File:HypComp3.png
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[edit]DescriptionHypComp3.png |
English: Images of unit circle under 3 maps Polski: obraz koła jednostkowego po zastosowaniu 3 funkcji |
Date | 11 October 2008 (upload date) |
Source | Own work by uploader in Maxima and Gnuplot with help of many people ( see references ) |
Author | Adam majewski |
Long description
[edit]conformal mapping from : the circle with center=0 and radius=1
given by equation :
where :
onto boundary of 3 period 3 hyperbolic components of Mandelbrot set[1] is given by 3 equations :
code of functions gamma by Robert P. Munafo[2].
These functions are computed in Maxima from boundary equation [3]
(%i3) b3:c^3+2*c^2+(1-P)*c+(P-1)^2=0$
(%i4) solve(b3,c);
(%o4) [c=(-(sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
(((sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P- 1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3,c=((sqrt(3)*%i)/2-1/2)*
(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
((-(sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-1)*sqrt(27*P^2-22*P+23)) /(6*sqrt(3))-
(27*P^2-36*P+25)/54)^(1/3))-2/3,c=(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
(3*P+1)/(9*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))
Purpose of this file
[edit]To show that every functions maps one half of one component and one half of other component.
Discussion
[edit]Similar problem is in Mathematica code[4]
It is possible to solve it another way : one function for one component[5]
Maxima source code
[edit]NmbrOfPnts:400; /* ------------------------ definitions -------------------------*/ /* code of functions gamma by Robert P. Munafo */ /* conformal maps from circle to period 3 components */ gamma3a(P):=(-(sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+ (((sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3; g3a(w):=gamma3a(w/8); gamma3b(P):=((sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+ ((-(sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P- 1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3; g3b(w):=gamma3b(w/8); gamma3c(P):=(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+(3*P+1)/(9*(((P-1)*sqrt(27*P^2-22*P+23))/ (6*sqrt(3))- (27*P^2-36*P+25) /54)^(1/3))-2/3; g3c(w):=gamma3c(w/8); /* exponential for of complex number with angle in turns */ l(t,r):=r*%e^(%i*t*2*%pi); /* ------------------------- */ GiveIntRayOfCircleL(angle,NmbrOfPnts):= makelist(rectform(ev(l(angle,j/NmbrOfPnts), numer)),j, 0,NmbrOfPnts); /* ---------------------- */ GiveUnitCircleL(NmbrOfPnts):= makelist(rectform(ev(l(j/NmbrOfPnts,1), numer)),j, 0,NmbrOfPnts); /* ------------------ computes points of circle and ray ------------------*/ circleList:GiveUnitCircleL(100); /* ------------------- conformal mapping from circle to 3 period 3 components ----------------*/ c3aList:map(g3a, circleList); c3bList:map(g3b, circleList); c3cList:map(g3c, circleList); /* ---------- ------------ drawing -------------------------*/ load(draw); /* Mario Rodríguez Riotorto http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html archive copy at the Wayback Machine */ draw(file_name = "hypComp3", pic_width=1000, pic_height= 500, terminal = 'png, columns = 2, gr2d(title = " unit circle D={w:abs(w)=1 } in standard plane ", points_joined =true, color = red, point_type = 0, points(map(realpart,circleList),map(imagpart,circleList)) ), gr2d(title = "3 images of circle under 3 maps gamma3(w) in parameter plane", points_joined =false, color = blue, point_type = 0, points(map(realpart,c3aList),map(imagpart,c3aList)), color = green, points(map(realpart,c3bList),map(imagpart,c3bList)), color = magenta, points(map(realpart,c3cList),map(imagpart,c3cList))) );
References
[edit]- ↑ Boundaries of 53 hyperbolic components of Mandelbrot set for periods 1-6 made with polynomial maps from the unit circle
- ↑ MuEncy by Robert P. Munafo
- ↑ Computing boundary of hyperbolic components from boundary equations
- ↑ Walter Hannah thesis, see page 11
- ↑ A Parameterization of the Period 3 Hyperbolic Components of the Mandelbrot Set Dante Giarrusso; Yuval Fisher Proceedings of the American Mathematical Society, Vol. 123, No. 12. (Dec., 1995), pp. 3731-3737
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current | 08:49, 11 October 2008 | 1,000 × 500 (7 KB) | Soul windsurfer (talk | contribs) | {{Information |Description= changed description of images |Source= |Date=11.10.2008 |Author= |Permission= |other_versions= }} | |
08:33, 11 October 2008 | 1,000 × 500 (7 KB) | Soul windsurfer (talk | contribs) | {{Information |Description={{en|1=Images of unit circle under 3 maps}} |Source=Own work by uploader |Author=Adam majewski |Date=11.10.2008 |Permission= |other_versions= }} <!--{{ImageUpload|full}}--> |
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