Toroidal blend of 8 octahedra

Toroidal blend of 8 octahedra
(3D model) (OFF file)
Rank	3
Type	Stewart toroid
Elements
Faces	4+4+4+4+4+4+8+8+8 triangles
Edges	2+2+4+4+4+8+8+8+8+8+8+8
Vertices	2+2+4+4+4+8
Vertex figures	2+4 squares, edge length 1
	4+8 [32.3]2
	2 [33.3]2
	4 [(32)3.32]
Measures (edge length 1)
Volume	${\frac {8{\sqrt {2}}}{3}}\approx 3.77124$
Surface area	$12{\sqrt {3}}\approx 20.78461$
Central density	0
Number of external pieces	48
Level of complexity	12
Related polytopes
Convex hull	Minkowski sum of octahedron with edge length 1 and rhombus with edge length √8/3 and diagonal ratio √2, where four coplanar edges of the octahedron are parallel to the X or Y axes, and the rhombus has a short diagonal parallel to the Z axis and a long diagonal parallel to the X or Y axis
Abstract & topological properties
Flag count	288
Euler characteristic	0
Surface	Torus
Orientable	Yes
Genus	1
Properties
Symmetry	K3, order 8
Convex	No
Nature	Tame

The toroidal blend of 8 octahedra is a Stewart toroid that consists of 48 triangles. It can be obtained by outer-blending eight octahedra together in a rhombus-shaped loop. The four octahedra at the vertices of the virtual rhombus are all oriented in the same way; the other four serve as triangular antiprisms.

Relations

Twelve copies of this toroid can be blended together, each blending pair coinciding at three octahedra with collinear centers, to form a toroidal blend of 38 octahedra with cubic symmetry and an appearance like the skeleton of a rhombic dodecahedron.

The following skew apeirohedron, upon close examination, has rings of octahedra that are similar to these toroids.

Versions of this toroid can be made out of copies of most of the Platonic, Archimedean, or Johnson solids that include the faceplanes of the octahedron, including the icosahedron and several others based on dodecahedral symmetry. All of the Platonic solids can form eight-member rings except for the tetrahedron.

If we create an isosceles trapezoidal loop of octahedra using two of the basic structures laid out here, we can blend two of these loops together to get a larger version of this toroid with an additional "bridge" of octahedra obliquely crossing it.

Gallery

• 

  The convex hull.

• 

  A rotating model.

External links

• Doskey, Alex. "Chapter 7 - Exploration of (R)(A) Toroids".
• Miyazaki, Koji and Takada, Ichiro. "Uniform Ant-hills in the World of Golden Isozonohedra", figure 16