Local Linearity and Mark Twain

Today is the birthday of Christine Mary Hamill born 1923 in London, England.  Hamill specialized in group theory and finite geometry.


I recently spent some time in the United Kingdom and would like to focus on the topic of local linearity and how this concept impacts our lives.  Local linearity is the term given when we zoom in on the graph of a differentiable function.   The function will look like a straight line. In fact, the graph is not exactly a straight line when we zoom in; however, its deviation from straightness is so small that it can't be detected by the naked eye.  Here is a few graphs that hopefully will illustrate the concept.



The above graph is called a parabola and is curved.  A point that exists on this curve is (1, 4.5).  The next graphs will zoom in at that point.

 
 
 

 As you can see when I zoom in a number of times at this particular point the curve "straightens".  The distance surrounding my given point is small.  A person living on this point would view their surroundings as flat, without curve.  Much like the prairies of Minnesota, South and North Dakota, within hundreds of miles the surface looks flat although those states are part of curved surface of the Earth.  To see what this graph really is, the graph must be zoomed out to fully appreciate its complexity and curvature.

My youngest son recently has spent two and half months in the United Kingdom.  He took classes in an university setting and in a pub setting.  I believe both were excellent classrooms.  Laughter and debate sometimes can occur in a formal setting, most often they occur over a pint of beer.  My son and I know he has changed and those changes will become more pronounced as he steps back and reflects in his local setting.  He has experienced a global perspective.  He has zoomed out and seen the complexity and curvature of the human spirit.  He recently quoted Mark Twain.

"Travel is fatal to prejudice, bigotry, and narrow-mindedness, and many of our people need it sorely on these accounts. Broad, wholesome, charitable views of men and things cannot be acquired by vegetating in one little corner of the earth all one's lifetime."

"A Madman Dreams of Turing Machines"

Today is the birthday of Jesse Douglas born 1897 in New York, New York.  Douglas worked on geometry, group theory and the calculus of variations.

I am currently reading the book "A Madman Dreams of Turing Machines" by Janna Levin published by Anchor Books.  The book is a fictional account of Kurt Godel and Alan Turing, mathematicians that made large contributions in their fields.  Godel was a logician and proposed the Incompleteness Theorem and Turing broke the code of the Germans' Enigma Machine during World War II.  These men lived tortured lives that unfortunately concluded in tragic deaths.  I also found a documentary produced by the BBC that chronicles their lives and achievements.  The documentary is called "Dangerous Knowledge" and can be viewed for free at: http://topdocumentaryfilms.com/dangerous-knowledge/.

The novel has many threads that could be discussed but there is one passage that for the moment I wish to focus on.  Godel was a member of The Vienna Circle which was an association of philosophers gathered around the University of Vienna in 1922.  The author describes Godel's anticipation for the gathering: "While he often loses Monday easily and tries to find root in Tuesday, and although Wednesday is a mere link between nights, he always knows Thursday.  He likes to arrive early and choose the same place each time, a dark wooden chair near the wall, almost hidden behind the floral arm of an upholstered booth, not too close to the center but not too far out where it might become crowded, people pressing in to warm themselves against the heat of argument emanating from the core.  Comfortably still, with an undisturbed tepid coffee he never intends to drink, he listens to the debates, the ideas, and the laughter, like a man marooned on an island tuning in to a distant radio broadcast.  Proof that there are others out there.  Proof that he is not alone.  Proof."

"Proof that there are others out there.  Proof that he is not alone.  Proof."  I recall when I was attending classes for my master's degree, I had the same anticipation.  I miss the discussions of mathematics, the arguments, and the laughter.  There was a group of us that were a core, moving from class to class, and in various stages of completion within our degrees.  I must now admit that I use my classroom in an attempt to revitalize that feeling.  There are moments where I acquire the same satisfaction.  However, I often feel alone.

I attempt to have these discussions with friends but their attention is much to short.  As soon as I use terms such as fraction, ellipse, multiple, or factorization, terms I view as rudimentary, their concentration dissipates.  I don't believe I am the same type of listener.  If there are legal terms, medical terms, or any technical terms dealing with my friends' occupations that I don't understand, I build a framework in which a substantial discussion may ensue.

I need a mathematics community!  I do have enlightening discussions with another mathematics teacher but these meetings are too few and too far in between. 


"Proof that there are others out there.  Proof that I am not alone.  Proof."

Event Horizon

Today is the birthday of Rene' Sluze born 1622 in Visé, Spanish Netherlands (now Belgium).  Sluze is best known for the curves called the Pearls of Sluze.  Pearls of Sluze are curves formed by the equation: yⁿ = k(a - x)^px^m   where n, p and m are integers.  The particular curves drawn below have n = 4, k = 2, a = 4, p = 3, and m = 2.





In a recent conversation with one of my sons, he expounded on the concept of an event horizon.  I had not heard of the concept and Wikipedia defines it in the following manner:  "In general relativity, an event horizon is a boundary in space-time beyond which events cannot affect an outside observer. In layman's terms, it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case of an event horizon is that surrounding a black hole. Light emitted from beyond the horizon can never reach the outside observer. Likewise, any object approaching the horizon from the observer's side appears to slow down and never quite pass through the horizon. The traveling object, however, experiences no strange effects and does, in fact, pass through the horizon in a finite amount of proper time."

 
My son was commenting about his own event horizon.  I have been dwelling on his comment for quite some time.  His event horizon will change him.  When he leaves, I will have an image in my mind and heart of who he is.  Those images will be frozen in time until I see him again.  I have been thinking of other event horizons.  Some of the images I have of my sons, my mother, my father, and my siblings are frozen in time.  The shift of those images at times can be shocking.  I wonder what event horizons I have gone through, may be none as drastic as my son's.  I do see event horizons that are coming for my parents and I wish time could freeze.  I have thought about this blog more than any other blog I have composed.  One of my friends reached his event horizon and I have not seen him for quite some time.  I wonder how my image of him will shift.  I am true to my words . . . I am rambling.