n-6-4 acrohedra

Rayne H. has in 2023 discovered an elegant family of n-6-4 acrohedra.  The family includes models for n < 12.

As in Rayne's family of n-6-3 acrohedra, construction varies slightly between even and odd values of n.

n is even

10-6-4 case 1
10-6-4 (case 1) non-coplanar
10-6-4 case 2
10-6-4 (case 2) non-coplanar

While solutions with even n-6-4 acrons not difficult to find, with n = 6, 8 and 10 are covered by the truncated octahedron, truncated cuboctahedron and truncated icosidodecahedron respectively.  These new constructions are still of interest and for the 10-6-4 case they do generate the minimal known acrohedra to date.

Construction for even n involves surrounding a base n-gon with alternate squares and complexes of three triangles (half hexagons) to form a coplanar figures which is n/2 gon of edge length 2.   Two such figures can then be joined by their n/2 sides. 

There are now two alternative constructions, for case 1 replace two joined sets of triangles with hexagons.  Two of the above figures can then be joined by these hexagons to form an n-6-4 polyhedron.  Remove the coplanarity from the triangular complexes by augmenting the central triangles with tetrahedra. (above left)

For case 2, replace just one triangular complex with a hexagon and augment it to join to the unmatched edges of the other triangular complex.  (above right).  This 'looping' of the hexagonal edges avoids the need to add a second n-gonal complex.

The above construction includes coplanar faces.  These can be resolved by the excavation of tetrahedra from the coplanar triangles to form figures such as those above.

n is odd

9-6-4 S T nc
9-6-4 S T non-coplanar

For odd n the above construction needs modifying as the ring of squares and triangular complexes does not fit around the n-gon.   There are two ways to resolve this:

Method S is to allow two adjacent square edges.  The remaining gap can be filled by a complex of 2*(n-1)/2 triangles which are faces of an n/2-gonal bipyramid with its apices at the junction of the two squares.  

Method T is to allow two adjacent triangular complexes.  Remove one triangle from each complex. 
The remaining gap can be filled by a complex of 2*(n-3)/2 triangles which are again faces of an n/2-gonal bipyramid its apices at the junction of the two triangles.

Method T results in fewer vertices than method S for coplanar models, but more triangles need to be augmented to remove the coplanarity leaving method S the minimal acrohedron in the non-coplanar case.  See  the acrohedra page for examples.

Again as in the even-n case there are options to replace either one or two triangular complexes with hexagons, perhaps the most interesting option is to replace 2 complexes (as in even case 1 above) but to have a method S polyhedron on one side, and a method T polyhedron on the other, this is the image above.

These 'ST' models have been generated for n = 7, 9 and 11.

'Case 2' models for odd values of n can be found on the acrohedra page.

These models
can be regarded as a form of 'eversion' of the n-4-3 acrohedra. 


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