Green's Rules has produced valid n-n-3 acrohedra for n = 4, 5, 6, 7, 8, 10, 5/2, 7/2, 8/3, and 10/3. All are orbiform.
Procedure[edit | edit source]
The procedure can be described as follows:
- Start with a virtual n-gon.
- Attach an n-gon to each edge.
- Connect the second open edge of each new n-gon to the second open edge of n-gon attached to the virtual n-gon two edges away.
- Add triangles to the triangular holes.
- If the remaining open edges can be closed with a regular polygon or regular polygon compound close it, otherwise add n n-gons and return to step 3.
Example[edit | edit source]
The following is an example of Green's rules applied where n = 10. The result is the gyrated blend of truncated dodecahedra.
Start with a virtual decagon.
Examples[edit | edit source]
The following table shows the results of Green's rule for small n. For even n Green's rule will result in a gyrated blend of .
|3-3-3||Generates a degenerate polyhedron topologically equivalent to a triangular bipyramid.|
|4-4-3||Gyrated blend of triangular prisms (tutrip)|
|5-5-3||Great dodecahedral cone (gadcone)|
|6-6-3||Gyrated blend of truncated tetrahedra (tutut)|
|8-8-3||Gyrated blend of truncated cubes (tutic)|
|9-9-3||Procedure does not generate a solution.|
|10-10-3||Gyrated blend of truncated dodecahedra (tutid)|
|11-11-3||Procedure does not generate a solution.|
|12-12-3||Gyrated blend of truncated hexagonal tiling||The 12-12-3 acron is planar and Green's rules result in a tiling of the plane.|
For n > 12 the n-n-3 acron is hyperbolic.