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) non-coplanar
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 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 then-4-3 acrohedra. .