Green's rules

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Green's rules are a procedure for generating n -n -3 acrohedra. The rules are based off of Mason Green's construction of a 7-7-3 acrohedron, the small supersemicupola.

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:

  1. Start with a virtual n -gon.
  2. Attach an n -gon to each edge.
  3. 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.
  4. Add triangles to the triangular holes.
  5. 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.

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 .

Polyhedra generated by Green's rules
Acron Name Picture Notes
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)
7-7-3 Small supersemicupola
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 tilings 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.

External links[edit | edit source]