic, and the "mathematical arts". "Mathematical arts" included mathematics, music, astronomy, metaphysics, natural philosophy, and ethics. He attended the University of Poitiers and obtained a law degree.

After graduation, he studied medicine and theology. He traveled and joined the army. During his stint in the army, he met scientist and philosopher, Isaac Beeckman. Beeckman convinced Descartes that he should apply mathematics and logic to understand the natural world.

Descartes is considered the father of modern philosophy. He believed that all truths are ultimately linked and that the rational structure of mathematics and science could be used to uncover the meaning of the natural world.

Descartes introduced Cartesian geometry which is a fusion of geometry and algebra. I call Cartesian geometry, coordinate geometry. I love coordinate geometry and I believe its application allowed me to have an easy transition in to trigonometry and polar coordinates. Descartes also developed laws of refraction and formed a practical understanding of rainbows. Pope Alexander VII placed Descartes' writings on the Index of Prohibited Works.

Deriving While Driving

Today is the birthday of Max Zorn born in 1906 in Krefeld, Germany. Zorn is responsible for Zorn's lemma which states that if a partial ordered set exists and any subset of that set has an upper bound then the original set has a maximum element.

The quote of the day is by Srinivasa Ramanujan. Replying to G.H Hardy's suggestion that the number of a taxi (1729) was "dull", "No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being (1 + 1728) and (729 and 1000)."

A week ago, our school held its commencement for the class of 2015. During the week prior to graduation, I spent a great deal of time with the seniors that were in my class and in the organizations that I advised. Our conversations were reflective on their growth and their excited and nervous anticipation of their upcoming educational journeys. Each class attempts to leave a legacy their senior year and some classes do have a more resilient legacy to the corrosiveness of time.

Each numerical year seems to have its own identity. My high school graduation year, 1976, was the bicentennial year. The year 1984, was wrapped in the Orwellian concept of Big Brother. 1999 was made famous by the singer, Prince. The year 2000 was the millennial year. 2008 had the 8 rotated so that it became the infinity symbol. Members of the class of 2012 asked the question, "Don't you want to be 1  2?" 2013 ends in 3 and is also divisible by 3! 2015 was the year of π because on March 14 at 9:26 am, the first eight digits of that irrational number occurred. 

I was contemplating these numerical wonderments at a stoplight on my way to school the day after the commencement ceremony and was drawn to the class of 2016. In my meandering thoughts, I jumped to the factorization of the 2016 and was shocked in my determination that 2016 has 36 factors. Factors are natural numbers that 2016 can be divided by without any remainder. The factors of 2016 are: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016. I started to wonder about the number of factors in each of the years that I have taught. The chart below depicts those graduation years and the number of their factors.

To determine the number of factors, I found the prime factorization of each year. The prime factorization of 2016 would be 2 x 2 x 2 x  2x 2 x 3 x 3 x 7 or in exponential form, 2^5 x 3^2 x 7^1. I used the exponents in the exponential form to find the total factors that exist in 2016. The total number factors can be found by using the following arithmetic: (5 + 1) x (2 + 1) x (1 + 1). Using my high school graduation year, 1976, I found that its prime factorization is 2^3 x 13^1 x 19^1. The exponents are 3, 1, and 1. The number of factors in 1976 is 16; [(3 +1) x (1 + 1) x (1 + 1)].

When I examined the list, I noticed there are five prime years, 1987, 1993, 1997, 2003, and 2011. I believe the next graduating class that will have exactly 36 factors will be the class of 3168. The prime factorization for 3168 is 2^5 x 3^2 x 11^1. The last graduating class to have 36 factors was the class of 1800, 2^3 x 3^2 x 5^2. I was not teaching in 1800 and am not planning on teaching in 3168!

2015 Calculus Presentations

Today is the birthday of James Hutton, born 1726, in Edinburgh, Scotland. Hutton is famous for his theory of the age of the Earth.

The math quote of the day is by Rene Descartes who said, "Perfect numbers like perfect men are very rare."

Every year the final for my calculus classes consists of finding a partner, agreeing on a topic of interest, finding the mathematics that exists in it, and creating a presentation on that topic. This year's classes formed twenty presentations.

I have linked the title of each presentation to its Youtube site or the reader may visit my Youtube site: Charles Kruger.

The topics are: Aquariums, Aviation, Car Audio, Computer Animation, Dance 1, Dance 2, Discus, Farm, Golf, Guitars, Hydroelectric Power, Luminance in Art, Natural Disasters, Photosynthesis, Rollerblading, Soccer, Swimming, Tangled, Theory of Relativity, and Track & Field.

I enjoy these presentations for many reasons. Students take ownership of their presentations. The students and I usually learn something new. The students enjoy finding the mathematics in their topics. The presentations allow the students to be creative. The presentations are a showcase or a capstone of what the students have learned. The students use skills they have learned in other classes.

Enjoy!

Math In A Pea Cart

Today is the birthday of Abraham de Moivre born 1667 in Vitry-le-Francois, Champagne, France. de Moivre was instrumental in the development of Analytical Geometry and Theoretical Probability.

The math quote of the day is by Theodore Von Karman who said, "The scientist describes what is, the engineer creates what never was."

When I was sixteen I started my employment with Lakeside Packing Company. This company was a cannery in my home town that processed peas and corn. I worked every summer from the age of sixteen to twenty-one. My hours of work were from 6 am to 6 pm, seven days a week for the entire summer. There were a few days during the "pea pack" that the combines could not enter into the fields due to a large amount of rainfall, but generally speaking, I worked every day. The greatest and perhaps only incentive for working at Lakeside was its pay structure. I was paid overtime, time and half, for each hour over 48 hours in a work week. On the rare occasion that the peas were canned with carrots (a commodity provided by the federal government), work after 40 hours was considered overtime. My base pay was the current minimum wage of $2.10/hour and with overtime pay the hourly wage was boosted to $3.15/hour. My gross weekly wage was: 12 hours x 7 = 84 hours; 48 x $2.10 = $100.80; 36 x $3.15 = $113.40; $100.80 + $113.40 = $214.20.

At Lakeside, I was allowed to go to the restroom when I needed to, take morning, lunch and afternoon breaks, and most importantly, socialize. What I wasn't allowed to do was think. My entry-level job was to push pea carts into lines that were sorted by pea's size and density (determined by the pea's ability to float in a brine). 

A pea cart was a trapezoidal prism on rusted casters whose momentum was often impeded by B6's, peas the size of #Triball 12 buckshot with the hardness of diamonds. As a side note, when I asked who ate these type of peas, I was told unsuspecting institutions such as schools and nursing homes. These carts were filled with peas from tubes extending from the ceiling. The tubes were the recipients of peas garnered from trucks that had dumped their cargo into washers. From the washers, the peas had made a journey to the attic of the factory where they were washed, sorted, and flung down the appropriate tubes to the carts below. On the image above, there would be a small door on the side opposite of the slanted side. Once the carts were filled, I would push the cart to an assigned line and another employee would roll the cart to a predetermined position, lift up the door, and feed the peas through the rectangular opening into a small, swirling, bubbling vortex of steaming water. This vortex would take the peas to the roof of the factory where the peas coursed their way through a series of pipes. This corkscrew of rides was called the blanching process that all vegetables went through prior to being canned.

Every day, 12 hours each day for about 3 months, I pushed a cart. When I went back to school, my mind was like undercooked oatmeal. Constructing a thought was like wading through wet cement, and as a result of this mental decay, my mathematical processes were slowed. Also, my vocabulary was, well... not creative. I had heard a particular word so often that its usage had to be considered a fundamental rite of passage in this work environment. I had heard this word used as a noun, a verb, an adjective, and an adverb in one sentence. With this one word, employees described their frustrations and their joys. This monolexemic sentence was not conducive in a family or academic setting. This poisonous mixture of mental atrophy and constricted diction reduced me to a zombie with a hunger for abstract thought.

My second year at Lakeside Packing Company, I was promoted to the job of emptying the peas from the carts into the boiling whirlpool of churning water. As much as I had hated last year's job, I hated this worse. The temperature was hot and humid and no one wanted to spend any time discussing teenage topics within my restricted work area.

I was concerned about my deteriorating mental state and I devised a plan to solve math problems. I wanted to know how many A3 peas were in my cart. I first measured the dimensions of my cart and the radii of randomly selected of peas. I averaged those radii and formed an estimate of the volume of an A3 pea.  My initial answer was computed by dividing the volume of the cart by the volume of the pea. I knew my quotient was higher than the actual number due to the empty space between the peas.

I devised a new plan based on average rates. My peas eventually made it to the "fillers". The "fillers" were machines that actually placed the peas into a can, sealed the cans, and sent the canned peas to the "cookers". "Cookers" were a holding location where the contents were "cooked" for an appropriate time and temperature. At the beginning of my shift, my "filler" had no peas. I emptied my cart and stopped. When my assigned "filler" received the peas, it started up but stopped again when it ran out of peas. I timed the start and stop interval of the "filler". I knew the number of cans per minute that the machine sealed. Quality control randomly selected five cans from the "filler" to examine their contents. I had them count the number of peas in each can and determined an average number of peas per can. My formula: Minutes x Cans/Minute x Peas/Can = Total Peas in a Cart. I was pleased with the results. Both methods resulted in close estimates with the latter being slightly lower than the former.

I can't remember how long this problem took me to solve. I don't think it took more than a couple of days. I was hoping for a problem that I could work on for the entirety of canning season. I do know I was sad when I completed it. I believe I worked on mental math problems the rest of the summer but those activities did not give me the same satisfaction. One of the reasons I enjoyed this problem is that I involved other workers. I had the people working on the fillers giving me information about their machine and taking the risk of stopping and starting their machine which was not a popular idea among the mechanics and our superiors. I also had to work in concert with the quality control and convince them of doing the added work of counting peas. Perhaps, the problem temporarily broke everyone from the constant boredom that encased their daily routine. Many people were curious on how many peas were in a cart during that brief moment in the summer of 1975. Yes, for a brief time, doing math made me cool :-)