Dihedral Angle

Author

Eric W. Weisstein
April 8, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SolidGeometry/DihedralAngle.nb.

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

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

Graphic

p=Normal[PolyhedronData["Tetrahedron","GraphicsComplex"]][[1]][[{1,2}]]

n=-PolygonNormal/@p

Graphics3D[{FaceForm[Green,Red],p,
Arrow[(Mean[p[[1,1]]]+#)&/@{{0,0,0},n[[1]]}],
Arrow[(Mean[p[[2,1]]]+#)&/@{{0,0,0},n[[2]]}]}]

So one of the normals must be flipped to prevent giving the complement of the dihedral angle

DihedralAngle@@p

N[%/Degree]

Example

Tetrahedron

DihedralAngles[PolyhedronData["Tetrahedron","GraphicsComplex"]]

N[%/Degree]

Cube

DihedralAngles[PolyhedronData["Cube","GraphicsComplex"]]

p=Normal[PolyhedronData["Cube","GraphicsComplex"]][[1]][[{1,2}]]

n=-PolygonNormal/@p

Graphics3D[{FaceForm[Green,Red],p,
Arrow[(Mean[p[[1,1]]]+#)&/@{{0,0,0},n[[1]]}],
Arrow[(Mean[p[[2,1]]]+#)&/@{{0,0,0},n[[2]]}]}]

DihedralAngle@@p

GyroelongatedPentagonalBicupola

PolyhedronData["GyroelongatedPentagonalBicupola"]

p=PolyhedronData["GyroelongatedPentagonalBicupola","Faces","Polygon"][[1]][[{17,34}]]

n=FullSimplify[-PolygonNormal/@p]

Graphics3D[{FaceForm[Green,Red],p,
Arrow[(Mean[p[[1,1]]]+#)&/@{{0,0,0},n[[1]]}],
Arrow[(Mean[p[[2,1]]]+#)&/@{{0,0,0},n[[2]]}]}]

DihedralAngle@@p/Degree//N

d1=N[DihedralAngles[PolyhedronData["GyroelongatedPentagonalBicupola","GraphicsComplex"]]/Degree]

d2=PolyhedronData["GyroelongatedPentagonalBicupola","DihedralAngles"]/Degree//N

PolyhedronData["GyroelongatedPentagonalBicupola","DihedralAngleRules"]//N

d3=DihedralAngles["GyroelongatedPentagonalBicupola"]/Degree//N

Subtract@@@Transpose[{d1,d2}]//Chop

d2-d3//Chop

PolyhedronFaces["GyroelongatedPentagonalBicupola"]==PolyhedronFaces[PolyhedronData["GyroelongatedPentagonalBicupola","GraphicsComplex"]]

PolyhedronVertices["GyroelongatedPentagonalBicupola"]==PolyhedronVertices[PolyhedronData["GyroelongatedPentagonalBicupola","GraphicsComplex"]]

PolyhedronData["GyroelongatedPentagonalBicupola","AdjacentFaceIndices"]

PolyhedronData["GyroelongatedPentagonalBicupola","EdgeIndices"]

PolyhedronData["GyroelongatedPentagonalBicupola","FaceIndices"]

AdjacentFaceIndices["GyroelongatedPentagonalBicupola"]==AdjacentFaceIndices[PolyhedronData["GyroelongatedPentagonalBicupola","GraphicsComplex"]]

Select[PolyhedronData[All],AdjacentFaceIndices[#]=!=AdjacentFaceIndices[PolyhedronData[#,"GraphicsComplex"]]&]

Position[i,{17,34}]

{d1[[15]],d2[[15]]}

2.77673/Degree

i1=PolyhedronData["GyroelongatedPentagonalBicupola","AdjacentFaceIndices"]

i2=AdjacentFaceIndices[PolyhedronData["GyroelongatedPentagonalBicupola","GraphicsComplex"]]

PolyhedronFaces[PolyhedronData["GyroelongatedPentagonalBicupola","GraphicsComplex"]]

Select[Transpose[{i1,i2}],#[[1]]=!=#[[2]]&]

AdjacentFaceIndices[PolyhedronData["GyroelongatedPentagonalBicupola","GraphicsComplex"]]