I am thankful that I have a High School level text along with this Uni level text because I learn myself new things in each edition of this textbook and that makes it rather difficult and unclear to students. So the High School text is always a primer of clarity. I am trying to make it all as simple as possible but when I run into something I just learned myself then I have these subsection wanderings.
I am now of the opinion that Calculus will enlighten us on the famous idea of Euclidean geometry in that it is the union of Elliptic geometry with that of Hyperbolic geometry. I believe calculus can demonstrate this:
Euclidean geometry = Elliptic geometry union Hyperbolic geometry
I believe that because most open curved lines are due to "e" and all closed curved lines are due to pi.
In Euclidean geometry there are no curved lines but rather straightlines or collections of straightlines forming a straightlinecurve.
Not that is still confusing, because all of us were brainwashed and polluted of mind when we learned geometry. We all thought that curves exist naturally as well as straigthlines.
But it turns out that when you have a borderline between finite and infinite, that creates holes or gaps of empty space between successive points of space. That empty space denies the existence of curves in geometry.
Now we can all believe we see curves such as a circle, but in Nature if we zoom in far enough, we find out that even circles are a collection of tiny straightlinesegments.
/ | \
That is three straightline segments that looks like a curve to us, similar to ( the half parenthesis.
So here is a Calculus means of proving my old nemesis, the famous formula above.
So now let us say we had a circle of radius 1 drawn in Cartesian Coordinate System and we sliced a piece of that circle that looks like this ( and we graphed that as a function in 1st quadrant. Now it really is not a curved line, this ( but is rather a collection of tiny line segments of length about 1*10^-603 or a bit larger. When you are at such a tiny length if you put two of them together, you would think you had a perfect circle, but instead you have a polygon of straightline segments. And the number pi is still relevant and important, but now we construct another curved line that is concave inwards ) instead of concave outwards (. Remember, pi governs this concave outward curve of ( for a circle is a closed line, but a hyperbola is a open line of Hyperbolic geometry and not governed by pi, but governed by "e".
So now, I want the student or reader to draw a circle on a graph paper, and then alongside the circle, intersecting the circle at one point I want drawn a logarithmic spiral. And I ask the question of whether we can make the log spiral curve of ) match the circle curve of (, so that the two placed together forms a )( and where one cancels the other, leaving us with |.
So that in my previous posts, when I said "e" was the pi of Hyperbolic geometry and pi is the engine that makes Elliptic geometry work, while "e" is the engine that makes Hyperbolic geometry work.
So to my delight, my old nemesis of the famous formula above is conquered because we can prove that for every concave outward curve of Euclidean geometry governed by pi, such as the trigonometry functions, we can construct a different curve of the logarithmic spiral, governed by "e", so that the one straightlinecurve cancels the other, yielding a straightline.
Now this is exciting in another frame of reference, that geometry really does not have 3 separate geometries but that all geometries are Euclidean and Euclidean can be broken into 2 duals of Elliptic versus Hyperbolic.
So in Old Math, they had the silly notion that geometry had three different and separate geometries, whereas in New Math, geometry is just one overall geometry but it can be made into two duals of Elliptic versus Hyperbolic.
So I am thankful to Calculus, that I have begun to conquer my old nemesis.
More than 90 percent of AP's posts are missing in the Google ?newsgroups author search archive from May 2012 to May 2013. Drexel ?University's Math Forum has done a far better job and many of those ?missing Google posts can be seen here: