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Difference between revisions of "User:Boris Tsirelson/sandbox2"

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<center><asy>
+
==Golden ratio==
int M=30;
 
real a = 0.07;
 
real a0 = 0.15;
 
real b = 0.02;
 
real c = 0.6;
 
real d = -0.2;
 
 
 
real x0 = -0.22;
 
real u = 0.2;
 
real v = 0.32;
 
 
 
draw ((-0.7,0)--(0.3,0),Arrow);
 
draw ((x0,-0.02)--(x0,1.2),Arrow);
 
 
 
label("$x$",(0.3,0),E);
 
label(rotate(90)*"$y$",(x0,1.2),N);
 
label("$x_0$",(x0,-0.02),S);
 
 
 
guide g1; guide g2; guide g3; guide g4; guide g5;
 
for (int k=floor(-0.7M); k<floor(0.3M); ++k) {
 
  real x = k/M;
 
  real z = 1+3*x^2;
 
  real y1 = 1/(z-2a-a0)+2b*(1+c*x)+d;
 
  real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
 
  real y3 = 1/(z-a0)+d;
 
  real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
 
  real y5 = 1/(z+2a-a0)-2b*(1+c*x)+d;
 
  g1=g1..(x,y1);
 
  g2=g2..(x,y2);
 
  g3=g3..(x,y3);
 
  g4=g4..(x,y4);
 
  g5=g5..(x,y5);
 
}
 
draw(g1,defaultpen+1);
 
draw(g2,defaultpen+1);
 
draw(g3,defaultpen+1);
 
draw(g4,defaultpen+1);
 
draw(g5,defaultpen+1);
 
 
 
real x = x0;
 
real z = 1+3*x^2;
 
real y1 = 1/(z-2a-a0)+2b*(1+c*x)+d;
 
real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
 
real y3 = 1/(z-a0)+d;
 
real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
 
real y5 = 1/(z+2a-a0)-2b*(1+c*x)+d;
 
path g = (x,y1)..(x-u,y2)..(x-v,y3)..(x-u,y4)..(x,y5);
 
draw( g );
 
 
 
pair w = (0.1,-0.6);
 
 
 
pair p = point(g,0.5);
 
dot ( p );
 
draw( p--p-0.5w, dashed );
 
label(rotate(90)*"$\Psi_{x_0}(y)$",p-0.5w,N);
 
 
 
draw( (x,y2)--(x-u,y2) );
 
draw( (x,y3)--(x-v,y3) );
 
draw( (x,y4)--(x-u,y4) );
 
 
 
draw( (x-u,y2+0.05)--(x-u,y4-0.05) );
 
draw( (x-v,y2+0.1)--(x-v,y4-0.1) );
 
 
 
real x = -0.15;
 
real z = 1+3*x^2;
 
real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
 
dot( (x,y4) );
 
draw( (x,y4)--(x,y4)+w, dashed );
 
label("$\scriptstyle \underline f_\alpha(x)$",(x,y4)+w,SE);
 
 
 
real x = -0.05;
 
real z = 1+3*x^2;
 
real y3 = 1/(z-a0)+d;
 
dot( (x,y3) );
 
draw( (x,y3)--(x,y3)+w, dashed );
 
label("$\scriptstyle \underline f_1(x)=\overline f_1(x)$",(x,y3)+w,SE);
 
 
 
real x = 0.05;
 
real z = 1+3*x^2;
 
real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
 
dot( (x,y2) );
 
draw( (x,y2)--(x,y2)+w, dashed );
 
label("$\scriptstyle \overline f_\alpha(x)$",(x,y2)+w,SE);
 
 
 
label("\small Fig. a4: Non-precise function",(x0,-0.2));
 
 
 
shipout(scale(250,120)*currentpicture);
 
</asy></center>
 
  
 +
Strangely, the figure in EoM is erroneous! ED=EB, not BD=EB.
  
 
<center><asy>
 
<center><asy>
import gsl;
 
 
int M=30;
 
 
picture whole;
 
 
draw ((-0.05,0)--(1.05,0),Arrow);
 
draw ((0,0)--(0,1.2),Arrow);
 
 
label("$x$",(1.05,0),E);
 
label("$\scriptstyle \xi_i(x)$",(0,1.2),N);
 
 
draw((-0.05,1)--(1.05,1));
 
label("$1$",(-0.05,1),W);
 
 
draw ((0.3,0)--(0.7,0),defaultpen+2);
 
draw ((0.3,0)--(0.3,1),dashed);
 
draw ((0.7,0)--(0.7,1),dashed);
 
 
label("$K_j$",(0.5,0),S);
 
 
guide g;
 
for (int k=floor(-3M); k<floor(3M); ++k) {
 
  real x = k/M;
 
  real y = sqrt(2pi) * pdf_gaussian(x);
 
  g=g..(x,y);
 
}
 
draw(shift(0.33,0)*scale(0.03,1)*g,defaultpen+1);
 
draw(shift(0.5,0)*scale(0.045,1)*g,defaultpen+1);
 
draw(shift(0.72,0)*scale(0.04,1)*g,defaultpen+1);
 
 
label("\small Fig. a3: Non-precise observations and a class of a histogram. is a class of a histogram and is a characterizing function ",(30,-20));
 
 
shipout(scale(170,50)*currentpicture);
 
</asy></center>
 
 
<center><asy>
 
import gsl;
 
 
int N=30;
 
 
picture whole;
 
picture common;
 
 
draw ((-0.3,0)--(1.45,0),Arrow);
 
draw ((0,0)--(0,1.2));
 
draw ((1,0)--(1,1.2));
 
 
label("$x$",(1.6,0),E);
 
 
add ( common, currentpicture );
 
erase();
 
 
 
add ( currentpicture, common );
 
 
guide g;
 
for (int k=floor(-0.2N); k<floor(1.2N); ++k) {
 
  real x = k/N;
 
  real y = cdf_gaussian_P(7*(x-0.5));
 
  y =0.85 y+0.15;
 
  g=g..(x,y);
 
}
 
draw(g,defaultpen+1.3);
 
 
label("$\scriptstyle g(x)$",(0.33,0.8));
 
 
add ( whole, shift(-60,0)*scale(40,36)*currentpicture );
 
erase();
 
 
 
add ( currentpicture, common );
 
 
draw((-0.2,1)--(1.2,1));
 
label("$1$",(-0.2,1),W);
 
 
guide g1;
 
guide g2;
 
for (int k=floor(-0.2N); k<floor(1.2N); ++k) {
 
  real x = k/N;
 
  real y = sqrt(2pi) * pdf_gaussian(7*(x-0.5));
 
  g1=g1..(x,y);
 
  g2=g2..(x,1.3y);
 
}
 
draw(g1,defaultpen+1.3);
 
draw(g2,defaultpen+1.3);
 
 
label("$\scriptstyle \xi(x)$",(0.5,0.4));
 
label("$\scriptstyle g'(x)$",(0.8,1.3));
 
 
add ( whole, shift(60,0)*scale(40)*currentpicture );
 
erase();
 
 
label(whole,"\small Fig. a2: Characterizing function obtained from a gray intensity ",(30,-20));
 
 
shipout(scale(1.2)*whole);
 
 
</asy></center>
 
 
 
<center><asy>
 
 
picture whole;
 
picture common;
 
 
draw ((-0.3,0)--(1.6,0),Arrow);
 
draw ((0,-0.2)--(0,1.3),Arrow);
 
draw((-0.2,1)--(1.5,1));
 
label("$x$",(1.6,0),E);
 
label("$\xi(x)$",(0,1.3),N);
 
label("$1$",(-0.2,1),W);
 
 
add ( common, currentpicture );
 
erase();
 
 
add ( currentpicture, common );
 
dot ((1,1),currentpen+5);
 
draw((1,0)--(1,1),dashed);
 
draw((-0.2,0)--(1.2,0),currentpen+1.5);
 
filldraw( shift(1,0)*scale(0.06)*unitcircle, white );
 
label("$x_0$",(1,0),S);
 
 
add ( whole, shift(-120,0)*scale(40)*currentpicture );
 
erase();
 
 
add ( currentpicture, common );
 
dot ((0.4,1),currentpen+5);
 
dot ((1,1),currentpen+5);
 
draw((0.4,0)--(0.4,1),dashed);
 
draw((1,0)--(1,1),dashed);
 
draw((-0.2,0)--(0.4,0),currentpen+1.5);
 
draw((1,0)--(1.3,0),currentpen+1.5);
 
draw((0.4,1)--(1,1),currentpen+1.5);
 
filldraw( shift(0.4,0)*scale(0.06)*unitcircle, white );
 
filldraw( shift(1,0)*scale(0.06)*unitcircle, white );
 
label("$a$",(0.4,0),S);
 
label("$b$",(1,0),S);
 
 
add ( whole, shift(0,0)*scale(40)*currentpicture );
 
erase();
 
 
add ( currentpicture, common );
 
draw((-0.2,0)--(0.3,0)--(0.5,1)--(0.7,1)--(1,0)--(1.3,0),currentpen+1.5);
 
 
label(whole,"Fig. a1:  Some characterizing functions",(30,-25));
 
add ( whole, shift(120,0)*scale(40)*currentpicture );
 
erase();
 
 
shipout(scale(1.2)*whole);
 
</asy></center>
 
 
<center><asy>
 
picture whole;
 
 
int N=3;
 
int M=30;
 
real c=0.6;
 
 
draw (arc((0,0),1,-90,90),defaultpen+2 );
 
 
guide g;
 
for (int k=-M*N+1; k<M*N; ++k) {
 
  real y=k/(M*N);
 
  pair z=(sqrt(1-y^2),y);
 
  pair w=(3z-z^3)/4;
 
  g=g..w;
 
  if (k%M==0) {
 
    draw((-0.5,y)--z);
 
    draw((-0.5,y)--(-0.1,y),Arrow);
 
    draw(z--z-c*z^2);
 
    draw(z--z-0.3c*z^2,Arrow);
 
  }
 
}
 
draw(g,defaultpen+1.3);
 
 
add ( whole, shift(-120,0)*scale(60)*currentpicture );
 
erase();
 
 
 
real n=1.3;
 
 
int N=7;
 
int M=10;
 
 
real c1=1.2;
 
real c2=0.6;
 
  
draw ((-1.4,0)--(1.4,0),defaultpen+2);
+
pair A=(-1,0);
 +
pair B=(0,0);
 +
pair E=(0,0.5);
 +
pair C=A+(0.5*(sqrt(5)-1),0);
 +
pair D=(-1/sqrt(5), 0.5*(1-1/sqrt(5)));
  
guide g;
+
draw( A--B--E--cycle,currentpen+1.5 );
for (int k=-M*N+1; k<M*N; ++k) {
+
dot(A,currentpen+3.5); dot(B,currentpen+3.5); dot(E,currentpen+3.5); dot(C,currentpen+3.5); dot(D,currentpen+3.5);
  real a=0.5*pi*k/(M*N);
 
  real s=sin(a);
 
  real t=tan(a);
 
  real x=(n^2-1)*t^3;
 
  if (n*abs(s)>=1) { continue; }
 
  real aux=(1-(n*s)^2)/(1-s^2);
 
  real y=-aux^1.5/n;
 
  g=g..(x,y);
 
  if (k%M==0) {
 
    draw((0,-1)--(t,0),Arrow(Relative(0.9)));
 
    draw((t,0)--(t,0)+c1*((x,y)-(t,0)),dashed);
 
    draw((t,0)--(t,0)+c2*((t,0)-(x,y)),Arrow(6,Relative(0.8)));
 
  }
 
}
 
draw(g,defaultpen+1.3);
 
  
dot((0,-1));  dot((0,-1/n));
+
draw( shift(E)*scale(0.5)*unitcircle,currentpen+1 );
label("$A$",(0,-1),W);
+
draw( shift(A)*scale(0.5*(sqrt(5)-1))*unitcircle,currentpen+1 );
label("$A'$",(0,-1/n),W);
 
  
add ( whole, shift(120,20)*scale(80)*currentpicture );
+
draw( shift(B)*scale(0.5)*unitcircle, dashed );
  
 +
clip(A+(-0.15,-0.15)--B+(0.15,-0.15)--E+(0.15,0.15)--A+(-0.15,0.15)--cycle);
  
label(whole,"Fig. a",(-100,-80));
+
label("$A$",A,S); label("$B$",B,S); label("$C$",C,S);
label(whole,"Fig. b",(120,-80));
+
label("$E$",E,N); label("$D$",D,N);
  
 +
label( "\small Golden Ratio construction", (-0.5,0.8) );
  
shipout(whole);
+
shipout(scale(100)*currentpicture);
 
</asy></center>
 
</asy></center>
  

Revision as of 16:12, 29 November 2014

Golden ratio

Strangely, the figure in EoM is erroneous! ED=EB, not BD=EB.





[Calculus: ] the art of numbering and measuring exactly a thing whose existence cannot be conceived. (Voltaire, Letter XVII: On Infinites In Geometry, And Sir Isaac Newton's Chronology)

And what are these fluxions? The velocities of evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities? (Berkeley, The Analyst)


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How to Cite This Entry:
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=35085