Tuesday, November 18, 2008

The Pleasure of Finding Things Out

I left a previous post saying I have a gut feeling that this can be solved for general cases (It was proved for triangles and I extended it to cyclic quadrilaterals). Now I got it. The attached equation shows the relationship between the Area and Perimeter for any convex polygon
as a function of the number of its sides (denoted by n)

In one form it amounts to saying that a circle is the most democratic of all convex shapes (every point is equidistant from the center..wish that was true for the states of a country).  Or to put it another way to max out the area you need all of your sides to be equal. Nothing else will do to max it out (maybe you knew that in grade school,). This time I convinced myself of its veracity.

"In the course of my law reading I constantly came upon the word "demonstrate". I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof? 

I consulted Webster's Dictionary. They told of 'certain proof,' 'proof beyond the possibility of doubt'; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. 

At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."
                                                                        -Abe Lincoln

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