Cantellation

Author

Eric W. Weisstein
June 28, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Examples

Graphics3D/@Table[CantellatedPolyhedron[Cube[],1/(Sqrt[2]),Method->meth],{meth,{"ConvexHullRegion","ExpandPolyhedron","Mangaldan"}}]

CantellatedPolyhedron[PolyhedronData["TruncatedIcosahedron","Polyhedron"],0.5,Method->"Mangaldan"]//Graphics3D

CantellatedPolyhedron[PolyhedronData["TruncatedIcosahedron","Polyhedron"],.9999]//Graphics3D

More than one face type (special treatment)

Usually just consider one face type at a time

Manipulate[Graphics3D@CantellatedPolyhedron[PolyhedronData["Cuboctahedron","Polyhedron"],r,n,Method->"ConvexHullRegion"],{{r,.6},-1,1},{n,{3,4}}]

Manipulate[Graphics3D@CantellatedPolyhedron[PolyhedronData["Cuboctahedron","Polyhedron"],r,Method->"ConvexHullRegion"],{{r,.6},0,1}]

Manipulate[Graphics3D@CantellatedPolyhedron[PolyhedronData["Cuboctahedron","Polyhedron"],s,Method->"Mangaldan"],{{s,.6},0,1}]

Platonic

PolyhedronData["Platonic"]

Cube (small rhombicuboctahedron)

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Cube","Polyhedron"],(2^(-1/2))]]

PolyhedronEdgeLengths[poly]//Union

Dodecahedron (small rhombicosidodecahedron)

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Dodecahedron","Polyhedron"],Root[5-20 ((#1)^2)+16 ((#1)^4)&,4]]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

Icosahedron (small rhombicosidodecahedron)

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Icosahedron","Polyhedron"],(Sqrt[(9/8)+((3 (Sqrt[5]))/8)])]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

Octahedron (small rhombicuboctahedron)

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Octahedron","Polyhedron"],Sqrt[3]/2]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

Tetrahedron (cuboctahedron)

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Tetrahedron","Polyhedron"],Sqrt[3/8]]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

Archimedean

PolyhedronData["Archimedean"]

Cuboctahedron (truncated octahedron, truncated octahedron, octahedron)

3

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Cuboctahedron","Polyhedron"],(Sqrt[3])/2,3]]

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

4 (cube)

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Cuboctahedron","Polyhedron"],(2^(-1/2)),4]]

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

4 (octahedron)

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Cuboctahedron","Polyhedron"],-(2^(-1/2)),4]]

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

GreatRhombicosidodecahedron

PolyhedronData["GreatRhombicosidodecahedron","Polyhedron"]//Graphics3D

Manipulate[Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["GreatRhombicosidodecahedron","Polyhedron"]//N,r,4]],{{r,.5},-1,1}]

GreatRhombicuboctahedron

Manipulate[Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["GreatRhombicuboctahedron","Polyhedron"]//N,r,4]],{{r,.5},-1,1}]

Icosidodecahedron

3

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Icosidodecahedron","Polyhedron"],(Sqrt[(9/8)+((3 (Sqrt[5]))/8)]),3]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

5

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["Icosidodecahedron","Polyhedron"],(Sqrt[(5/8)+((Sqrt[5])/8)]),5]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

SmallRhombicosidodecahedron

4

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],GoldenRatio,4]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

SmallRhombicosidodecahedron

Manipulate[Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"]//N,r,4]],{{r,.5},-1,1}]

SmallRhombicuboctahedron

Manipulate[Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"]//N,r,4]],{{r,.5},-1,1}]

3

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],-(Sqrt[3])/2,3]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

Non-Perpendicular Squares

g=Select[Polyhedron[SmallRhombicuboctahedron],Length[#[[1]]]==4&&Count[Cross[#[[1,2]]-#[[1,1]],#[[1,-1]]-#[[1,1]]],0]<2&];

Show[Graphics3D[g],Boxed->False]

Show[Graphics3D[p=Expansion[g,1]],Boxed->False]

Union[FullSimplify/@PolyhedronEdgeLengths[p]]

TruncatedCube

Triangles

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedCube","Polyhedron"],-Sqrt[3]/2,3]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

Octagons

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedCube","Polyhedron"],(2^(-1/2)),8]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

TruncatedDodecahedron

3

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedDodecahedron","Polyhedron"],-(Sqrt[(9/8)+((3 (Sqrt[5]))/8)]),3]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

10

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedDodecahedron","Polyhedron"],(Sqrt[(5/8)+((Sqrt[5])/8)]),10]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

TruncatedIcosahedron

5

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedIcosahedron","Polyhedron"],-(Sqrt[(5/8)+((Sqrt[5])/8)]),5]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

6

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedIcosahedron","Polyhedron"],(Sqrt[(9/8)+((3 (Sqrt[5]))/8)]),6]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

TruncatedOctahedron

4

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedOctahedron","Polyhedron"],-(2^(-1/2)),4]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

6

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedOctahedron","Polyhedron"],Sqrt[3]/2,6]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

TruncatedTetrahedron

3

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedTetrahedron","Polyhedron"],-Sqrt[3/8],3]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

6

Graphics3D[poly=CantellatedPolyhedron[PolyhedronData["TruncatedTetrahedron","Polyhedron"],Sqrt[3/8],6]]//Quiet

ListSimplify[PolyhedronEdgeLengths[poly]]//Union//Quiet

Nested TruncatedPolyhedron

From Charles, June 28, 2021

cantellatedPolyhedron[poly_] := TruncatedPolyhedron[TruncatedPolyhedron[poly, .5], .5]

chamfering is BeveledPolyhedron.

You can find all Conway operators in System` and Region`Polyhedron contexts. Look at Applications > Polyhedron Operations section in TruncatedPolyhderon reference pages.

???

TruncatedPolyhedron[TruncatedPolyhedron[Cube[],.5],.5]//Graphics3D

Manipulate[Graphics3D@TruncatedPolyhedron[TruncatedPolyhedron[Cube[],r],s],{{r,.5},0,1},{{s,.5},0,1}]