Vertex Figure

Author

Eric W. Weisstein
October 27, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/VertexFigure.nb.

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

Polygons

poly[n_]:=Table[{Cos[2Pi i/n],Sin[2Pi i/n]},{i,0,n}];

V5

Show[GraphicsArray[Table[Graphics[{Line[poly[n]],Thickness[.025],Line[Plus@@Partition[Take[poly[n],3],2,1]/2],
PointSize[.08],Point[poly[n][[2]]]},
AspectRatio->Automatic,PlotRange->All
],{n,3,6}]]]

V6

g1=GraphicsArray[Table[Graphics[{Line[poly[n]],Thickness[.025],Line[Plus@@Partition[Take[poly[n],3],2,1]/2],
PointSize[.08],Point[poly[n][[2]]]}
],{n,3,6}]]

Cube

V5

p=Table[{-0.52,0.52},{3}];
cube=Orient[Cube[[1]]];

trcube={Polygon[{{0,-0.5,0.5},{0.5,-0.5,0},{0.5,-0.5,-0.5},{-0.5,-0.5,-0.5},{-0.5,-0.5,0.5}}],Polygon[{{0.5,0.5,0.5},{0.5,0,0.5},{0,-0.5,0.5},{-0.5,-0.5,0.5},{-0.5,0.5,0.5}}],Polygon[{{-0.5,-0.5,0.5},{-0.5,-0.5,-0.5},{-0.5,0.5,-0.5},{-0.5,0.5,0.5}}],Polygon[{{0.5,-0.5,-0.5},{0.5,0.5,-0.5},{-0.5,0.5,-0.5},{-0.5,-0.5,-0.5}}],Polygon[{{0.5,0.5,0.5},{0.5,0.5,-0.5},{0.5,-0.5,-0.5},{0.5,-0.5,0},{0.5,0,0.5}}],Polygon[{{-0.5,0.5,0.5},{-0.5,0.5,-0.5},{0.5,0.5,-0.5},{0.5,0.5,0.5}}],
Polygon[{{0,-0.5,0.5},{0.5,0,0.5},{0.5,-0.5,0}}]
};
GraphicsArray[{
Show[Graphics3D/@{PointSize[.08],Point[{0.5,-0.5,0.5}],cube,VertexFigures[cube][[1]]},Boxed->False,PlotRange->p,DisplayFunction->Identity],
Show[Graphics3D/@{trcube,VertexFigures[cube][[1]]},Boxed->False,PlotRange->p,DisplayFunction->Identity],
Show[Graphics3D[Polygon[{{0,-0.5,0.5},{0.5,0,0.5},{0.5,-0.5,0}}]],PlotRange->p,DisplayFunction->Identity]
}]]

V6

p=Table[{-0.52,0.52},{3}];
cube=PolyhedronData["Cube","Faces"];

trcube={Polygon[{{0,-0.5,0.5},{0.5,-0.5,0},{0.5,-0.5,-0.5},{-0.5,-0.5,-0.5},{-0.5,-0.5,0.5}}],Polygon[{{0.5,0.5,0.5},{0.5,0,0.5},{0,-0.5,0.5},{-0.5,-0.5,0.5},{-0.5,0.5,0.5}}],Polygon[{{-0.5,-0.5,0.5},{-0.5,-0.5,-0.5},{-0.5,0.5,-0.5},{-0.5,0.5,0.5}}],Polygon[{{0.5,-0.5,-0.5},{0.5,0.5,-0.5},{-0.5,0.5,-0.5},{-0.5,-0.5,-0.5}}],Polygon[{{0.5,0.5,0.5},{0.5,0.5,-0.5},{0.5,-0.5,-0.5},{0.5,-0.5,0},{0.5,0,0.5}}],Polygon[{{-0.5,0.5,0.5},{-0.5,0.5,-0.5},{0.5,0.5,-0.5},{0.5,0.5,0.5}}],
Polygon[{{0,-0.5,0.5},{0.5,0,0.5},{0.5,-0.5,0}}]
};
g2=GraphicsArray[{
Graphics3D[{PointSize[.08],Point[{0.5,-0.5,0.5}],cube,VertexFigures[cube][[1]]},Boxed->False,PlotRange->p],
Graphics3D[{trcube,VertexFigures[cube][[1]]},Boxed->False,PlotRange->p],
Graphics3D[Polygon[{{0,-0.5,0.5},{0.5,0,0.5},{0.5,-0.5,0}}],PlotRange->p]
}]

Combined Figure

Grid[List/@{g1,g2}]

GraphicsGrid[List/@{g1,g2},Spacings->{Automatic,-240}]

Platonics

<<Platonic`;

Show[GraphicsArray[Map[Graphics3D[#,Boxed->False]&,
{
p=Table[Platonic[n],{n,5}],
VertexFigures/@p,
VertexFigures[#,Convex->True]&/@p
},
{2}]
]]//Timing

Archimedean

<<Archimedean`;

Table[ArchimedeanName[n],{n,5}]

Show[GraphicsArray[Map[Graphics3D[#,Boxed->False]&,
{
p=Table[Archimedean[n],{n,5}],
VertexFigures/@p,
VertexFigures[#,Convex->True]&/@p
},
{2}]
],GraphicsSpacing->-.2]//Timing

Show[GraphicsArray[Map[Graphics3D[#,Boxed->False]&,
{
p=Table[Archimedean[n],{n,6,10}],
VertexFigures/@p,
VertexFigures[#,Convex->True]&/@p
},
{2}]
],GraphicsSpacing->-.2]//Timing

Show[GraphicsArray[Map[Graphics3D[#,Boxed->False]&,
{
p=Table[Archimedean[n],{n,11,13}],
VertexFigures/@p,
VertexFigures[#,Convex->True]&/@p
},
{2}]
],GraphicsSpacing->-.2]//Timing