Dottie Number

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Dottie Number

Postby dcb2011 » Sun May 01, 2016 10:28 pm UTC

A few years ago, via these forums, I learned about a Dottie Number Blog by a member (username: Zach 739085133).
If you do a search for "Dottie Number" or this username, you will find mention of the topic.

That blog shut down some time ago, which is unfortunately, because I found it to be well-written and with much information that I never saw anywhere else. For example, it included data from a Russian researcher who had written a Java program that computed the Dottie Number to a million (?) decimal places. And it included several approximations for the Dottie Number, as well as mention of the fact it was the solution to a particular chord of a circle. I am going by memory, which is why I am posting this question. Anybody know where I can find similar info? I notice "Zach 739085133" hasn't posted on these boards for several years. Anybody know what happened to him? I am particularly interested in the material he posted about the Dottie Number being the same as the solution for the chord of a circle.

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Re: Dottie Number

Postby >-) » Mon May 02, 2016 2:08 am UTC

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Re: Dottie Number

Postby PsiCubed » Mon May 02, 2016 3:14 pm UTC

I don't know about the blog, but computing the Dottie Number is very easy. Using Newton's method you simply do:

x0 = 1
xn+1 = (cos(xn)-xn)/(1+sin(xn))

Each iteration doubles the number of correct digits, so x20 would be correct to a million digits.

Writing a program that does the above isn't too difficult. You'd probably need a specific "Big Numbers' library to actually do the calculations, though.

(Or, if you want only 30 digits or so: Just open the Windows 8 Calculator App, make sure you're in radian mode, and stick your finger on the "O" key until the numbers stop changing. Took me about 11 seconds to get 0.73908513321516064165531208767387 :) )

Posts: 24
Joined: Thu Nov 03, 2011 10:19 pm UTC

Re: Dottie Number

Postby dcb2011 » Wed May 04, 2016 3:35 am UTC

>-) wrote:Here's an archive of the blog
. . .

Thanks, ">-)".

This is exactly what I am looking for.

I wonder why more of this blog's information hasn't been copied over to the Wikipedia or Mathworld sites; it has a lot of info that isn't found elsewhere. It would be a shame for it to be lost permanently if the archive site ever shuts down.

Anybody know the story about why Zach decided to pull the plug on this blog? (Perhaps somebody else could have kept it going.)

PsiCubed wrote:I don't know about the blog, but computing the Dottie Number is very easy.
. . .

Thanks, "PsiCubed".

I have sufficient digits of the Dottie Number to work with. It is the conditions under which it arises that interest me.

I am doing some informal work with Kepler's Equation of Elliptical Motion (equal area swept out in equal time.)

E - e sin(E) = M

One avenue of my exploration has been to define an approximating function for sin(E) in terms of cos(M) and sin(M). For example:

Say sin(E) = f[cos(M), sin(M)], where f is the approximating function.

To simplify this exploration, my first step was to consider the solution at the quarter period. At the quarter period, cos(M) = 0, so the approximating function is solely in terms of sin(M). Kepler's Equation becomes

E = M + e sin(E)

sin(E) = sin[M + e sin(E)]
sin(E) = sin(M) cos(e sin(E)) + cos(M) sin(e sin(E))
sin(E) = cos(e sin(E)), at t = T/4

For e = 0, sin(E) = 1
For e = 1, sin(E) = D, where D is the Dottie Number.

A graph for the range of e is included on my blog:

Keplers Equation: Eccentric Anomaly Values at the Quarter-period

At this point, I have created a couple functions that nicely bracket sin(E) at the quarter period, but I am now circling back to take up work and try to improve it. I had hoped one of the techniques Zach discusses in his blog would provide a hint for further direction in this work--hence my post.

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