Toroidal blend of 8 dodecahedra

Toroidal blend of 8 dodecahedra
(3D model)
Rank	3
Type	Stewart toroid
Elements
Faces	4+4+4+8 +4+4+4+8 +4+4+8+8+8+8 pentagons
Edges	2+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
Vertices	4+4+4+4+8+8+8 +4+4+4+4+8+8+8 +8+8+8+8+8
Vertex figures	4+4+4+4+4+4+4+8+8+8+8+8+8+8 triangles, edge length (1+√5)/2
	4+4+8+8+8 [5.5.5.5]
	4 [5.5.5.5.5]
Measures (edge length 1)
Volume	$2\left(15+7{\sqrt {5}}\right)\approx 61.30495$
Surface area	$20{\sqrt {25+10{\sqrt {5}}}}\approx 137.63819$
Central density	0
Number of external pieces	80
Level of complexity	20
Related polytopes
Convex hull	Minkowski sum of dodecahedron with edge length 1 and rhombus with edge length ${\frac {\sqrt {10\left(25+11{\sqrt {5}}\right)}}{5}}\approx 4.45407$ and diagonal ratio (1+√5)/2, where the diagonals are parallel to edges of the dodecahedron
Abstract & topological properties
Flag count	800
Euler characteristic	0
Surface	Torus
Orientable	Yes
Genus	1
Properties
Symmetry	K3, order 8
Convex	No
Nature	Tame

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

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.

External links

• Doskey, Alex. "Chapter 7 - Exploration of (R)(A) Toroids".
• Parker, Matt. “Polygons of New York”.