Supergolden Rectangle

Author

Eric W. Weisstein
February 16, 2025

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

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

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

Sources

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

Wikipedia

https://en.wikipedia.org/wiki/Supergolden_ratio#Supergolden_rectangle

yellow={{0,0},{(psi^(-2)),0},{(psi^(-2)),1-(psi^(-1))},{0,1-(psi^(-1))}};
green={{(psi^(-2)),0},{psi,0},{psi,1-(psi^(-1))},{(psi^(-2)),1-(psi^(-1))}};
blue={{(psi^(-2)),1-(psi^(-1))},{psi,1-(psi^(-1))},{psi,1},{(psi^(-2)),1}};
red={{0,1-(psi^(-1))},{(psi^(-2)),1-(psi^(-1))},{(psi^(-2)),1},{0,1}};
Graphics[{
EdgeForm[Black],
FaceForm[Yellow],Polygon[yellow],
FaceForm[Darker[Green,.5]],Polygon[green],
FaceForm[Darker[Blue,.4]],Polygon[blue],
FaceForm[Darker[Red,.4]],Polygon[red],
Text[Superscript[ψ,-2],{0,1-(psi^(-1))}/2,{2,0}],
Text[Superscript[ψ,-2],{(psi^(-2)),0}/2,{0,2}]
},
BaseStyle->Directive[14,FontFamily->"Times"]]

RectangleDimensions/@{yellow,green,blue,red}

3//RootReduce

Pegg

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

Figure

v={{0,0},{psi-(psi^(-2)),0},{psi,0},{psi,(psi^(-3))},{psi,1},{psi-(psi^(-2)),1},{0,1},{psi-(psi^(-2)),(psi^(-3))}};

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

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

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

RectangleDimensions/@rects

With[{psi=},
Graphics[{
FaceForm[LightYellow],EdgeForm[Black],Polygon/@rects,
Text[1,{0,1/2},{3,0}],
Text[1,{1/2,0},{0,2.5}],
Text[Superscript[ψ,-2],{psi-(psi^(-2))/2,0},{0,2}],
Text[Superscript[ψ,-3],{psi,(psi^(-3))/2},{-2,0}],
Text[Superscript[ψ,-1],{psi,(psi^(-3))+(psi^(-1))/2},{-2,0}],
Text[ψ,{psi/2,1},{0,-2}],
Arrow[{{psi/2-.05,1.065},{0,1.065}}],Arrow[{{psi/2+.04,1.065},{psi,1.065}}]
},
BaseStyle->Directive[14,FontFamily->"Times"]]
]

width=1+(psi^(-2))//RootReduce

height=(psi^(-1))+(psi^(-3))//RootReduce