| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
Here's a problem that I got from Stan. He got it from Andy Odlyzko. Define a sequence as follows: x[1]=1, x[n+1] = Sqrt( 3-x[n]^2 ) - x[n]. The problem is to plot the pairs (x[n],x[n+1]) for the first 10,000 or so values and to then explain the unusual result. You get a very unusual figure (something like a non-convex 17-gon with curved edges). Note: Stan was reciting this problem from memory and there is a slight possibility that the coefficients are wrong; he says he gave the correct coefficients to Peter Gilbert who can therefore issue a correction if the above statement is wrong.
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1108.1 | BEING::POSTPISCHIL | Always mount a scratch monkey. | Mon Aug 14 1989 10:55 | 5 | |
x[6] is complex. Does that indicate an error in the coefficients or
should only the real part be plotted?
-- edp
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| 1108.2 | Sorry, Don't Have the Date Handy | DRUMS::FEHSKENS | Mon Aug 14 1989 17:57 | 6 | |
Isn't this just a variant of the iteration that was presented in
the issue of Scientific American with the cover art on "Wallpaper
for the Mind"?
len.
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| 1108.3 | Corrected equations | 4GL::GILBERT | Ownership Obligates | Tue Aug 15 1989 13:46 | 5 |
x[0]=1, x[1]=2, x[n+1] = Sqrt( 3 + x[n]^2 ) - x[n]. | |||||
| 1108.4 | AITG::DERAMO | Daniel V. {AITG,ZFC}:: D'Eramo | Tue Aug 15 1989 15:03 | 4 | |
That looks like it will be rather boring, x[n] converges
rather rapidly to 1.
Dan
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| 1108.5 | Try these! | RDGENG::HALL | Wed Aug 16 1989 05:54 | 34 | |
Try
x[0]=1,
x[1]=2,
x[n+1] = Sqrt( 3 + x[n]^2 ) - x[n-1].
You can vary x[0] and x[1], and also the 3, to get various forms of
the basic pattern.
Another interesting recurrance is:
x[0]=1.1
x[1]=1
x[n+1] = C1*x[n] + C0*x[n-1]
If you plot x[n+1] against x[n] with C0 = -1.03, C1 = -1.99 the
resulting pattern is a rather attractive expanding star. And with
the same C0, but C1 = 0.1 you get a rotating, expanding box.
In fact, varying x[0], x[1], C0, C1 provides a large range of
interesting graphics. And still further variations if you plot x[n+a]
against x[n].
What intrigues me is that such simple expressions can produce such
surprising and complex patterns. I'd like to hear of any interesting
results!
Martin.
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| 1108.6 | KOBAL::GILBERT | Ownership Obligates | Wed Aug 16 1989 13:01 | 1 | |
Oops. Ignore the equations in .3; Martin has them correct in .5. | |||||