The in itself rotated Golden Rectangle
by Klaus-Anton, Jan 11, 2024, 1:58 PM
The Golden Rectangle has the proportion of the Golden Ratio. The Golden Ratio is normally defined as:

As proportion it is:

The Domino-Triangle - which is a right triangle - has as the sides proportion: one to two to squareroot of five:

I translate into English: The blue spirals are different from the right-angled triangles composed than the red ones. There is a cath of the red spirals successor triangle on the hypotenuse of the predecessor triangle. With the blue ones in the spirals it is the other way round.
I have sent Figure 9 into Loop for the in itself rotated Polygon post #13.
Inspiration:
Hans Walser 2022. Spiralen, Schraubenlinien und spiralartige Figuren. Mathematische Spielereien in zwei und drei Dimensionen. Springer Spektrum. Berlin, Germany. Abb. 5.15 a) und b) Rechtwinklige Dreiecke. Seite 75. Und Kapitel 5.4.2 Goldenes Rechteck. Seite 75 f. Darin inbesondere Abb. 5.16 Im Goldenen Rechteck. Seite 76. video direct Link Abb. 5.16 Im Goldenen Rechteck. Availability as eBook.

As proportion it is:

![[asy]
import geometry;
//usepackage("xcolor");
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;// the same as: sqrt(5)/+.5
pair A,B,C,D,myE,F,G,H,I,J;
A=(0,0); B=(GR,0); C=(GR,1); D=(0,1);
myE=(GR-.5,0);//(.5,1);
F=(.5,1);//F=(GR-.5,0);
// F=(gr+.5,0);
G=foot(myE,A,F);
H=foot(F,C,myE);
path Golden_Rectangle=A--B--C--D--cycle;
fill(A--F--D--cycle^^myE--B--C--cycle,red);
fill(A--myE--G--cycle^^C--F--H--cycle,blue);
draw(Golden_Rectangle);
draw(A--F);
draw(myE--C);
draw(myE--G);
draw(F--H);
/*
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",myE,S); dot("",myE,SE);
label("$F$",F,N); dot("",F,NW);
label("$H$",H,E); dot("",H,NW);
label("$G$",G,E); dot("",G,NW);
//dot((sqrt(5)/2,0),cyan);
*/
/*
distance(rotate(990)*"$\color{red}1$",B,C,8mm);
distance("$\Phi$",A,B,8mm);
distance("$\color{blue}\sqrt5/2$",A,myE,16mm);
distance("$\color{red}0.5$",myE,B,24mm);
*/
perpendicularmark(B,NW,white,PenMargins);
perpendicularmark(D,SE,white,PenMargins);
shipout(bbox(2mm,Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/1/9/f/19f1d04dc2923b85b5407906e232634a62e9ea49.png)
Figure 1: The Golden Rectangle. - In red & blue: Domino-Triangles
The Domino-Triangle - which is a right triangle - has as the sides proportion: one to two to squareroot of five:

![[asy]
import geometry;
//usepackage("xcolor");
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;// the same as: sqrt(5)/+.5
pair A,B,C,D,myE,F,G,H,I,J;
A=(0,0); B=(GR,0); C=(GR,1); D=(0,1);
myE=(GR-.5,0);//(.5,1);
F=(.5,1);//F=(GR-.5,0);
// F=(gr+.5,0);
G=foot(myE,A,F);
H=foot(F,C,myE);
path Golden_Rectangle=A--B--C--D--cycle;
fill(A--F--D--cycle^^myE--B--C--cycle,red);
fill(A--myE--G--cycle^^C--F--H--cycle,blue);
draw(Golden_Rectangle);
draw(A--F);
draw(myE--C);
draw(myE--G);
draw(F--H);
/*
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",myE,S); dot("",myE,SE);
label("$F$",F,N); dot("",F,NW);
label("$H$",H,E); dot("",H,NW);
label("$G$",G,E); dot("",G,NW);
//dot((sqrt(5)/2,0),cyan);
*/
distance(rotate(990)*"$\color{red}1$",B,C,8mm);
distance("$\Phi$",A,B,8mm);
distance("$\color{blue}\sqrt5/2$",A,myE,16mm);
distance("$\color{red}0.5$",myE,B,24mm);
shipout(bbox(2mm,Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/0/4/f/04fc15b5cbd3356dc56cdd8aa2f0c08d2f68a0c5.png)
Figure 2: The Golden Rectangle. - The lengths here used.
![[asy]
import geometry;
//usepackage("xcolor");
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;// the same as: sqrt(5)/+.5
pair A,B,C,D,myE,F,G,H,I,J;
A=(0,0); B=(GR,0); C=(GR,1); D=(0,1);
myE=(GR-.5,0);//(.5,1);
F=(.5,1);//F=(GR-.5,0);
// F=(gr+.5,0);
G=foot(myE,A,F);
H=foot(F,C,myE);
I=intersectionpoint(F--myE,G--H);//dot(I);
path Golden_Rectangle=A--B--C--D--cycle;
fill(A--F--D--cycle^^myE--B--C--cycle,red);
fill(A--myE--G--cycle^^C--F--H--cycle,blue);
path arc_I=arc(I,.1,degrees(dir(I--myE)) ,degrees(dir(I--H)), CCW );
draw(arc_I);
real my_dir=degrees(dir(I--midpoint(arc_I), dir(I--myE,I--H)));
label(rotate(my_dir)*
"$\arctan(2)$", I,5*dir(I--midpoint(arc_I), dir(I--myE,I--H)));
draw(Golden_Rectangle);
draw(A--F);
draw(myE--C);
draw(myE--G);
draw(F--H);
draw(F--myE^^G--H);
/*
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",myE,S); dot("",myE,SE);
label("$F$",F,N); dot("",F,NW);
label("$H$",H,E); dot("",H,NW);
label("$G$",G,E); dot("",G,NW);
//dot((sqrt(5)/2,0),cyan);
*/
/*
distance(rotate(990)*"$\color{red}1$",B,C,8mm);
distance("$\Phi$",A,B,8mm);
distance("$\color{blue}\sqrt5/2$",A,myE,16mm);
distance("$\color{red}0.5$",myE,B,24mm);
*/
real alpha=aTan(2);
path path_alpha=arc(A,.2,0,alpha);
draw(path_alpha,white,PenMargins);
//label("$\arctan(2)$",path_alpha,white);// autosizes on AoPS
label("$\arctan(2)$",midpoint(path_alpha),dir(A--midpoint(path_alpha)),white);
path path_gamma=arc(C,.2,180,180+alpha);
draw(path_gamma,white,PenMargins);
label(format("$\approx%.3f^\circ$",alpha),midpoint(path_gamma),dir(C--midpoint(path_gamma)),white);
shipout(bbox(2mm,Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/texer/l/lwsxckqv.png)
Figure 3: The Golden Rectangle. - Angles.
![[asy]
usepackage("fourier");
import geometry;
//usepackage("xcolor");
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;// the same as: sqrt(5)/+.5
pair A,B,C,D,myE,F,G,H,I,J;
A=(0,0); B=(GR,0); C=(GR,1); D=(0,1);
myE=(GR-.5,0);//(.5,1);
F=(.5,1);//F=(GR-.5,0);
// F=(gr+.5,0);
G=foot(myE,A,F);
H=foot(F,C,myE);
I=intersectionpoint(F--myE,G--H);//dot(I);
path Golden_Rectangle=A--B--C--D--cycle;
fill(A--F--D--cycle^^myE--B--C--cycle,red);
fill(A--myE--G--cycle^^C--F--H--cycle,blue);
path arc_I=arc(I,.1,degrees(dir(I--myE)) ,degrees(dir(I--H)), CCW );
draw(arc_I);
real my_dir=degrees(dir(I--midpoint(arc_I), dir(I--myE,I--H)));
label(rotate(my_dir)*
"$\arctan(2)$", I,5*dir(I--midpoint(arc_I), dir(I--myE,I--H)));
//draw(A--F^^myE--C,white+linewidth(GR),PenMargins);// not accepted. Why not?
draw(F--myE,PenMargins);
draw(G--H);
draw(A--F,white+linewidth(GR)+squarecap+beveljoin);
draw(myE--C,white+linewidth(GR)+squarecap+beveljoin);
//draw(myE--C);
draw(myE--G,white+linewidth(GR)+squarecap+beveljoin);
draw(F--H,white+linewidth(GR)+squarecap+beveljoin);
//draw(F--myE^^G--H,PenMargins); not accepted
//draw(Golden_Rectangle,linewidth(GR)+white+squarecap+miterjoin);
/*
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",myE,S); //dot("",myE,SE);
label("$F$",F,N); //dot("",F,NW);
label("$H$",H,E); //dot("",H,NW);
label("$G$",G,E); //dot("",G,NW);
//dot((sqrt(5)/2,0),cyan);
*/
/*
distance(rotate(990)*"$\color{red}1$",B,C,8mm);
distance("$\Phi$",A,B,8mm);
distance("$\color{blue}\sqrt5/2$",A,myE,16mm);
distance("$\color{red}0.5$",myE,B,24mm);
*/
real alpha=aTan(2);
path path_alpha=arc(A,.2,0,alpha);
draw(path_alpha,white+linewidth(1));
//label("$\arctan(2)$",path_alpha,white);// autosizes on AoPS
label("$\arctan(2)$",midpoint(path_alpha),dir(A--midpoint(path_alpha)),white);
label("$\alpha$", A, 3*dir(A--midpoint(path_alpha)),white);
label("$\alpha$", C, -3*dir(A--midpoint(path_alpha)),white);
pair upi=plain.I;
label("$\alpha$", I, 2*dir(I--foot(I,A,F)*dir(15)));
draw(arc(I,GR/12//.15/1.25
,degrees(dir(I--F))
,degrees(dir(I--G))
));
path path_gamma=arc(C,.2,180,180+alpha);
draw(path_gamma,white,PenMargins);
label(format("$\approx%.3f^\circ$",alpha),midpoint(path_gamma),dir(C--midpoint(path_gamma)),white);
perpendicularmark(D,SE,white+linewidth(1)+extendcap+miterjoin);
perpendicularmark(B,NW,white+linewidth(1));
shipout(bbox(2mm,Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/f/4/f/f4f5611c55b99a67fecb76d9c6af30b8b58440c4.png)
Figure 4: The Golden Rectangle. - Angles. - Sharper Contrasts.
![[asy]
usepackage("fourier");
import geometry;
//usepackage("xcolor");
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;// the same as: sqrt(5)/+.5
pair A,B,C,D,myE,F,G,H,I,J,K;
A=(0,0); B=(GR,0); C=(GR,1); D=(0,1);
myE=(GR-.5,0);//(.5,1);
F=(.5,1);//F=(GR-.5,0);
// F=(gr+.5,0);
G=foot(myE,A,F);
H=foot(F,C,myE);
I=intersectionpoint(F--myE,G--H);//dot(I);
real my_real=.5/GR;
J=relpoint(F--H, my_real);//dot(J,linewidth(10));
K=relpoint(myE--G, my_real);//dot(K,linewidth(10));
pair L=foot(J,K,H);//dot(L,linewidth(10));
fill(J--L--H--cycle,blue);
triangle t1=triangle(A,myE,G);//draw(t1,linewidth(4));
path path_t1=A--myE--G--cycle;
triangle t2=triangle(A,F,D);//draw(t2,linewidth(4));
fill(J--F--G--cycle^^K--myE--H--cycle,red);
//draw(shift(G)*scale(gr)*rotate(degrees(dir(G--myE)))*t1,green);//okay
fill(shift(G)*scale(gr)*rotate(degrees(dir(G--myE)))*path_t1,blue);//okay
//fill(shift(G)*scale(gr)*rotate(degrees(dir(G--myE)))*t1,blue);
//fill((path)shift(G)*scale(gr)*rotate(degrees(dir(G--myE)))*t1,blue);//
//path my_path=shift(G)*scale(gr)*rotate(degrees(dir(G--myE)))*t1;// cannot cast 'triangle' to 'path'
//path my_path=(path)(shift(G)*scale(gr)*rotate(degrees(dir(G--myE)))*t1);// cannot cast 'triangle' to 'path'
/*
dot("",J,magenta);
dot("",K,white);
dot("",L,green);
dot("",M);
*/
pair M=foot(K,J,G);
draw(J--G,linewidth(GR)+white+beveljoin);
draw(K--M,linewidth(GR)+white+beveljoin);
draw(K--H,linewidth(GR)+white+beveljoin);
path Golden_Rectangle=A--B--C--D--cycle;
fill(A--F--D--cycle^^myE--B--C--cycle,red);
fill(A--myE--G--cycle^^C--F--H--cycle,blue);
path arc_I=arc(I,.1,degrees(dir(I--myE)) ,degrees(dir(I--H)), CCW );
//draw(arc_I);
real my_dir=degrees(dir(I--midpoint(arc_I), dir(I--myE,I--H)));
/*
label(rotate(my_dir)*
"$\arctan(2)$", I,5*dir(I--midpoint(arc_I), dir(I--myE,I--H)));
*/
//draw(A--F^^myE--C,white+linewidth(GR),PenMargins);// not accepted. Why not?
draw(A--F,white+linewidth(GR)+squarecap+beveljoin);
draw(myE--C,white+linewidth(GR)+squarecap+beveljoin);
//draw(myE--C);
draw(myE--G,white+linewidth(GR)+squarecap+beveljoin);
draw(F--H,white+linewidth(GR)+squarecap+beveljoin);
//draw(F--myE^^G--H,PenMargins); not accepted
/*
draw(F--myE);
draw(G--H);
*/
//draw(Golden_Rectangle,linewidth(GR)+white+squarecap+miterjoin);
/*
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",myE,S); //dot("",myE,SE);
label("$F$",F,N); //dot("",F,NW);
label("$H$",H,E); //dot("",H,NW);
label("$G$",G,E); //dot("",G,NW);
//dot((sqrt(5)/2,0),cyan);
*/
/*
distance(rotate(990)*"$\color{red}1$",B,C,8mm);
distance("$\Phi$",A,B,8mm);
distance("$\color{blue}\sqrt5/2$",A,myE,16mm);
distance("$\color{red}0.5$",myE,B,24mm);
*/
real alpha=aTan(2);
path path_alpha=arc(A,.2,0,alpha);
//draw(path_alpha,white,PenMargins);
//label("$\arctan(2)$",path_alpha,white);// autosizes on AoPS
//label("$\arctan(2)$",midpoint(path_alpha),dir(A--midpoint(path_alpha)),white);
path path_gamma=arc(C,.2,180,180+alpha);
//draw(path_gamma,white,PenMargins);
//label(format("$\approx%.3f^\circ$",alpha),midpoint(path_gamma),dir(C--midpoint(path_gamma)),white);
//dot(relpoint(B--A,my_real));// okay
shipout(bbox(2mm,Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/texer/y/yiddduno.png)
Figure 5: The Golden Rectangle. - In itself rotated, the next Step.
Hans Waler 2022 wrote:
Dabei sind die blauen Spiralen anders aus den rechtwinkligen Dreiecken zusammengesetzt als die roten. Bei den roten Spiralen liegt eine Kathete des Nachfolgdreieckes auf der Hypotenuse des Vorgängerdreieckes. Bei den blauen Spiralen ist es umgekehrt.
I translate into English: The blue spirals are different from the right-angled triangles composed than the red ones. There is a cath of the red spirals successor triangle on the hypotenuse of the predecessor triangle. With the blue ones in the spirals it is the other way round.
![[asy]
import geometry;
//usepackage("xcolor");
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;// the same as: sqrt(5)/+.5
pair A,B,C,D,myE,F,G,H,I,J;
A=(0,0); B=(GR,0); C=(GR,1); D=(0,1);
myE=(GR-.5,0);//(.5,1);
F=(.5,1);//F=(GR-.5,0);
// F=(gr+.5,0);
G=foot(myE,A,F);
H=foot(F,C,myE);
path Golden_Rectangle=A--B--C--D--cycle;
picture pic;
fill(pic,A--F--D--cycle^^myE--B--C--cycle,red);
fill(pic,A--myE--G--cycle^^C--F--H--cycle,blue);
draw(Golden_Rectangle);
draw(A--C^^B--D);
draw(F--myE^^G--H,dotted);
draw(A--F);
draw(myE--C);
draw(myE--G);
draw(F--H);
shipout(bbox(2mm,Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/a/8/c/a8c9184099f42b8aee34aa7077ce3afb4f04201c.png)
Figure 6: The Golden Rectangle. - In itself rotated.
The Intersection Point of the Diagonals is the Rotation Center.
![[asy]
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;
pair A,B,C,D,myE,F,G,H,O=(0,0);
A=(-.5*GR,-.5);
B=(+.5*GR,-.5);
C=(+.5*GR,+.5);
D=(-.5*GR,+.5);
myE=B-.5;
F=D+.5;
G=foot(myE,A,F);
H=foot(F,C,myE);
picture pic1;
filldraw(pic1, A--F--D--cycle^^B--C--myE--cycle,red,white+linewidth(GR)+beveljoin);
filldraw(pic1, myE--G--A--cycle^^F--H--C--cycle,blue,white+linewidth(GR)+beveljoin);
add(pic1);
add(scale(gr)*rotate(-1*(90-aTan(2)))*pic1);
shipout(bbox(2mm,Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/texer/e/eamhyiyw.png)
Figure 7: The Golden Rectangle. - In itself rotated.
Shortened Code
![[asy]
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;
pair A,B,C,D,myE,F,G,H,O=(0,0);
A=(-.5*GR,-.5);
B=(+.5*GR,-.5);
C=(+.5*GR,+.5);
D=(-.5*GR,+.5);
myE=B-.5;
F=D+.5;
G=foot(myE,A,F);
H=foot(F,C,myE);
picture pic1;
filldraw(pic1, A--F--D--cycle^^B--C--myE--cycle,red,white+linewidth(GR)+beveljoin);
filldraw(pic1, myE--G--A--cycle^^F--H--C--cycle,blue,white+linewidth(GR)+beveljoin);
add(pic1);
add(scale(gr)*rotate(-1*(90-aTan(2)))*pic1);
add(scale(gr^2)*rotate(-2*(90-aTan(2)))*pic1);dot(O,invisible);// invisible dot here only because of AoPS-asy
shipout(bbox(2mm,Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/d/c/f/dcf3b8e7120ad211588a2beb55157920f91b7b44.png)
Figure 8: The Golden Rectangle. - In itself rotated.
The next Step
![[asy]
unitsize(5cm);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)+1)/2;
pair A,B,C,D,myE,F,G,H,O=(0,0);
A=(-.5*GR,-.5);
B=(+.5*GR,-.5);
C=(+.5*GR,+.5);
D=(-.5*GR,+.5);
myE=B-.5;
F=D+.5;
G=foot(myE,A,F);
H=foot(F,C,myE);
picture pic;
filldraw(pic, A--F--D--cycle^^B--C--myE--cycle,red,white+linewidth(GR)+beveljoin);
filldraw(pic, myE--G--A--cycle^^F--H--C--cycle,blue,white+linewidth(GR)+beveljoin);
for(int n=0;n<6;++n){
add(scale((gr^n))*rotate((-n)*(90-aTan(2)))*pic);
}
pen GoldenYellow=rgb(1.0, 0.87, 0.0); // https://latexcolor.com/
shipout(bbox(2mm,Fill(GoldenYellow)));
[/asy]](http://latex.artofproblemsolving.com/1/f/6/1f62784a57b8103fe66ee754bcddc1f72e2640b0.png)
Figure 9: The Golden Rectangle. - In itself rotated.
Further Steps - With a Loop
The Picture can remember you Elephants kissing each other with their truncs.
I have sent Figure 9 into Loop for the in itself rotated Polygon post #13.
Inspiration:
Hans Walser 2022. Spiralen, Schraubenlinien und spiralartige Figuren. Mathematische Spielereien in zwei und drei Dimensionen. Springer Spektrum. Berlin, Germany. Abb. 5.15 a) und b) Rechtwinklige Dreiecke. Seite 75. Und Kapitel 5.4.2 Goldenes Rechteck. Seite 75 f. Darin inbesondere Abb. 5.16 Im Goldenen Rechteck. Seite 76. video direct Link Abb. 5.16 Im Goldenen Rechteck. Availability as eBook.
This post has been edited 25 times. Last edited by Klaus-Anton, Jan 13, 2024, 4:52 PM