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Summary

Description
English: 3D diagram of square root of real number ( positive and negative)
Polski: Wykres 3D kwadratowego pierwiastka z liczby rzeczywistej ( dodatniej i ujemnej)
Date
Source Own work
Author Adam majewski

Summary

This diagram shows function (red 3D curve):

Gray 2D lines are projections of 3D red curve on 2D planes.

One can see that red curve consist of 2 parabolas rotated with respect to themselves ( 90 degrees)

Software needed

The image was made using

Tested on versions:

  • wxMaxima 0.7.6
  • Maxima 5.16.3
  • Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
  • Gnuplot Version 4.2 patchlevel 3

After running below batch file image was created on the screen. It was manually rotated, copied to clipboard, pasted to Paint program. Some interrupted gray lines were added.

Discusion

Is it a explanation of period doubling ?

"It is an explanation of the shape of the curves in period doubling, but there is more behind period doubling. E.g., when the multiplier of an n-periodic point z is -1, then the equation for period 2*n has a multiple solution. " Wolf Jung

Maxima source code

 /* functions for computing  */ 
 give_beta(_c):= sqrt(_c) $  

 give_alfa(_c):= -sqrt(_c) $ 

 /* c */
 xMax:1;
 xMin:-1;

 /* re(s) */
 yMin:-2;
 yMax:2;

 /* im(s) */
 zMin:-3;
 zMax:3; 
 iXmax:1000;
 dx:(xMax-xMin)/iXmax;
 p1_alfa:[];
 p1_beta:[];
 re_c:[];
 im_c:[];
 re_im:[];

 for c:xMin step dx thru xMax do 
 (
 alfa:give_alfa(c),
 p1_alfa:cons([c,realpart(alfa),imagpart(alfa)],p1_alfa),
 re_c:cons([c,realpart(alfa),zMin],re_c),
 im_c:cons([c,-2,imagpart(alfa)],im_c),
 re_im:cons([-1,realpart(alfa),imagpart(alfa)],re_im),
 beta:give_beta(c),
 p1_beta:cons([c,realpart(beta),imagpart(beta)],p1_beta),
 re_c:cons([c,realpart(beta),zMin],re_c),
 im_c:cons([c,-2,imagpart(beta)],im_c),
 re_im:cons([-1,realpart(beta),imagpart(beta)],re_im)
 ); 

 load(draw);
 draw3d(
  user_preamble="set ticslevel 0;",
  terminal = screen,
  pic_height= iXmax,
  title       = "square roots s= sqrt(c) of real number c ",
  ylabel     = "Re(sqrt(c))",
  zlabel ="Im(sqrt(c))",
  xlabel     = "c",
  yrange = [yMin,yMax],
  zrange = [zMin,zMax],
  point_type    = filled_circle,
  /* */
  points_joined = false,
  color=gray,
  point_size    = 0.1,
  points(re_c),
  points(im_c),
  points(re_im),
  /* */
  point_size    = 0.2,
  points_joined = true,
  color  = red,
  points(p1_beta),
  points(p1_alfa),
  /* grid and tics */
  xtics      = {-2,-1,0,1,2}, 
  xtics_rotate = true,
  ytics      = {-2,-1,0,1,2},
  ztics		={-2,-1,0,1,2},
  grid       = true /* draw grid*/
 )$

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
w:en:Creative Commons
attribution share alike
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Under the following conditions:
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  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
You may select the license of your choice.

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11 July 2009

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Date/TimeThumbnailDimensionsUserComment
current18:05, 11 July 2009Thumbnail for version as of 18:05, 11 July 20091,072 × 621 (11 KB)Soul windsurfer{{Information |Description={{en|1=3D diagram of square root of real number ( positive and negative)}} {{pl|1=Wykres 3D kwadratowego pierwiastka z liczby rzeczywistej ( dodatniej i ujemnej)}} |Source=Own work by uploader |Author=[[User:Adam majewski|Adam m
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