Readin, Writin, and Rithmetic

Today is the birthday of Henry White born 1861, Cazenovia, New York.  White worked on invariant theory, the geometry of curves and surfaces, algebraic curves and twisted curves.  Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds.

 

Above is a triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.

At the end of the school year my Advanced Algebra students and I had a discussion on what we were going to do during summer break.  I told them that I would be reading several books, writing on my blog, and doing mathematics.  After I listed the topics I would be reading: science fiction, fantasy, history, and mathematics, they were interested in my blog.  I explained that I love to write and I needed to write more to be a better writer.  Some of my students scoffed at the mention that I do math for fun.  I asked them would they be surprised if an English teacher wrote.  English teachers are not reading teachers, however, reading is an inherent part of Language Arts instruction and many English teachers read.  Industrial Technology teachers may build furniture.  Why shouldn't mathematics teachers do math?  I believe everyone should be engage in the three R's.  The mathematics can start with simple logic puzzles, a sudoku, or kenken.  Expand you mind, read a little math.  I suggest starting with Flatland by Edwin Abbott or Zero: The Biography of A Dangerous Idea by Charles Seife.  Read the biography of Alan Turing.  My brain as well as my body needs exercise.

Father's Day and Prime Numbers

The Fundamental Theorem of Arithmetic states that any natural number (except for 1) can be expressed as the product of primes and that expression is unique.  The proof of this theorem is one of my favorites.  The Fundamental Theorem of Arithmetic consists of two statements.:  1)  The factorization of any natural number as a product of primes and 2)  The uniqueness of the factorization.  The following website has an excellent proof of the theorem.
http://www.algebra.com/algebra/homework/divisibility/Proof-of-Fundamental-Theorem-of-Arithmetic.lesson

Any natural number greater than one has its own code and that code is unique.  I view this theorem as the declaration that there is a DNA so to speak in this set of numbers.  When a natural number is multiplied by another natural number, the new natural number has its own unique factorization but its prime factors are inherent to its "parents".

Today is Father's Day and I recently had a number of conversations with my dad.  My dad is a good conversationalist but I have found I need to listen carefully to gather tidbits of his past.  We were talking about a book that he enjoyed.  The book had a sad ending and both my parents admitted that they had wept at the end.  My father also confessed that he wept a number of times during the book and attributed to his strong sentimentality towards life.  This is the man who told me as a child "men don't cry."  I am also extremely sentimental and will weep at the strangest times.  As a child and a young man, my father was stoic and tough but when he became a grandfather a gentler side emerged.  That side was probably always there.

During his conversation, he showed me a small book that listed the titles of the books he had been listening to.  I flipped the pages of that small book past the titles and noticed that years were listed with small brief entries - Tuesday, April 5, "It rained"; Thursday, May 6, "Mowed lawn."  These were all entries in the year 1979 . . . Friday, March 3, "Chuck came home from Bemidji"; Friday, March 31, "Bill left for school"; Sunday, February 4, "Bruce moved to Rochester."  The year 1979 was the only year in which brief quips were entered... Monday, February 12, "Mom left for San Fransisco"; Tuesday, February 13, "Shoveled snow."  When inquired about the entries, he just stated matter-of-factly, "Well, your Aunt Lucille kept a book."  WHAT?!  Just then I realized that 1979 was the when he became an empty nester.

This is my empty nest year and I started a blog.  I am, we are a complex mixture of nature and nurture.  I am, my siblings are a product of DNA and an environment created by 2 (not 1) individuals.  Do I have all the attributes my father wished for me? Probably not for I am also my mother's son and I have my own uniqueness.

In my sons, I see a strong sense of independence, various degrees of sentimentality, a great deal of passion, and a sense of adventure.  Not all these are my attributes and the environment in which they grew was in flux as 2 (not 1) learned how to be parents.  We are still learning but we now learn from a distance and how parenting responsibilities have changed.

Prime factorization . . . sons, daughters, mothers, fathers . . . Happy Father's Day :-)

3

My favorite number is 3.  I have some personal facts about 3.  I was born on the third day of the month.  I am the third child in my family.  My first name starts with the letter C which is the third letter of the alphabet.  My last name starts with the letter K which takes three line segments to make.  I have three children.

Here are a some other facts about the number 3.

Three is the first odd prime number.
It is both the first Fermat prime (2²ⁿ + 1) and the first Mersenne prime (2ⁿ − 1), the only number that is both.

It is the first lucky prime.  A lucky number is a natural number in a set which is generated by a "sieve".  Begin with a list of integers starting with 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,  Every second number (all even numbers) is eliminated, leaving only the odd integers:  1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,  The second term in this sequence is 3. Every third number which remains in the list is eliminated: 1, 3, 7, 9, 13, 15, 19, 21, 25,
The next surviving number is now 7, so every seventh number that remains is eliminated: 1, 3, 7, 9, 13, 15, 21, 25, When this procedure has been carried out completely, the survivors are the lucky numbers: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, ... .
A lucky prime is a lucky number that is prime.  The first few are 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193.

It is the only prime triangular number.


Three is the only prime which is one less than a perfect square. Any other number which is n² − 1 for some integer n is not prime, since it is (n − 1)(n + 1).

Three non-collinear points determine a plane and a circle.

Gauss proved that every integer is the sum of at most 3 triangular numbers.  An example would be 27 = 21 + 6.

Most elbows consist of three bones, the only joint in the human body where three articulations are surrounded by one capsule.

Earth is the third planet in its local Solar System.

Atoms consist of three constituents: protons, neutrons, and electrons.

We perceive our universe to have three spatial dimensions.

Many world religions contain triple deities or concepts of trinity, including: the Christian Holy Trinity, the Hindu Trimurti, the Hindu Tridevi, the Three Jewels of Buddhism, the Three Pure Ones of Taoism, and the Triple Goddess of Wicca.

 In Christianity, three people (including Jesus) were crucified at the Crucifixion.  Jesus laid dead in a tomb for three days before his resurrection.  The threefold office of Christ is a Christian doctrine that Christ performs the functions of prophet, priest, and king.

In Islam, during wudhu, the hands, arms, face and feet are each washed three times.  According to the prophet Muhammad, there are three holy cities of Islam (to which pilgrimage should be made): Mecca, Medina, and Jerusalem.

Common Sense

"Common sense is the collection of prejudices acquired by age eighteen"  Albert Einstein

“Common sense is not so common.”  Voltaire, A Pocket Philosophical Dictionary

“Nowadays most people die of a sort of creeping common sense, and discover when it is too late that the only things one never regrets are one's mistakes.” Oscar Wilde, The Picture of Dorian Gray

“The three great essentials to achieve anything worthwhile are, first, hard work; second, stick-to-itiveness; third, common sense.” Thomas A. Edison 

“It is the obvious which is so difficult to see most of the time. People say 'It's as plain as the nose on your face.' But how much of the nose on your face can you see, unless someone holds a mirror up to you?” Isaac Asimov, I, Robot

“Common sense is the most widely shared commodity in the world, for every man is convinced that he is well supplied with it.” René Descartes  

“A long habit of not thinking a thing wrong, gives it a superficial appearance of being right, and raises at first a formidable outcry in defense of custom. But the tumult soon subsides. Time makes more converts than reason.”  Thomas Paine, Common Sense

“Common sense ain't common.” Will Rogers

“Common sense is seeing things as they are; and doing things as they ought to be.” Harriet Beecher Stowe

 “The cradle rocks above an abyss, and common sense tells us that our existence is but a brief crack of light between two eternities of darkness.” Vladimir Nabokov

Common sense is defined by Merriam-Webster as, "sound and prudent judgment based on a simple perception of the situation or facts."    The Cambridge Dictionary defines it as, "the basic level of practical knowledge and judgment that we all need to help us live in a reasonable and safe way".

I don't believe that common sense exists.  I tend to agree with Einstein's and Roger's definitions.  I find comments such as "He may have book sense but he doesn't have common sense" places the recipients of those comments into defined boxes rather attempting to understanding the complexities that exists in all of us.

Common sense keeps us from asking what if, challenging the status quo, and restricts our imagination.  My case in point is the parallel postulate.  The parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements.  It states that, in two-dimensional geometry:  If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.  Probably the best known equivalent of Euclid's parallel postulate is Playfair's axiom, named after the Scotsman John Playfair, which states:  At most one line can be drawn through any point not on a given line parallel to the given line in a plane.

Two mathematicians, Nikolai Lobachevsky and János Bolyai, created two geometries, hyperbolic and elliptic, by challenging the notion of the parallel postulate.  The discovery of these alternative geometries that may correspond to the structure of the universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world.  

 

We build, construct, and teach an Euclidean world and in a local view that is true.  A global view does have longitudinal lines intersecting the equator at 90° angles but intersecting at the poles forming triangles whose interior angles exceed the sum of 180°.  An universe view may point to a hyperbolic geometry.

An attorney friend mentions me in most of his closing arguments as the disbeliever of common sense.  I think he pleads the jurors to apply theirs.  A colleague commented on her shock that the parallel lines she was taught do not exist on our globe.  Perhaps common sense is not so common nor should it be.

 

The Problem

In April of 2011, I read an article in the Mathematics Magazine published by the Mathematical Association of America titled Ellipse to Hyperbola: “With This String I Thee Wed” by Tom M. Apostol and Mamikon A. Mnatsakanian pp. 83–97.  In this article the authors introduce a method of tracing an ellipse and a hyperbola using string and a straw.  An ellipse is the set of all points in a plane whose distances to two fixed points (the foci) is a constant sum.  A hyperbola is the set of points in a plane whose distances to two fixed points (the foci) is a constant difference. 

I replicated their process using a construction software called The Geometer's Sketchpad^®.  I created a circle and defined my foci as the points (0, 0) and (1, 0).  A point E is a point that traverses the circle.  A line l is constructed passing through point E and one of the foci (0, 0).  A segment is constructed connecting point E and the remaining focus (1, 0).  A perpendicular bisector of that segment is formed.  The intersection of that perpendicular bisector and line l is point F.  As point E traverses the circle, point F traces an ellipse.

The proof that point F forms an ellipse is as follows:  AE is the length of the radius which in this example is 2.  A segment FB can be formed creating triangle BDF.  Angles BDF and EDF are both right angles because of the perpendicular bisector and for the same reason segments BD and ED are the same length.  Since triangles BDF and EDF share segment FD, the two triangles are congruent thus segments EF and BF are congruent.  So the sum of the lengths of AF and BF will always equal to the length of the radius and by definition of an ellipse that is what is formed by point F as E traverses the circle.

If one of the foci is moved outside the circle, a hyperbola is formed.  In the following diagram, I moved the focus (1, 0) to (3, 0) to illustrate that fact.




I am fascinated by the traces that are formed as E traverses a pathway and I changed E's pathway to create the following graphs.



 

I believe to do an analysis of these graphs I need to have an algebraic model for this construction.  Given a pathway, P(x), a traversing point on P(x), two foci (a, b) and (c, d), a line l passing through (a, b) and the traversing point on P(x), and a perpendicular bisector of the segment with endpoints (c, d) and the traversing point, determine the intersection of line l and the perpendicular bisector.

I will share my work thus far . . . later. :-)