I thought it might be painful in recovering and extracting this information I wrote many years ago, but Google search in newsgroups makes it fast and easy, thanks Google!
World's finest explanation of the number "e"
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Archimedes Plutonium<
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Feb 10, 2015, 3:00:18 AM
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Newsgroups: sci.math
Date: Sun, 8 Feb 2015 02:11:23 -0800 (PST)
Subject: discovery of "e" is the pi of log spirals Re: Geometry meaning of "e"
and how we get pi+e as upper bound barrier
From: Archimedes Plutonium <
plutonium....@gmail.com>
Injection-Date: Sun, 08 Feb 2015 10:11:23 +0000
discovery of "e" is the pi of log spirals Re: Geometry meaning of "e" and how we get pi+e as upper bound barrier
Now when I went to college in 1968-1972 was the first time I learned of "e" and its special significance for rate of change. I learned it in Calculus class. But I never really felt fully comfortable with that understanding. Perhaps because it was not
geometrical enough for me. So that whenever I needed to explain "e", I had to go through all those hurdles of a algebra explanation. When explaining pi to a new student, the explanation is so super easy and the student immediately sees the connections of
circumference and diameter that it satisfies 100%, both student and teacher. But never the case with "e" with its long drawn out explaining and having to use silly things like bank account money.
So, why has no-one ever tried explaining "e" purely geometrical? Perhaps because no-one before was smart enough in math. Because I have done a search and found no-one has ever taught "e" as the "pi of the golden mean log spiral". If you take the
Rectangles of whirling squares of the Fibonacci sequence 1,1, 2, 3, 5, 8, 13, 21, 34, 55, etc and take a fixed amount of squares as a turn, we easily can get "e" to be the "pi" of a diameter.
So, why in the history of mathematics was no-one ever able to see that before I discovered it some years past? Why? It is not difficult. The reason I believe, is that no-one ever asked if "e" was more geometrical than actually algebraic? If you never ask
yourself the question-- where does "e" appear in pure geometry, then you never will discover that "e" is the pi of log spirals.
AP
Newsgroups: sci.math
Date: Sun, 8 Feb 2015 04:30:06 -0800 (PST)
Subject: "e" is to the golden mean log-spiral what pi is to a circle
From: Archimedes Plutonium <
plutonium....@gmail.com>
Injection-Date: Sun, 08 Feb 2015 12:30:06 +0000
"e" is to the golden mean log-spiral what pi is to a circle
Looking through a Google search it appears
Years 2010-2013 and on July15, 2013, I wrote this about "e":
Only it does not show the Fibonacci sequence of squares as shown in this picture from "Mathematics: A Human Endeavor" by Harold Jacobs, 1970.
Now in that Jacobs text on page 291 is a picture of a logarithmic spiral inside a rectangle of whirling squares. It is probably on the Internet somewhere, but I want the student or reader to photocopy that page of Jacobs and then get a piece of flexible
wire, and cut the wire of a length that matches the radius as shown on that page of length 55. Now, three and a tiny bit more of those 55 lengths should be as long as two of those arcs in the 55 square shown, (that is a semicircle). Now, however, using
that same wire track down the length of the wire that it takes to cover the 55 square and the 34 square and finally the 8 square, note that the 8 square has to be extended over to the right inside the 21 square.
What the student or reader should find is that it takes roughly 2.71 of the wire to cover that arc.
So here we learn the best meaning of the number "e". The number "e" is pi in hyperbolic geometry where circles are not closed but are open and spiraling outwards.
--- end quoting old post of mine ---
Back in 2013, I did not need precision and used a stiff wire to measure. But now I need more precision.
If we look at the squares of 8, 5, and 3:
For 8 square 1/4 x 3.1 x 16 = 12.4 arclength
For 5 square 1/4 x 3.1 x 10 = 7.75 arclength
For 3 square 1/4 x 3.1 x 6 = 4.65 arclength
Summing those arclengths gives 24.8
Taking the radius of the log-spiral in squares 8, 5, 3 as that of 8+1 for the center square gives a radius of 9. And 9 x 2.7 = 24.3. Not bad. So let us try larger squares of 55, 34, 21
For 55 square 1/4 x 3.1 x 110 = 85.25 arclength
For 34 square 1/4 x 3.1 x 68 = 52.7 arclength
For 21 square 1/4 x 3.1 x 42 = 32.5 arclength
Summing those results gives 170.5 and the radius for these three squares is 55+8 = 63.
Now 63x 2.7 = 170.1. So we see a convergence of three squares involving a center of the log-spiral.
So the idea here is that "e" is to the golden mean log spiral that pi is to a circle.
AP
Newsgroups: sci.math