Toroidal blend of 8 dodecahedra

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Toroidal blend of 8 dodecahedra
TypeStewart toroid
Faces4+4+4+8 +4+4+4+8 +4+4+8+8+8+8 pentagons
Edges2+2+4+4+8+8+8+8+8+8 +2+2+4+4+8+8+8+8+8+8 +4+4+8+8+8+8+8+8+8+8+8
Vertices4+4+4+4+8+8+8 +4+4+4+4+8+8+8 +8+8+8+8+8
Vertex figures4+4+4+4+4+4+4+8+8+8+8+8+8+8 triangles, edge length (1+5)/2
 4+4+8+8+8 []
 4 []
Measures (edge length 1)
Surface area
Central density0
Number of external pieces80
Level of complexity20
Related polytopes
Convex hullMinkowski sum of dodecahedron with edge length 1 and rhombus with edge length and diagonal ratio (1+5)/2, where the diagonals are parallel to edges of the dodecahedron
Abstract & topological properties
Flag count800
Euler characteristic0
SymmetryK3, order 8

The toroidal blend of 8 dodecahedra is a Stewart toroid that consists of 80 pentagons. It can be obtained by outer-blending eight dodecahedra together in a rhombus-shaped loop. The four dodecahedra at the vertices of the virtual rhombus are all oriented in the same way.

Relations[edit | edit source]

The shape of this toroid differs from the toroidal blend of 8 octahedra: the latter's virtual rhombus has a diagonal ratio of 2, while this one's has diagonal ratio (1+5)/2.

Other versions of the toroid can be made out of copies of most of the Archimedean or Johnson solids that share the faceplanes of the dodecahedron.

Thirty copies of the toroid can be blended together, each blending pair coinciding at three dodecahedra with collinear centers, to form a toroidal blend of 92 dodecahedra with dodecahedral symmetry and an appearance like the skeleton of a rhombic triacontahedron.

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