Hal Lane wrote:
Paul, Re: Your "...somebody else might like playing with [the equation]."
I rendered the "Farting Monkey equation:"
f[x,c]=c*x*(x^2 - 2*x + 1)/(x^3 - x^2 - 2*x + 1) Domain:{x,y}={{-2,3},{-2.5,2.5}}
as...
FartMonk { ; Roger Bagula via Paul N. Lee z2=z*z zt2=z*2 z3=z2*z c=pixel z=pixel: a=c*z*(z2-zt2+1) b=z3-z2-zt2+1 z=a/b |z|<4 }
...and get a humdrum circle with a few minor non-circular areas. Have I missed something? Anyone have any ideas? <---<<
Well, I have not really had any time to explore this as yet, and may not be able to for several days, at least until my current situation settles down to allow the time. But you could take this up directly with Roger, who has been posting this to more than one forum: news:sci.fractals http://groups.yahoo.com/group/fractals/ Roger sent this yesterday evening: ----------------------------------------------------------------------- From: Roger Bagula <rlbagula@sbcglobal.net> Subject: derivation of the equation Date: Tue, 19 Sep 2006 17:49:23 -0700
Roger Bagula wrote:
I call it the "Farting Monkey equation" after a birthday greeting card: f[x,c]=c*x*(x^2 - 2*x + 1)/(x^3 - x^2 - 2*x + 1) Doman:{x,y}={{-2,3},{-2.5,2.5}}
It's a fairly serious derivation: Born -Von Karman equations: ( phono / sound in crystals) d^2u[n]/dt^2=(beta/ mu[i])*(u[n+1]-2*u[n]+u[n-1])
Fermi-Pasta-Ulam ( mostly Ulam type equations) ( Solitons in crystals) d^2u[n]/dt^2=(u[n+1]-2*u[n]+u[n-1])+alpha*((u[n+1]-u[n])^2+(u[n]-u[n-1)^2)
Elemination of the second derivative (d^2u[n]/dt^2): (-1/alpha)*(beta/mu(i]-1)*(u[n+1]-2*u[n]+u[n-1])=((u[n+1]-u[n])^2+(u[n]-u[n-1)^2)
substitution of u[n+1]=x*u[n] u[n+1]=u[n]/x to give: ( for example) u[n+1]-^2+u[n]^2+u[n-1]^2=u[n]^2*(1+x^2+1/x^2-2/x-2*x)
I get: u[n]=(-1/alpha)*(beta/mu(i]-1)*(x+1/x-2)/(1+x^2+1/x^2-2/x-2*x)
Which after some algebra gives: ( did this all by hand so it would be nice if somebody would check it!) f[x,c]=c*x*(x^2 - 2*x + 1)/(x^3 - x^2 - 2*x + 1)
Sincerely, P.N.L. ------------------------------------------------- http://home.att.net/~Paul.N.Lee/PNL_Fractals.html http://www.Nahee.com/Fractals/