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Re: [ontolog-forum] Beauty in mathematics -- detected in brain scans

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: John F Sowa <sowa@xxxxxxxxxxx>
Date: Fri, 21 Feb 2014 09:54:07 -0500
Message-id: <5307688F.4090305@xxxxxxxxxxx>
On 2/20/2014 7:57 PM, Rich Cooper wrote:
> Found a Wikipedia page on it:
> http://en.wikipedia.org/wiki/Euler%27s_identity    (01)

That's one of the best Wikipedia pages I've seen:  it explains
Euler's identity in a concise way that both a novice mathematician
and anybody who teaches (or has taught) math could enjoy.    (02)

The graphic animation on the upper right side of the page illustrates
the beauty of the abstract math.  It shows the result of repeated
multiplications for computing (1 + iπ/N)^N as N goes to infinity.    (03)

But a bit more explanation might make it clearer:    (04)

  N=1.  The value of (1 + iπ/1)^1 is the endpoint of a line that starts
        at (1, 0) and goes straight up to (1, 3.14159).    (05)

  N=2.  The value of (1 + iπ/2)^2 is (1 + iπ/2)*(1 + iπ/2).  That is
        represented by the endpoint of a bent line that goes from (1, 0)
        to (1, 1.57) and then to (-1.467, 3.14).    (06)

  N=3.  The value of (1 + iπ/3)^3 is represented by the endpoint of
        a bent line with three segments.    (07)

As N goes to infinity, the graph converges to a semicircle that
goes from (1, 0) to (-1, 0).    (08)

John    (09)

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