Golden Rectangle

Author

Eric W. Weisstein
February 16, 2025

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polygons/GoldenRectangle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GoldenRectangle.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=-a/2≤x≤a/2&&-GoldenRatio a/2≤y≤GoldenRatio a/2;

implreg=ImplicitRegion[ineq,{x,y}];

reg=Rectangle[-{1,GoldenRatio}a/2,{1,GoldenRatio}a/2];

assum=a>0;

Figures

Golden rectangle

1×(ϕ^(1/2)) rectangle dissection

https://blog.wolfram.com/2019/03/07/shattering-the-plane-with-twelve-new-substitution-tilings-using-2-phi-psi-chi-rho/

phi=GoldenRatio;
v={{0,0},{(phi^(1/2)),0},{(phi^(1/2))+(phi^(3/2)),0},{0,1},{(phi^(1/2)),1},{0,1+phi},{(phi^(1/2)),1+phi},{(phi^(1/2))+(phi^(3/2)),1+phi}};

Graphics[MapIndexed[Text[#2[[1]],#]&,v]]

rects={v[[{1,3,8,6}]],v[[{1,2,5,4}]],v[[{4,5,7,6}]]};

RectangleDimensions[l_]:=Module[{len=EuclideanDistance@@@Partition[l,2,1,1]//RootReduce},
NumericalSort[DeleteDuplicates@RootReduce[len/Min[len]]]
]

RectangleDimensions/@rects

RootReduce[(GoldenRatio^(1/2))]

RootReduce[1/((1+phi)/((phi^(1/2))+(phi^(3/2))))]

(phi^(1/2))+(phi^(3/2))==(phi^(5/2))//RootReduce

((ϕ^(5/2))/(ϕ^2))

With[{phi=GoldenRatio},
Graphics[{
FaceForm[LightYellow],EdgeForm[Black],Polygon/@rects,
Text[(ϕ^(5/2)),{(phi^(5/2))/2,1+GoldenRatio},{0,-2}],
Text[1,{0,1/2},{2.5,0}],
Text[ϕ,{0,1+phi/2},{2.5,0}],
Text[Superscript[ϕ,Row[{1,"/",2}]],{(phi^(1/2))/2,0},{0,2.3}],
Text[Superscript[ϕ,Row[{3,"/",2}]],{(phi^(1/2))+(phi^(3/2))/2,0},{0,2.3}],
Arrow[{{(phi^(5/2))/2-.15,1.15+GoldenRatio},{0,1.15+GoldenRatio}}],Arrow[{{(phi^(5/2))/2+.15,1.15+GoldenRatio},{(phi^(5/2)),1.15+GoldenRatio}}],
Text[(ϕ^2),{(phi^(5/2)),(phi^2)/2},{-2.5,0}],
Arrow[{{(phi^(5/2))+.15,(phi^2)/2+.15},{(phi^(5/2))+.15,(phi^2)}}],
Arrow[{{(phi^(5/2))+.15,(phi^2)/2-.15},{(phi^(5/2))+.15,0}}]
},
BaseStyle->Directive[14,FontFamily->"Times"]]
]

width=(phi^(1/2))+(phi^(3/2))//RootReduce

N[Log[phi,%],20]

(phi^(5/2))//RootReduce

height=1+phi//RootReduce

phi^2//RootReduce

Plots

Diagram

LaminaData["FilledGoldenRectangle","Diagram"]

DiscretizeRegion

Block[{a=1},Show[DiscretizeRegion[reg],ImageSize->Small]]

RegionPlot

Block[{a=1},RegionPlot[ineq,{x,-a/2,a/2},{y,-a GoldenRatio/2,a GoldenRatio/2},AspectRatio->Automatic,ImageSize->Small]]

Graphics

GoldenRectangle::usage="GoldenRactangle[n] plots n iterations of the golden rectangle
and the corresponding logarithmic spiral."

rect[0]:={{0,0},{1,-1}}

rect[n_]:=Module[{m=Mod[n,4],phi=1/GoldenRatio,x1,x2,y1,y2},{{x1,y1},{x2,y2}}=rect[n-1];
Which[m==0,{{x1,y2},{x1+phi^n,y2-phi^n}},m==1,{{x2,y2+phi^n},{x2+phi^n,y2}},m==2,{{x2-phi^n,y1+phi^n},{x2,y1}},m==3,{{x1-phi^n,y1},{x1,y1-phi^n}}]]

square[{{x1_,y1_},{x2_,y2_}}]:=Line[{{x1,y1},{x2,y1},{x2,y2},{x1,y2},{x1,y1}}]

GoldenRectangle[n_]:=Module[{i,a=(4/5)^(1/4) GoldenRatio^(ArcTan[2]/Pi),b=2Log[GoldenRatio]/Pi,x0,y0},{x0,y0}={(5+3Sqrt[5])/10,(Sqrt[5]-5)/10};
ParametricPlot[Evaluate[{x0,y0}+{-Cos[t],Sin[t]}a Exp[b t]],{t,-100,.25},AspectRatio->Automatic,Ticks->None,Axes->False,PlotRange->{{-.1,2},{-1,0}},Prolog->Table[square[rect[i]],{i,0,n}]]]

Straightedge and compass construction

https://m.wikihow.com/Construct-a-Golden-Rectangle

Properties

Area

Area

Assuming[assum,Simplify@Area[reg]]

Integrate

Assuming[a>0,FullSimplify[Area[ineq[x,y],{x,y}]]]/.(1+(Sqrt[5]))->2GoldenRatio

RegionMeasure

Assuming[assum,Simplify@RegionMeasure[reg]]

AreaInertiaTensor

Assuming[a>0,FullSimplify[AreaInertiaTensor[ineq[x,y],{x,y}]]]/.(Sqrt[5])->2(GoldenRatio-1)//FullSimplify

Centroid

RegionCentroid

Assuming[assum,Simplify@RegionCentroid[reg]]

Convex

TimeConstrained[Region`ConvexRegionQ[reg],600]//Timing

TimeConstrained[Region`ConvexRegionQ[implreg],600]//Timing

Circumcircle

FullSimplify[(1/2)(Sqrt[(a^2)+(((a GoldenRatio))^2)]),a>0]

DiagonalLengths

FullSimplify[(Sqrt[(a^2)+(((a GoldenRatio))^2)]),a>0]

Perimeter

Length[p={a,GoldenRatio a,a,GoldenRatio a}]

Total[p]//FullSimplify

RadiiOfGyration

Assuming[a>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]/.(Sqrt[5])->2(GoldenRatio-1)//FullSimplify

Spiral

{x0,y0}={(5+3Sqrt[5])/10,(Sqrt[5]-5)/10}

{x0,y0}//N

th[{x_,y_}]:=ArcTan[-(x-x0),y-y0]
theqn[{x_,y_}]:=Module[{t=th[{x,y}]},
		{x==x0-a Cos[t]Exp[b t],
		y==y0+a Sin[t]Exp[b t]}
	]

FullSimplify[theqn/@{{0,0},{1,-1}} /. n->0]//Flatten

FindRoot[Evaluate[Take[%324,2]//N],{a,1.,1.1},{b,.3,.31},MaxIterations->25000]

%324 /. Equal->List /. %339

%324 /. {Equal->List,a->1.12053,b->.306349}

Square[{{x1_,y1_},{x2_,y2_}}]:=Line[{{x1,y1},{x2,y1},{x2,y2},{x1,y2},{x1,y1}}]

rect[0]:={{0,0},{1,-1}}
rect[n_]:=Module[{m=Mod[n,4],phi=1/GoldenRatio,
		x1,x2,y1,y2},
		{{x1,y1},{x2,y2}}=rect[n-1];
		Which[
			m==0,{{x1,y2},{x1+phi^n,y2-phi^n}},
			m==1,{{x2,y2+phi^n},{x2+phi^n,y2}},
			m==2,{{x2-phi^n,y1+phi^n},{x2,y1}},
			m==3,{{x1-phi^n,y1},{x1,y1-phi^n}}]
	]
{x0,y0}={(5+3Sqrt[5])/10,(Sqrt[5]-5)/10};

GoldenRectangle[n_]:=Module[{i},
		ParametricPlot[
	Evaluate[{x0,y0}+{-Cos[t],Sin[t]}1.12053Exp[.306349t]],{t,-100,.25},
AspectRatio->Automatic,Ticks->None,Axes->False,
PlotRange->{{-.1,2},{-1,0}},Prolog->Table[Square[rect[i]],{i,0,n}]]
	]

rect[0]:={{0,0},{1,-1}}
rect[n_]:=Module[{m=Mod[n,4],phi=1/GoldenRatio,
		x1,x2,y1,y2},
		{{x1,y1},{x2,y2}}=rect[n-1];
		Which[
			m==0,{{x1,y2},{x1+phi^n,y2-phi^n}},
			m==1,{{x2,y2+phi^n},{x2+phi^n,y2}},
			m==2,{{x2-phi^n,y1+phi^n},{x2,y1}},
			m==3,{{x1-phi^n,y1},{x1,y1-phi^n}}]
	]

{x0,y0}={(5+3Sqrt[5])/10,(Sqrt[5]-5)/10}

th[{x_,y_}]:=ArcTan[-(x-x0),y-y0]
theqn[{x_,y_}]:=Module[{t=th[{x,y}]},
		{x==x0-a Cos[t]Exp[b t],
		y==y0+a Sin[t]Exp[b t]}
	]

FullSimplify[theqn/@{{0,0},{1,-1}} /. n->0]//Flatten

FindRoot[Evaluate[Take[%324,2]//N],{a,1.,1.1},{b,.3,.31},MaxIterations->25000]

%324 /. Equal->List /. %339

%324 /. {Equal->List,a->1.12053,b->.306349}

square[{{x1_,y1_},{x2_,y2_}}]:=Line[{{x1,y1},{x2,y1},{x2,y2},{x1,y2},{x1,y1}}]

ParametricPlot[
	Evaluate[{(5+3Sqrt[5])/10,(Sqrt[5]-5)/10}+{-Cos[t],Sin[t]}1.12053Exp[.306349t]],{t,-100,.25},
AspectRatio->Automatic,Ticks->None,Axes->False,PlotStyle->Red,
PlotRange->{{-.2,2},{-1,0}},Prolog->Table[square[rect[i]],{i,0,10}]]

{a,b}={1.1205291652509972363051964665659,0.30634896253003312211567570119977}

ParametricPlot[
	Evaluate[{(5+3Sqrt[5])/10,(Sqrt[5]-5)/10}+{-Cos[t],Sin[t]}a Exp[b t]],{t,-100,.25},
Ticks->None,Axes->False,PlotPoints->400,(*AspectRatio->Automatic,*)
PlotRange->{{.999,1.003},{-.20,-.26}},Prolog->Table[square[rect[i]],{i,0,20}]]

l={{-ArcTan[2-Sqrt[5]],-ArcTan[2-Sqrt[5]],-ArcTan[2+Sqrt[5]]},
	{(5+3Sqrt[5])/Sqrt[50+20Sqrt[5]],(5-Sqrt[5])/Sqrt[50-20Sqrt[5]],
	(3Sqrt[5]-5)/Sqrt[50-20Sqrt[5]]}}

f[i_]:=-l[[2,i]]+1.12053Exp[l[[1,i]] .306349]

Table[f[i],{i,3}]//N

N[ExponentialFit@@l-ExponentialWeightedFit@@l,100]

N[ExponentialFit@@l,20]

ExponentialWeightedFit@@l//N

FullSimplify[ExponentialFit@@l]

Expand[%49[[1]]]

Exp[(-1/2 ArcTan[2+Sqrt[5]]Log[(50-20Sqrt[5])/(5-Sqrt[5])^2]+
			ArcTan[2-Sqrt[5]]Log[(50-20Sqrt[5])/(3Sqrt[5]-5)^2]+
						ArcTan[2+Sqrt[5]]Log[(5+3Sqrt[5])/Sqrt[50+20Sqrt[5]]])/Pi]//N

FullSimplify[(ArcTan[2+Sqrt[5]]Log[(5+Sqrt[5])/5]+ArcTan[2-Sqrt[5]]Log[(5+Sqrt[5])/4])/Pi]

Expand[(5+Sqrt[5])^2/20]

Exp[ArcTan[2+Sqrt[5]]Log[(3+Sqrt[5])/2]/Pi-Log[(5+Sqrt[5])/4]/2]

FullSimplify[%]

<<SqrtSimplify`

ArcTan[2+Sqrt[5]]/Pi //N

2ArcTan[2+Sqrt[5]]/Pi//N

Sqrt[25-5Sqrt[5]]//N

GoldenRatio^(2ArcTan[2+Sqrt[5]]/Pi)Sqrt[25-5Sqrt[5]]/5//N

(3/2+Sqrt[5]/2)^(ArcTan[2+Sqrt[5]]/Pi)Sqrt[25-5Sqrt[5]]/5//N

ArcTan[2+Sqrt[5]]//N

Tan[

N[Log[GoldenRatio]2/Pi,20]

(5+Sqrt[5])/4//N

Table[GoldenRatio^n//N,{n,5}]

(3+Sqrt[5])/2//N

1/Tan[Pi/5]//N

Log[GoldenRatio]//N

1/Sqrt[5+Sqrt[5]]//N

Tan[3Pi/10.]

(ArcTan[2.+Sqrt[5]]/Pi)

1/Tan[3.Pi/10]

N[((((4/5)))^(1/4))(GoldenRatio^(ArcTan[2]/π)),10]

Sums

(∑_(n=0)^∞)(-(1/(ϕ^(4 n+2)))-(1/(ϕ^(4 n+3)))+(1/(ϕ^(4 n)))+(1/(ϕ^(4 n+1))))==(1/10) (3 (Sqrt[5])+5)

Sum[(1/(GoldenRatio^(4n)))+(1/(GoldenRatio^(4n+1)))-(1/(GoldenRatio^(4n+2)))-(1/(GoldenRatio^(4n+3))),{n,0,∞}]

FullSimplify[%]

FullSimplify[%-((2GoldenRatio+1)/(GoldenRatio+2))]

((1/ϕ)-(1/(ϕ^2))-(1/(ϕ^3))+1) (∑_(n=0)^∞)(1/(ϕ^(4 n)))==(1/10) (3 (Sqrt[5])+5)

(1+(GoldenRatio^(-1))-(GoldenRatio^(-2))-(GoldenRatio^(-3)))Sum[(1/(GoldenRatio^(4n))),{n,0,∞}]

FullSimplify[%]

(∑_(n=0)^∞)((1/(ϕ^(4 n+2)))-(1/(ϕ^(4 n+3)))-(1/(ϕ^(4 n)))+(1/(ϕ^(4 n+1))))==(1/10) ((Sqrt[5])-5)

Sum[-(1/(GoldenRatio^(4n)))+(1/(GoldenRatio^(4n+1)))+(1/(GoldenRatio^(4n+2)))-(1/(GoldenRatio^(4n+3))),{n,0,∞}]

FullSimplify[%]