Skew Polygon Expanded Polyhedra
Certain Platonic and Archimedean Polyhedra contain skew polygons other than their Petrie polygon. These can also be used as a basis for expansion.
Icosahedron and Truncated Icosahedron
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The icosahedron contains a skew hexagon (above left), this is the Petrie polygon of the great dodecahedron which shares a set of edges with the icosahedron. Expansion across these 10 polygons results in the figure above centre (F,V,E)=(110,240,132) containing 20 triangular faces, 60 golden rhombi and 30 distorted hexagons (a=138.19° as in the Petrie expanded great stellated truncated dodecahedron). The skew hexagon around the icosahedron also appears in truncated form around the truncated icosahedron. Expansion across these 10 polygons results in the figure above right (F,V,E)=(122,240,360) containing 12 pentagons, 20 regular hexagons, 30 distorted hexagons [bronze] (a=138.19°) and 60 distorted hexagons [gold] (a=109.471° as found in the Petrie expanded cube)
Dodecahedron and Truncated Dodecahedron
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The dodecahedron contains a skew dodecagon (above left), this is the truncated Petrie polygon of the small stellated dodecahedron. Expansion across these 10 polygons results in the figure above right (F,V,E)=(162,200,360) containing 12 pentagonal faces, 60 squares, 60 golden rhombi and 30 distorted hexagons (a=41.81° as in the Petrie expanded great truncated icosahedron). The skew dodecagon around the dodecahedron also appears in truncated form around the truncated dodecahedron. Expansion across these 10 polygons results in the figure above right (F,V,E)=(182,300,480) containing 12 decagons, 20 triangles, 30 distorted hexagons [bronze] (a=41.81°) and 60 distorted hexagons [gold] (a=109.471° as found in the Petrie expanded cube).
Icosidodecahedron
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The icosahedron contains a skew dodecagon (above left). Expansion across these 10 polygons results in the figure above centre (F,V,E)=(122,180,300) containing 12 pentagons, 20 triangles, 30 rhombi (a=41.81° as found in the medial rhombic triacontahedron) and 60 distorted hexagons (a=109.471° as found in the Petrie expanded cube).
Truncated Tetrahedron
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The truncated tetrahedron contains a skew enneagon (above left). Expansion across these 10 polygons results in the figure above centre (F,V,E)=(32,48,78) containing 4 triangles, 12 squares, 4 regular hexagons and 12 distorted hexagons (a=70.529° as found in the Petrie expanded octahedron).
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