Polyhedral Curvature

MATH 261 Computational Geometry

Polyhedra on the front of the worksheet

Graphics3D[{EdgeForm[Thick],FaceForm[Opacity[0.9]],PolyhedronData["GreatRhombicuboctahedron","GraphicsComplex"]},Boxed->False]

𝒫3=Polyhedron[
{{0,0,0},{3,0,0},{3,3,0},{0,3,0},{0,0,1},{3,0,1},{3,3,1},{0,3,1},{1,1,2},{2,1,2},{2,2,2},{1,2,2},{1,1,1},{2,1,1},{2,2,1},{1,2,1}},
{{1,2,6,5},{2,3,7,6},{3,4,8,7},{4,1,5,8},{1,2,3,4},{9,10,14,13},{10,11,15,14},{15,16,12,11},{16,13,9,12},{5,6,14,13},{6,7,15,14},{7,8,16,15},{8,5,13,16},{9,10,11,12}}
];
Graphics3D[{Opacity[0.9],EdgeForm[Thick],𝒫3},Boxed->False]

𝒫3=Polyhedron[
{{0,0,0},{3,0,0},{3,3,0},{0,3,0},{0,0,1},{3,0,1},{3,3,1},{0,3,1},{1,1,0},{2,1,0},{2,2,0},{1,2,0},{1,1,1},{2,1,1},{2,2,1},{1,2,1}},
{{1,2,6,5},{2,3,7,6},{3,4,8,7},{4,1,5,8},{1,2,10,9},{2,3,11,10},{3,4,12,11},{4,1,9,12},{9,10,14,13},{10,11,15,14},{15,16,12,11},{16,13,9,12},{5,6,14,13},{6,7,15,14},{7,8,16,15},{8,5,13,16}}
];
Graphics3D[{Opacity[0.85],EdgeForm[Thick],𝒫3},Boxed->False]

Graphics3D[{EdgeForm[Thick],FaceForm[Opacity[0.9]],Dodecahedron[]},Boxed->False]

𝒫4=Polyhedron[
{{0,0,0},{0,3,0},{0,3,3},{0,0,3},{0,1,1},{0,2,1},{0,2,2},{0,1,2},{3,0,0},{3,3,0},{3,3,3},{3,0,3},{3,1,1},{3,2,1},{3,2,2},{3,1,2},{1.5,1,1},{1.5,2,1},{1,1,2},{2,1,2},{2,2,2},{1,2,2},{1,1,3},{2,1,3},{2,2,3},{1,2,3}},
{{19,22,26,23},{17,20,24,23,19},{18,21,25,26,22},{20,21,25,24},{20,16,15,21},{8,19,22,7},{5,17,18,6},{17,13,14,18},{6,18,22,7},{18,14,15,21},{5,17,19,8},{17,13,16,20},
{4,12,24,23},{12,11,25,24},{11,3,26,25},{3,4,23,26},{1,2,6,5},{2,3,7,6},{3,4,8,7},{4,1,5,8},{9,10,14,13},{10,11,15,14},{11,12,16,15},{12,9,13,16},{2,10,11,3},{1,9,12,4},{1,9,10,2}}
];
Graphics3D[{Opacity[0.6],EdgeForm[Thick],𝒫4},Boxed->False]

Graphics3D[{EdgeForm[Thick],FaceForm[Opacity[0.9]],PolyhedronData["StellatedTetrahedron","GraphicsComplex"]},Boxed->False]

Subregions of polyhedra on the back of the worksheet

Graphics3D[{
EdgeForm[Thick],FaceForm[Opacity[0.7]],Tetrahedron[],
Red,Arrowheads[0.08],Arrow[Tube[{{((-(Sqrt[3]))/8),((-3)/8),0},{((Sqrt[3])/4),0,0}},0.01]],Arrow[Tube[{{((Sqrt[3])/4),0,0},{((-(Sqrt[3]))/8),(3/8),0}},0.01]],Arrow[Tube[{{((-(Sqrt[3]))/8),(3/8),0},{((-(Sqrt[3]))/8),((-3)/8),0}},0.01]]
},Boxed->False]

𝒫3=Polyhedron[
{{0,0,0},{3,0,0},{3,3,0},{0,3,0},{0,0,1},{3,0,1},{3,3,1},{0,3,1},{1,1,2},{2,1,2},{2,2,2},{1,2,2},{1,1,1},{2,1,1},{2,2,1},{1,2,1}},
{{1,2,6,5},{2,3,7,6},{3,4,8,7},{4,1,5,8},{1,2,3,4},{9,10,14,13},{10,11,15,14},{15,16,12,11},{16,13,9,12},{5,6,14,13},{6,7,15,14},{7,8,16,15},{8,5,13,16},{9,10,11,12}}
];
Graphics3D[{
Opacity[0.9],EdgeForm[Thick],𝒫3,
Red,Arrowheads[0.08],Arrow[Tube[{{0,0,1},{3,0,1}},0.04]],Arrow[Tube[{{3,0,1},{3,3,1}},0.04]],Arrow[Tube[{{3,3,1},{0,3,1}},0.04]],Arrow[Tube[{{0,3,1},{0,0,1}},0.04]]
},Boxed->False]

Graphics3D[{
Opacity[0.9],EdgeForm[Thick],𝒫3,
Red,Arrowheads[0.08],Arrow[Tube[{{0,0,1},{3,0,1}},0.04]],Arrow[Tube[{{3,0,1},{2,1,1}},0.04]],Arrow[Tube[{{2,1,1},{2,1,2},{2,2,1},{3,3,1}},0.04]],Arrow[Tube[{{3,3,1},{0,3,1}},0.04]],Arrow[Tube[{{0,3,1},{0,0,1}},0.04]]
},Boxed->False]