Truncation

Author

Eric W. Weisstein
December 19, 2003

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Graphics`Polyhedra`

f2[x_?NumericQ]:=
(
Print[x];
Min[Abs[#]&/@PolyhedronEdgeLengths[TruncateByPlanes[g,x]]]
)

Platonic Truncated Series

<<Graphics`Polyhedra`

Graphics`Polyhedra`Truncate is limited to values of r≤1/2, so I should really write my own…

Show[GraphicsArray[Graphics3D[Truncate[Polyhedron[Cube],#][[1]],Boxed->False]&/@{0,1/3,1/2}]]

Show[GraphicsArray[Graphics3D[Truncate[Polyhedron[Dodecahedron],#][[1]],Boxed->False]&/@{0,1/3,1/2}],ImageSize->200]

Show[GraphicsArray[Graphics3D[Truncate[Polyhedron[Icosahedron],#][[1]],Boxed->False]&/@{0,1/3,1/2}]]

Show[GraphicsArray[Graphics3D[Truncate[Polyhedron[Tetrahedron],#][[1]],Boxed->False]&/@{0,1/3,1/2}]]

Show[GraphicsArray[Graphics3D[Truncate[Polyhedron[Octahedron],#][[1]],Boxed->False]&/@{0,1/3,1/2}]]

Cuboctahedron

Show[Graphics3D[TruncateByPlanes[Cuboctahedron,#],Boxed->False,PlotLabel->#]]&/@Range[1,1/2,-1/12]

Show[Graphics3D[p=TruncateByPlanes[Cuboctahedron,3/4]],Boxed->False]

Union[FullSimplify[PolyhedronEdgeLengths[p]]]

Show[Truncate[Graphics3D[Polyhedron[Cuboctahedron]],.5],Boxed->False]

Icosidodecahedron

Show[Graphics3D[Truncate[Graphics3D[Polyhedron[Icosidodecahedron]],#][[1]],Boxed->False,PlotLabel->#]]&/@Range[0,.5,.05]

Rhombic Dodecahedron

Union[Norm/@PolyhedronVertices[RhombicDodecahedron]//N]

Show[GraphicsArray[Graphics3D[Polyhedron[#],Boxed->False]&/@{RhombicDodecahedron,SmallRhombicuboctahedron}]
]

Sqrt[3]/2.

g=RhombicDodecahedron;

FindMinimum[f2[x],{x,.84,.85},WorkingPrecision->20]

Show[Graphics3D[TruncateByPlanes[RhombicDodecahedron,.845299],Boxed->False]]

Rhombic Triacontahedron

Show[Graphics3D[Truncate[Graphics3D[Polyhedron[RhombicTriacontahedron]],#][[1]],Boxed->False,PlotLabel->#]]&/@Range[0,.5,.05]