# Toroidal blend of 8 octahedra

Toroidal blend of 8 octahedra
Rank3
TypeStewart toroid
Elements
Faces4+4+4+4+4+4+8+8+8 triangles
Edges2+2+4+4+4+8+8+8+8+8+8+8
Vertices2+2+4+4+4+8
Vertex figures2+4 squares, edge length 1
4+8 [32.3]2
2 [33.3]2
4 [(32)3.32]
Measures (edge length 1)
Volume${\displaystyle {\frac {8{\sqrt {2}}}{3}}\approx 3.77124}$
Surface area${\displaystyle 12{\sqrt {3}}\approx 20.78461}$
Central density0
Number of external pieces48
Level of complexity12
Related polytopes
Convex hullMinkowski 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 count288
Euler characteristic0
SurfaceTorus
OrientableYes
Genus1
Properties
SymmetryK3, order 8
Flag orbits36
ConvexNo
NatureTame

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[1], 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.