## Platonische Körper

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### Platonische Körper

Polyhedron: Rhombentriakontaeder, subtraktive Herstellung durch Herausschneiden aus einem Kuboid

Code: Alles auswählen

`(* Syntax: Wolfram, Code: Simon Tyran, Vienna, yukterez.net *) ecken={{0,0,1/2 (-1-Sqrt[5])},{0,0,1/2 (1+Sqrt[5])},{1/10 (5-Sqrt[5]),Root[1-5 #1^2+5 #1^4&,1],1/10 (5+3 Sqrt[5])},{1/10 (5-Sqrt[5]),Sqrt[1/10 (5+Sqrt[5])],1/10 (5+3 Sqrt[5])},{2/Sqrt[5],0,1/10 (5+3 Sqrt[5])},{1/10 (5+3 Sqrt[5]),Root[1-5 #1^2+5 #1^4&,1],1/10 (5+Sqrt[5])},{1/10 (5+3 Sqrt[5]),Root[1-5 #1^2+5 #1^4&,1],1/10 (-5+Sqrt[5])},{1/10 (5+3 Sqrt[5]),Sqrt[1/10 (5+Sqrt[5])],1/10 (5+Sqrt[5])},{1/10 (5+3 Sqrt[5]),Sqrt[1/10 (5+Sqrt[5])],1/10 (-5+Sqrt[5])},{-(2/Sqrt[5]),0,1/10 (-5-3 Sqrt[5])},{-(1/Sqrt[5]),-Sqrt[1+2/Sqrt[5]],1/10 (5+Sqrt[5])},{-(1/Sqrt[5]),-Sqrt[1+2/Sqrt[5]],1/10 (-5+Sqrt[5])},{-(1/Sqrt[5]),Sqrt[1+2/Sqrt[5]],1/10 (5+Sqrt[5])},{-(1/Sqrt[5]),Sqrt[1+2/Sqrt[5]],1/10 (-5+Sqrt[5])},{1/Sqrt[5],-Sqrt[1+2/Sqrt[5]],1/10 (5-Sqrt[5])},{1/Sqrt[5],-Sqrt[1+2/Sqrt[5]],1/10 (-5-Sqrt[5])},{1/Sqrt[5],Sqrt[1+2/Sqrt[5]],1/10 (5-Sqrt[5])},{1/Sqrt[5],Sqrt[1+2/Sqrt[5]],1/10 (-5-Sqrt[5])},{-1-1/Sqrt[5],0,1/10 (5+Sqrt[5])},{-1-1/Sqrt[5],0,1/10 (-5+Sqrt[5])},{1/10 (-5-Sqrt[5]),Root[1-5 #1^2+5 #1^4&,2],1/10 (5+3 Sqrt[5])},{1/10 (-5-Sqrt[5]),Sqrt[2/(5+Sqrt[5])],1/10 (5+3 Sqrt[5])},{1/10 (5+Sqrt[5]),Root[1-5 #1^2+5 #1^4&,2],1/10 (-5-3 Sqrt[5])},{1/10 (5+Sqrt[5]),Sqrt[2/(5+Sqrt[5])],1/10 (-5-3 Sqrt[5])},{1+1/Sqrt[5],0,1/10 (5-Sqrt[5])},{1+1/Sqrt[5],0,1/10 (-5-Sqrt[5])},{1/10 (-5-3 Sqrt[5]),Root[1-5 #1^2+5 #1^4&,1],1/10 (5-Sqrt[5])},{1/10 (-5-3 Sqrt[5]),Root[1-5 #1^2+5 #1^4&,1],1/10 (-5-Sqrt[5])},{1/10 (-5-3 Sqrt[5]),Sqrt[1/10 (5+Sqrt[5])],1/10 (5-Sqrt[5])},{1/10 (-5-3 Sqrt[5]),Sqrt[1/10 (5+Sqrt[5])],1/10 (-5-Sqrt[5])},{1/10 (-5+Sqrt[5]),Root[1-5 #1^2+5 #1^4&,1],1/10 (-5-3 Sqrt[5])},{1/10 (-5+Sqrt[5]),Sqrt[1/10 (5+Sqrt[5])],1/10 (-5-3 Sqrt[5])}}; polygon={{16,15,11,12},{14,13,17,18},{10,28,20,30},{8,5,6,25},{12,28,31,16},{32,30,14,18},{6,3,11,15},{8,17,13,4},{11,21,19,27},{13,29,19,22},{7,16,23,26},{24,18,9,26},{12,11,27,28},{30,29,13,14},{7,6,15,16},{18,17,8,9},{2,22,19,21},{23,1,24,26},{3,2,21,11},{4,13,22,2},{16,31,1,23},{1,32,18,24},{31,28,10,1},{10,30,32,1},{6,5,2,3},{8,4,2,5},{28,27,19,20},{20,19,29,30},{26,25,6,7},{9,8,25,26}}; x1=Max[ecken[[All, 1]]];x2=Max[ecken[[All, 2]]];x3=Max[ecken[[All, 3]]]; f[α_,β_,δ_]:=RotationMatrix[α π/180,{0,1,0}].(RotationMatrix[β π/180,{0,0,1}].(RotationMatrix[δ π/180,{1,0,0}]));solve[p_]:=Quiet[N[Reduce[x==pts[1, p, 1]]]]; plane[p_,c_]:=InfinitePlane[Evaluate[Table[c ecken[[polygon[[p,k]]]],{k,2,4}]]];cube[b_]:=Cuboid[b{-x1,-x2,-x3},b{+x1,+x2,+x3}];pts[b_, p_,c_]:=Normal[Evaluate[Solve[{q,y,z}\[Element]plane[p,c]&&{q,y,z}\[Element]cube[b],{q,y,z},Reals]]][[1,1,2]];opar=1; isize=400; plot[vp_,n_,ζ_,ς_,p_,s_,b_,r_,c_,ξ_,h_,α_,β_,δ_]:=Show[ Graphics3D[{Opacity[n],EdgeForm[Thickness[0.003]],Rotate[Rotate[Rotate[GraphicsComplex[ecken,Polygon[polygon]],α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}]},SphericalRegion->True], Graphics3D[{Opacity[ς],EdgeForm[Thickness[0.001]],FaceForm[Darker[Blue]],Rotate[Rotate[Rotate[InfinitePlane[Evaluate[Table[c ecken[[polygon[[p,k]]]],{k,2,4}]]],α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}]},SphericalRegion->True], Graphics3D[{Opacity[1],EdgeForm[Thickness[0.003]],FaceForm[Darker[Red]],Rotate[Rotate[Rotate[GraphicsComplex[s ecken,Polygon[polygon[[p]]]],α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}]},SphericalRegion->True], Graphics3D[{Opacity[1],EdgeForm[Thickness[0.003]],FaceForm[Darker[Red]],Rotate[Rotate[Rotate[GraphicsComplex[s ecken,Polygon[polygon[[p]]]],α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}]},SphericalRegion->True], Graphics3D[{Opacity[ζ],EdgeForm[Thickness[0.003]],Rotate[Rotate[Rotate[cube[b],α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}]},SphericalRegion->True], If[1.0 α==0.,If[1.0 β==0.,If[1.0 δ==0.,If[h>0,If[p>2,Quiet[ContourPlot3D[Evaluate[q==pts[1,p,1]],{q,-x1,+x1},{y,-x2,+x2},{z,-x3,+x3},ContourStyle->Directive[Orange,Opacity[h],Specularity[White,30]]]],{}],{}],{}],{}],{}], If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],EdgeForm[Thickness[0.003]],Blue,Arrow[{{0,0,0},{2.1,0,0}}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}], If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],EdgeForm[Thickness[0.003]],Red,Arrow[{{0,0,0},{0,2.1,0}}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}], If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],EdgeForm[Thickness[0.003]],Darker[Green],Arrow[{{0,0,0},{0,0,2.1}}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}],If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],EdgeForm[Thickness[0.003]],Blue,Line[{{0,0,0},{-2,0,0}}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}], If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],EdgeForm[Thickness[0.003]],Red,Line[{{0,0,0},{0,-2,0}}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}], If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],EdgeForm[Thickness[0.003]],Darker[Green],Line[{{0,0,0},{0,0,-2}}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}],If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],PointSize[0.02],Blue,Point[{-2,0,0}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}],If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],PointSize[0.02],Red,Point[{0,-2,0}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}],If[ξ>0,Graphics3D[Rotate[Rotate[Rotate[{Opacity[opar],PointSize[0.02],Darker[Green],Point[{0,0,-2}]},α π/180,{0,1,0}],β π/180,{0,0,1}],δ π/180,{1,0,0}],SphericalRegion->True],{}], PlotRange->r,ViewPoint->vp,SphericalRegion->True,ImageSize->isize,Boxed->False];VP={{0,0,Infinity},{0,Infinity,0},{Infinity,0,0},{0,0,-Infinity},{0,-Infinity,0},{-Infinity,0,0}};X0=x1; Y0=x2; Z0=x3;T[text_, color_] := Style[text, FontSize->11, color]; construct=Manipulate[Grid[{{Rasterize[plot[w VP[[1]],n,ζ,ς,p,s,b,r,c,ξ,h,α,β,δ], ImageSize->isize], Rasterize[plot[w VP[[2]],n,ζ,ς,p,s,b,r,c,ξ,h,α,β,δ], ImageSize->isize]},{Rasterize[plot[w VP[[3]],n,ζ,ς,p,s,b,r,c,ξ,h,α,β,δ], ImageSize->isize], plot[1 VP[[6]],n,ζ,ς,p,s,b,r,c,ξ,h,0,0,0]},{Grid[{{T["x"==x/.Solve[solve[p],x],Black]},{T["y"==y/.Solve[solve[p],y],Black]},{T["z"==z/.Solve[solve[p],z],Black]}}],Grid[{{{T["α"->α,Black]},{T["β"->β,Black]},{T["δ"->δ,Black]}},{{T["±X"->x1 1., Blue]}, {T["±Y"->x2 1., Red]}, {T["±Z"->x3 1., Darker[Green]]}}}]}}],{{n,1},0,1},{{ζ,0.1},0,1},{{ς,0.1},0,1/2},{{r,2.7},1/3,10},{{p,4},1,Length[polygon],1},{{s,1.01},1,2},{{b,1},0,1,1},{{ξ,1},0,1,1},{{c,-10},-10,1,11},{{h,0.4},0,1,0.2},{{w,1},-1,1,2},{α,0,360,1},{β,0,360,1},{δ,0,360,1}]; construct`

Triacontahedron: Wenn die Kantenlängen A..B=1, A..C=2·Sin(ArcTan(2/(1+√5)))=1.05146 und B..D=1.7013 (B..D=A..C·Ф wobei Ф der goldene Schnitt ist), so sind die Längen des Kuboids {x, y, z}={2+2/√5, 2·√(1+2/√5), 1+√5}={2.89443, 2.75276, 3.23607}; der dihedrale Winkel ist 144° und die Rhombuswinkel 148.2825° & 31.7175°. Ecken (Vertices): 32, Kanten (Edges): 60, Gesichtsflächen (Rhomben): 30.

Netz aus 30 gleichseitigen Rhomben:

Animation mit den 5 wichtigsten Gesichtern und ihren Schnittintersektionen mit dem umhüllenden Kuboid:

Videos, Top Front Links Rechts Ansicht:

Schnittintersektionen für alle Rhomben:

Rhombus 1, Kuboidausrichtung 0° bzw. 180°:

Rhombus 2, Kuboidausrichtung 162° bzw. 342°:

Rhombus 3, Kuboidausrichtung 54° bzw. 234°:

Rhombus 4, Kuboidausrichtung 54° bzw. 234°:

Rhombus 5, Kuboidausrichtung 36° bzw. 216°:

Beispiel: Berechnung der Schnitthöhen am Kuboid:

Mit der Lösung für x, y & z aus der linken unteren Spalten des oberen Plots erhalten wir die gelb unterlegten Schnittstellen:

Dabei ist wieder die längere Schnittmarkierung 1.34165 auf der z-Kante des Kuboids der goldene Schnitt Φ mal der kürzeren Markierung 0.829184 auf der x-Kante. Interaktives Arbeitsblatt: triacontahedron.cdf (kann mit dem kostenlosen cdf-Player abgespielt werden). Version 2: icosahedron.vs.dodecahedron.vs.triacontrahedron.compound.nb

by Simon Tyran, Vienna @ youtube || rumble || odysee || minds || wikipedia || stackexchange || License: CC-BY 4 ▣ If images don't load: [ctrl]+[F5]

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