Toroidal blend of 8 octahedra
Toroidal blend of 8 octahedra  

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 [3^{2}.3]^{2}  
2 [3^{3}.3]^{2}  
4 [(3^{2})^{3}.3^{2}]  
Measures (edge length 1)  
Volume  
Surface area  
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  K_{3}, 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 outerblending eight octahedra together in a rhombusshaped 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[edit  edit source]
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 eightmember 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.
Gallery[edit  edit source]

The convex hull.

A rotating model.
External links[edit  edit source]
 Doskey, Alex. "Chapter 7  Exploration of (R)(A) Toroids".
 Miyazaki, Koji and Takada, Ichiro. "Uniform Anthills in the World of Golden Isozonohedra", figure 16
References[edit  edit source]
 ↑ Chains of antiprisms (PDF)