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."